PHYSICAL REVIEW D 95, 045004 (2017) One-dimensional semirelativistic Hamiltonian with multiple Dirac delta potentials
Fatih Erman* Department of Mathematics, İzmir Institute of Technology, Urla 35430, İzmir, Turkey † Manuel Gadella Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Campus Miguel Delibes, Paseo Belén 7, 47011 Valladolid, Spain ‡ Haydar Uncu Department of Physics, Adnan Menderes University, 09100 Aydın, Turkey (Received 18 August 2016; published 17 February 2017) In this paper, we consider the one-dimensional semirelativistic Schrödinger equation for a particle interacting with N Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an N × N matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most N bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a self- adjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for N centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two-center case. We observe the so-called threshold anomaly for two symmetrically located centers. The semirelativistic version of the Kronig-Penney model is shortly discussed, and the band gap structure of the spectrum is illustrated. The bound state and scattering problems in the massless case are also discussed. Furthermore, the reflection and the transmission coefficients for the two delta potentials in this particular case are analytically found. Finally, we solve the renormalization group equations and compute the beta function nonperturbatively.
DOI: 10.1103/PhysRevD.95.045004
I. INTRODUCTION Moreover, pointlike Dirac delta potentials in two and In nonrelativistic quantum mechanics, Dirac delta poten- three dimensions are known as simple pedagogical toy tials are one class of exactly solvable models, and they are models in understanding several nontrivial concepts, useful to describe very short interactions between a single originally introduced in quantum field theory, namely particle and a fixed heavy source. For this reason, they are dimensional transmutation, regularization, renormalization, – also called contact or point interactions if Dirac delta asymptotic freedom, etc. [6 14]. It is also a nontrivial function is pointlike. It is a good approximation to use subject from a purely mathematical point of view. One them when the wavelength of the particle is much larger approach to define them properly is based on the theory of than the range of the potential. Besides their simplicity, they self-adjoint extensions of symmetric operators. This allows have a vast amount of applications for modeling real us to define rigorously the formal Hamiltonian for Dirac physical systems (see the recent review [1] and the books delta potentials as a self-adjoint extension of the local free [2,3] and references therein). A well-known model utilizing kinetic energy operator [3,15]. Dirac delta potentials in nonrelativistic quantum mechanics As is well known, the relativistic extensions of the is the so-called Kronig-Penney model [4], and it is actually Schrödinger equation, namely the Klein-Gordon and the a reference model in describing the band gap structure of Dirac equations, require the introduction of antiparticles, metals in solid state physics [5]. so they are inconsistent with the single particle theory. However, they describe the dynamics of quantum fields whose excitations are bosons or fermions. In other words, *[email protected] the Klein-Gordon and the Dirac equations indeed belong † [email protected] to the domain of quantum field theory. In contrast to the ‡ [email protected] Klein-Gordon and the Dirac equations, the eigenvalue
2470-0010=2017=95(4)=045004(30) 045004-1 © 2017 American Physical Society FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) equation for the semirelativistic kinetic energy operator with these heavy particles through the Dirac delta potentials pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 þ m2 does not require antiparticles since it has only at those points. positive energy solutions. Historically, it appeared as an This is a very toy model of a relativistic particle trapped approximation to the Bethe-Salpeter formalism [16,17] in in one dimension, and it interacts with some impurities (in describing the bound states in the context of relativistic the massless case, it could be the photons trapped in one quantum field theory. For this reason, this Hamiltonian dimension that interact with the impurities). Similar to the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 þ m2 is known as the free spinless Salpeter one-center (one delta potential) case, this problem must Hamiltonian. Moreover, widely used and rather successful also require renormalization. Our approach here is to find − −1 ’ models in phenomenological meson physics have been the formal resolvent ðH EÞ or Green s function expres- constructed by considering spinless Salpeter Hamiltonians sion (see [27] for the nonrelativistic case) of the above with several potentials [18–20]. It is important to empha- Hamiltonian (1.2) by renormalizing the coupling constant size that only the relativistic dispersion relation is imposed through the heat kernel techniques with emphasis on some ’ here, whereas relativistic invariance is not fully required general results on the spectrum of the problem. Green s (e.g., all the momentum integral measures are just dp). function approach is rather useful since it includes all the Therefore, the Salpeter Hamiltonian is a good approxima- information about the spectrum of the Hamiltonian. The tion to relativistic systems in the domain, where the particle method we use here has been constructed in the non- creations and annihilations are not allowed. On the other relativistic version of the model on two- and three- hand, the use of potentials for the interaction of two or more dimensional manifolds [28,29] and in the nonrelativistic particles violates the principle of relativity even at the many-body version of it in [30]. A one-dimensional non- classical level. This is due to the fact that the message in the relativistic many-body version of the model (1.2), where change of the position of the particle has to be received the particles are interacting through the two-body Dirac instantaneously by the other particle [21,22]. Nevertheless, delta potentials, is known as the Lieb-Liniger model [31] it has been proposed in [23] that Dirac delta potentials and has been studied in great detail in the literature [32–35]. could be an exception, and the following one-dimensional It is well known that the heat kernel is a very useful tool Salpeter Hamiltonian is considered: in studying one-loop divergences, anomalies, asymptotic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expansions of the effective action, and the Casimir effect in 2 2 quantum field theory [36] and also in quantum gravity [37]. H ¼ P þ m − λδðxÞ: ð1:1Þ Here, we claim that it can be used as a regularization of the Here λ is the coupling constant or the strength of the above formal Hamiltonian (1.2). This is essentially due to interaction, and the nonlocal kinetic energy operator (free the fact that the heat kernel Ktðx; yÞ converges to the Dirac delta function in the distributional sense so that the part of the above Hamiltonian)pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is defined in momentum 2 2 Hamiltonian can be regularized by replacing it with the space as multiplication by p þ m [24]. Similar to its heat kernel. One advantage of using the heat kernel is it nonrelativistic version in higher dimensions, this model may allow possible extensions to consider more general has been used in order to illustrate some quantum field elliptic pseudodifferential free Hamiltonians (it may even theoretical concepts in a simpler relativistic quantum include some regular potentials) since the only requirement mechanics context [23]. This model was actually first to remove the divergent part is to have the information of discussed from the mathematical point of view as a self- short time asymptotic expansion of the heat kernel [38].By adjoint extension of pseudodifferential operators in [25]. renormalizing the coupling constant after the heat kernel Moreover, an extension of the method developed in [23] to regularization through the resolvent formalism, we obtain the derivative of the Dirac delta potentials has been studied an explicit expression for the resolvent—a kind of Krein’s in [26]. formula [15]. It is given in terms of an N × N holomorphic In this paper, we study the generalization of the work (analytic) matrix [on the region RðEÞ 045004-2 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) physically meaningful results at the end. It is not obvious symmetrically around the origin, the results for the reflec- that the renormalization procedure guarantees that the tion and the transmission coefficients obtained from the ground state energy of our model is bounded from below. asymptotic approximation is completely consistent with the Here, we show that this is indeed the case (this is necessary one obtained numerically. We see that the reflection and for every physical system [40]) and prove that there exists a transmission coefficients behave like those in the non- unique self-adjoint operator associated with the resolvent relativistic case. For example, the transmission coefficient formula we find. The issues about the self-adjointness TðkÞ has some sharp peaks around certain values of k where can also be shown in the more abstract self-adjoint it becomes unity. These peaks have been interpreted as extension theory in mathematics literature (for one center, resonances by some authors [45–47] in the nonrelativistic see [25,41]). case. However, they should not be confused with reso- The discrete or the bound state spectrum of the one- nances as unstable states in quantum mechanics [48]. dimensional spinless free Salpeter Hamiltonian perturbed Furthermore, we realize one novel behavior of the by one and two Dirac delta potentials has been rigorously reflection coefficient near very small values of k=m.It discussed in [41]. We obtain essentially the same results is surprising that the reflection coefficient suddenly on the bound state spectrum for the two-center case. vanishes as the kinetic energy of the incoming particles Additionally, we show some general results on the number goes to zero for a certain choice of the parameters. This of bound states for an arbitrary number of centers and study phenomenon is actually known as the threshold anomaly how the bound state energies change with respect to the in one-dimensional nonrelativistic quantum mechanics parameters in the model by working out the principal [49]. The underlying reason for such an anomaly is matrix. We find an explicit expression for the bound state essentially the appearance of a bound state very close wave function for an arbitrary number of centers. Actually, to the threshold energy (starting point of the continuum no matter how many Dirac delta potentials there are in our spectrum). The reflection coefficient generally goes to system, the bound state wave function is calculated from unity as we decrease the energy of the incoming particles. the contour integration of the resolvent around its isolated However, if physically meaningful continuum wave simple poles. This wave function is shown to be pointwise functions can be constructed for all values of x,andthe bounded except at the location of the centers, as expected potential VðxÞ is symmetric [Vð−xÞ¼VðxÞ]andvanishes for any system in quantum mechanics [42]. Although it outside a finite region, and if it supports a bound state at diverges at the points where Dirac delta potentials are threshold, then the reflection coefficient goes to zero at the located, it is still square integrable. However, the expect- threshold. This is stated as a theorem in [49] and is valid ation value of the free Hamiltonian for the bound states is only for the above class of potentials in the nonrelativistic divergent. This is not surprising, and it basically tells us that quantum mechanics. Here, we show that the threshold the bound state wave function of the system does not anomaly also appears even in the semirelativistic case, belong to the domain of the free Hamiltonian. This gives us where renormalization is required. We find numerically an intuitive idea why these interactions are defined through and approximately (through asymptotic expansion) those the self-adjoint extension theory. Although we do not critical values of the parameters that lead to the threshold expect any degeneracy for bound states in one-dimensional anomaly. We show that these critical values of the quantum mechanics [43], this may not be true for singular parameters are those values for which the second bound potentials [44] and for the semirelativistic Salpeter equa- state appears at threshold energy E ¼ m. In the massless tion. Therefore, the nondegeneracy of the ground states in case m ¼ 0, we can analytically obtain the reflection and the context of Salpeter Hamiltonians is not obvious. In this transmission coefficients and show that the threshold paper, we show that the ground state of our model is anomaly occurs precisely at those values of the parameters nondegenerate as long as the distance between the centers for which the new bound state at threshold (E ¼ 0 in the is finite; then, the wave function for the ground state can be massless case) appears. However, this anomaly in the chosen to be positive. massless case is slightly different from the nonrelativistic We solve the semirelativistic Lippmann-Schwinger and semirelativistic massive case. The reflection coeffi- equation for the scattering problem of the N Dirac delta cient always approaches zero as the energy of the potential. This is one of the main results of the paper. The incoming particles goes to zero no matter which values reflection coefficient RðkÞ and transmission coefficient of the parameters are chosen. In any case, an anomalous TðkÞ are explicitly calculated in closed analytical forms. behavior is observed as a sudden change in the reflection We find the behavior of the reflection and transmission coefficient. Moreover, we consider the semirelativistic coefficients as functions of the energy of the incoming version of the Kronig-Penney model, and the band gaps in particle numerically. We also make an asymptotic approxi- the spectrum are illustrated by examining the transmission mation and obtain an analytical expression for the reflec- coefficient. We also study the nonrelativistic limits of the tion and transmission coefficients when kjai − ajj is bound state and scattering solutions, and these limits are sufficiently large. In particular, for two centers located consistent with the nonrelativistic results. 045004-3 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) The model under this study is shown to be asymptoti- threshold anomaly and show that it occurs near the border cally free; that is, the scattered particle becomes free as its of the continuum energy spectrum. In Sec. X, we analyti- energies become higher and higher. In the massless case, cally study the bound state and scattering problem in the the Hamiltonian initially does not contain any intrinsic massless case. Finally, we derive the renormalization group energy scale due to the dimensionless coupling constants in equations and compute the β function for the model in natural units. However, a new set of parameters, namely the Sec. XI and shortly introduce a possible extension of the bound state energies to each center, is introduced after the model in Sec. XII. Section XIII contains our conclusions, renormalization procedure. The appearance of the dimen- and Appendix A includes the proof of the analyticity of the sional parameters is called dimensional transmutation and principal matrix. fixes the energy scale of the system. This can be interpreted as the simplest example of anomaly or quantum mechanical II. RENORMALIZATION OF RELATIVISTIC symmetry breaking, as in the nonrelativistic version of the FINITELY MANY DIRAC DELTA POTENTIALS problem [12]. We also derive the renormalization group THROUGH HEAT KERNEL equations and find the fixed points of the single beta function for the full system. All these issues have already We consider the time-independent Schrödinger equation been addressed in the context of a single Dirac delta (also called Salpeter equation) for the Hamiltonian (1.2) potential, and studying such nontrivial concepts in quantum XN field theory in a single particle relativistic theory has been x H ψ x H0 ψ − λ δ x − a ψ x one of the main motivation in [23] from pedagogical h j j i¼h j j i i ð iÞ ð Þ i¼1 reasons. We expect that extending the single center problem XN to many centers may help to understand some nontrivial ¼hxj H0 − λ ja iha j jψi¼EψðxÞ; ð2:1Þ concepts in quantum field theory and bridge the huge gap i i i i¼1 between the quantum field theory and relativistic quantum ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mechanics. p 2 2 where H0 ¼ P þ m and the kets ja i are the eigenkets The paper is organized as follows. In Sec. II, we derive a i of the position operator with eigenvalue ai. The second resolvent formula for our model in terms of an N × N equality in Eq. (2.1) is just the consequence of the property matrix (called principal matrix) by using the heat kernel as of Dirac delta function, δðx − aiÞψðxÞ¼δðx − aiÞψðaiÞ. a regularization. In Sec. III, we discuss the bound state We will use the units such that ℏ ¼ c ¼ 1 throughout the spectrum and prove that the eigenvalues of the principal paper. We first find the regularized resolvent for the matrix are decreasing functions of energy, and we show that regularized version of the above Hamiltonian. We propose we have at most N bound states. We also study how the that the regularized Hamiltonian is eigenvalues of the principal matrix change with respect to the bound state energy of the ith delta center (coming from XN the renormalization condition) and with respect to the − λ ϵ ϵ ϵ Hϵ ¼ H0 ið Þjai ihai j; ð2:2Þ distance between the centers. In Sec. IV, the ground state i¼1 energy is shown to be bounded from below using the Geršgorin theorem. Then, we briefly give a proof that there where we have introduced short “time” cutoff ϵ through the ϵ exists a unique self-adjoint operator associated with the heat kernel Kϵ=2ðx; aiÞ¼hxjai i and made the coupling resolvent formula obtained in the renormalization pro- constants explicitly dependent on ϵ. The heat kernel is cedure (technical details are given in Appendix B). In defined as the fundamental solution to the following heat Sec. VI, we find the bound state wave function by equation [24]: computing the contour integral of the resolvent around one of its isolated simple poles and discuss its nonrelativ- ∂Ktðx; yÞ H0K ðx; yÞ¼− : ð2:3Þ istic limit. In Sec. VII, we show that the bound state wave t ∂t function is exponentially pointwise bounded and diverges ϵ ϵ ’ at the location of the centers. Then, we point out that the The expression jai ihai j written in Dirac s bra-ket notation wave function is square integrable, but the expectation is just the projection operator onto the space spanned by ϵ 2 R value of the free Hamiltonian in the bound state is jai i in L ð Þ. The reason why the heat kernel works for the divergent. In Sec. VIII, we prove that the ground state is regularization of the problem is based on the fact that it nondegenerate unless the centers are infinitely far away converges to the Dirac delta function in the distributional ϵ → from each other, and the wave function for the ground state sense as the cutoff is removed, i.e., hxjai i hxjaii¼ þ can be chosen strictly positive. In Sec. IX, we exactly solve δðx − aiÞ as ϵ → 0 . In other words, we recover the the semirelativistic Lippmann-Schwinger equation for the original Hamiltonian when the cutoff goes to zero. −1 problem and find an explicit expression for the reflection To find the regularized resolvent RϵðEÞ¼ðHϵ − EÞ , and transmission coefficients. Furthermore, we discuss the we will solve the following inhomogenous equation: 045004-4 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) Z XN ∞ −1 −tðH0−EÞ − λ ϵ ϵ ϵ − ψ ρ ðH0 − EÞ ¼ dt e : ð2:11Þ H0 ið Þjajihajj E j i¼j i; ð2:4Þ 0 j¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2 2 −t P2þm2 −mt ϵ For H0 ¼ P þ m , we have ‖e ‖ ≤ e for all assuming complex number E ∉ SpecðH0Þ. Let jf i¼ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi i t ≥ 0. Then, the integral (2.11) exists if RðEÞ ϵ −1 ϵ where we have used the semigroup property of the heat 1 − hf jðH0 − EÞ jf i if i ¼ j T ϵ;E i i 2:7 kernel ijð Þ¼ − ϵ − −1 ϵ ≠ ð Þ hfi jðH0 EÞ jfji if i j: Z ∞ ϵ ψ dz Kt1 ðx; zÞKt2 ðz; yÞ¼Kt1þt2 ðx; yÞ; ð2:13Þ By solving hfjj i from the above matrix equation (2.6) and −∞ substituting it into Eq. (2.5), we obtain the regularized resolvent for all x, y and t1, t2 ≥ 0. If we now take the limit ϵ → 0þ, before taking the integral aboveR with respect to −1 −1 ∞ tE RϵðEÞ¼ðH0 − EÞ þðH0 − EÞ x and y, and assume that the function 0 dt e Ktðx; yÞ belongs to some class of test functions of each variable x XN ϵ −1 ϵ ϵ −1 ϵ ϵ − −1 and y for RðEÞ 045004-5 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) pffiffiffiffiffiffi ∼ m π −tm Therefore, for bound state problems, it is very conven- behavior of the Bessel function K1ðmtÞ π 2mte as t → ∞ [53]. Moreover, the exponential upper bound of the ient to choose the renormalization scale to be the bound 1 λR 0 Bessel function state energy by setting = i ¼ so that we eliminate the unphysical scale Mi. 1 1 −x=2 If we apply the same argument to the several center case, K1ðxÞ 0 −1 −1 for all x> , which was given in [29] by using its RðEÞ¼ðH0 − EÞ þðH0 − EÞ integral representation, guarantees that the integral R XN ∞ tE 0 dt Ktðai;ajÞ e is finite. However, the integral in the Φ−1 − −1 × jaii½ ðEÞ ijhajj ðH0 EÞ ; ð2:23Þ diagonal part of matrix (2.9) is divergent due to the i;j¼1 asymptotic behavior where 1 K1ðmtÞ ∼ ; ð2:17Þ R mt ∞ tEi tE 0 dt Ktðai;aiÞðe B − e Þ if i ¼ j Φ ðEÞ¼ R → 0 ij ∞ tE as t [53]. − 0 dt Ktðai;ajÞ e if i ≠ j; Let us temporarily consider the one-center case (N ¼ 1) 2:24 for simplicity. Suppose that the ith center is isolated from ð Þ all other centers. Then the regularized principal matrix is R just a single function for the ith center and reads defined on the complex E plane, where ðEÞ 045004-6 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) where RðEÞ 8 pffiffiffiffiffiffiffiffiffiffi R μ2− 2 > 1 ∞ −μja −a j m > − dμ e i j 2 2 2 R E < 0 < π m μ −m þE if ð Þ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Φ 2− 2 − R 2 2 ijðEÞ¼ i E m jai ajj 1 μ −m ð2:29Þ > e qffiffiffiffiffiffiffiffi ∞ −μjai−ajj > −i − dμ e 2 2 2 if RðEÞ > 0; > 2 π m μ −m þE : m 1− 2 E where i ≠ j and IðEÞ > 0. Here the integral over the be the eigenvalue equation for the principal matrix. variable μ comes from the integration over the branch For real values of E, the principal matrix is Hermitian cut along ½im; i∞Þ. Expressing the integral in the off- due to the symmetry property of the heat kernel diagonal part of the principal matrix (2.27) by Ktðai;ajÞ¼Ktðaj;aiÞ so all its eigenvalues are real valued Eq. (2.29) is very useful when we study the spectrum and depend on the real variable E. We are now going to of the problem. show that the eigenvalues of the principal matrix are decreasing functions of E. For simplicity, we will show III. ON THE BOUND STATE SPECTRUM this fact for the nondegenerate case without loss of generality (it can be generalized to the degenerate case Since the bound state spectrum can be found from the as well). To prove this, we first need to show that the poles of resolvent, the bound states energies should only principal matrix is holomorphic (analytic) on the complex come from the points of the real E axis such that the plane RðEÞ 045004-7 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) Z Z Z XN ∂ωkðEÞ ∞ ∞ ∞ − k t1E t2E k ¼ dx ðAi ðEÞÞ dt1 e Kt1 ðx; aiÞ dt2 e Kt2 ðx; ajÞ Aj ðEÞ ∂E −∞ 0 0 i;j¼1 Z Z ∞ XN ∞ 2 − k tE 0 ¼ dx Ai ðEÞ dt Ktðx; aiÞ e < : ð3:6Þ −∞ 0 i¼1 The above fact implies that the eigenvalues of the principal emphasize that the diagonal part of the principal matrix is matrix are decreasing functions of E. As a consequence of positive when E 045004-8 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) Em Em 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.0 EB 1 EB 1 m 2 0.5 0.5 m 50 E E 2 2 0.0 0.0 E, E, 1 1 0.5 0.5 1.0 1.0 ω ω 1 2 FIG. 1. Eigenvalues 1 and 2 as a function of E=m for different values of EB=m (assuming that delta centers are twin, i.e., EB ¼ EB for simplicity) and 2ma ¼ 1. Here a1 ¼ −a and a2 ¼ a. is worthwhile going through the proof in our simple XN Φ Φ semirelativistic system where a single particle interacts j iiðEÞj > j ijðEÞj; ð4:2Þ ≠ with N external Dirac delta potentials. A much more i j interesting case is, of course, associated with the model for all E 0.5 2 ma 5 0.5 E 2 E , 0.0 2 E , 0.0 1 E 1 0.5 0.5 1.0 1.0 FIG. 2. Eigenvalues ω1 and ω2 as a function of 2ma for the FIG. 3. Eigenvalues ω1 and ω2 as a function of E=m for values EB=m ¼ 1=2 and E=m ¼ 1=2. 2ma ¼ 5 and EB=m ¼ 1=2. 045004-9 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) E m Em 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.4 2 ma 1 0.2 0.2 0.0 E E 0.2 0.0 det det 0.4 0.2 0.6 2 ma 0.775 0.8 0.4 1.0 Em Em 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.4 0.2 1 0.0 2 E 0.2 det det E 0.4 3 0.6 2 ma 110 2 ma 35 4 0.8 1.0 5 FIG. 4. The determinant of the principal matrix as a function of E=m for different values of 2ma. Here EB=m ¼ 1=2. (4.2) is satisfied. Hence, if we find a lower bound for jΦiij Now we follow the above line of arguments until we and an upper bound for jΦijj, namely obtain an analytical solution. For that purpose, let us find a lower bound for minijΦiiðEÞj and an upper bound for Φ ≥ Φ j iiðEÞj min j iiðEÞj; max jΦ j. Using the integral representation of the Bessel 1≤i≤n j ij function K1 [53] XN jΦ ðEÞj ≤ ðN − 1Þ max jΦ ðEÞj; ð4:4Þ Z ij 1≤ ≤ ij ∞ ≠ j N −x cosh t i j K1ðxÞ¼ dt cosh te ; ð4:7Þ 0 the condition (4.2) is implied by the stronger requirement t 1þet and the bounds cosh t ≥ e =2, and cosh t ≤ 2 ,wehave min jΦiiðEÞj > ðN − 1Þ max jΦijðEÞj: ð4:5Þ 1≤i≤n 1≤j≤N Z ∞ t t −x=2 e −xe K1ðxÞ ≥ e dt e 2 : ð4:8Þ Once we obtain the value of E, which saturates this 0 2 inequality, it is satisfied for all E below this critical value. Consequently, there cannot be any solution beyond By making the change of variables u ¼ et, we obtain a this critical value, and the ground state energy must be lower bound for the Bessel function μ i larger than that critical value. Let ¼ miniEB and d ¼ Z −x=2 ∞ −x minjjai − ajj for all i. Then, e −xu e K1ðxÞ ≥ du e 2 ¼ ; ð4:9Þ Z 2 1 x ∞ Φ tμ − tE minj iiðEÞj ¼ dt Ktðai;aiÞðe e Þ; 0 i 0 for all x> . Using the upper bound of the Bessel function Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ mt (2.16),wehave Φ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 tE maxj ijj¼ dt p 2 2 K1 m d þ t e ; j 0 π d þ t 1 m − E minjΦiiðEÞj > log ; ð4:10Þ ð4:6Þ i π m − μ for E<μ. where E 045004-10 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) Z XN ∞ pffiffiffiffiffiffiffiffiffi the Hamiltonian after the renormalization procedure is mt −m 2 2 Φ ≤ − 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi tE 2 d þt j ijðEÞj ðN Þ dt p 2 2 e e very crucial from the physical point of view since only ≠ 0 π d þ t i j self-adjoint operators are observables and the self- 1 1 adjoint Hamiltonian generates the unitary time evolution × pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ : ð4:11Þ m d2 þ t2 2 [59,60]. Our proof is based on the following corollary pffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi −m 2 2 −m p 2 2 (Corollary 9.5 in [50]), and it is essentially first used in Since e 2 d þt ≤ e 2 t and d t ≥ t for all t, we get þ [61] for proving the existence of the self-adjoint Z Hamiltonian of the nonrelativistic Dirac delta potentials XN 1 ∞ − m− Φ ≤ − 1 tð 2 EÞ in two- and three-dimensional manifolds and of the j ijðEÞj ðN Þ 2 dt t e π 0 i≠j d relativistic (Klein-Gordon) Dirac delta potentials on Z ∞ two-dimensional manifolds [62] (also includes the Lee m − m− þ dt t e tð 2 EÞ model). For the sake of completeness, let us restate this 2πd 0 corollary here. 1 1 m Let Δ be a subset of the complex plane and E ∈ Δ.A < ðN − 1Þ þ : ðE − mÞ2 πd2 2πd family JðEÞ of bounded linear operators on the Hilbert space H under consideration, which satisfies the resolvent ð4:12Þ identity This leads to the need for imposing the following JðE1Þ − JðE2Þ¼ðE1 − E2ÞJðE1ÞJðE2Þð5:1Þ inequality: for E1;E2 ∈ Δ is called a pseudoresolvent on Δ [50]. 1 m − E 1 1 m Let Δ be an unbounded subset of C that does not coincide log < ðN − 1Þ þ : π m − μ ðE − mÞ2 πd2 2πd with the spectrum of A and JðEÞ be a pseudoresolvent on Δ. If there is a sequence E ∈ Δ such that E → ∞ as ð4:13Þ k j kj k → ∞ and The value of E that saturates this inequality can be found lim − EkJðEkÞx ¼ x; ð5:2Þ analytically now, so that we conclude for all N ≥ 1 that k→∞ for all x ∈ H, then JðEÞ is the resolvent of a unique densely 2πðN − 1ÞCðm; dÞ 1=2 E ≥ m − ; 4:14 defined closed operator A. gr 2πðN−1ÞCðm;dÞ ð Þ W 2 ð ðm−μÞ Þ We are not going to give the first part of the proof here again since it is exactly given in [62] and the reader can where W is the Lambert W function [58], defined by the easily go through it by reading the relevant section y 1 m given there. solution of ye ¼ x and Cðm; dÞ¼ðπd2 þ 2πdÞ. If we choose the sequence Δ ¼fEkjEk ¼ −kjE0j; k ¼ 1; 2; …g, where E0 is below the lower bound on the V. THE HAMILTONIAN AFTER ground state energy that has been found in Sec. IV, the RENORMALIZATION resolvent (2.23) is a pseudoresolvent on the above set. Although we do not know what the form of the As for the second part of the proof, it is more Hamiltonian after the renormalization procedure is, we involved and technical. Since the proof is not essential can ask whether there is a self-adjoint operator asso- to be able to follow the rest of the paper, we give it in ciated with the resolvent formula. We show that there Appendix B. exists a unique self-adjoint Hamiltonian associated with VI. THE BOUND STATE WAVE FUNCTION the resolvent formula (2.23). This problem has been FOR N CENTERS discussed from the self-adjoint extension point of view in [25] andcouldalsobeprovedbyothermethods. The projection operator onto the subspace spanned by Here, our approach is to renormalize the model by heat the eigenfunctions corresponding to the kth isolated eigen- kernel techniques, formally obtain an explicit formula k value (bound state energy Ebound) is given by the following for the resolvent, and then show that this formula for the contour integral [42]: resolvent corresponds to a unique densely defined self- I adjoint Hamiltonian without going into rather technical 1 P ψ k ψ k − hxj kjyi¼ BðxÞð BðyÞÞ ¼ dE Rðx; yjEÞ; domain issues of unbounded operators. We think this 2πi Γ proof can be useful if we extend this model into many- k body or field theoretical models. The self-adjointness of ð6:1Þ 045004-11 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) k −1 XN where Rðx; yjEÞ¼hxjRðEÞjyi is the resolvent kernel ∂ω ðEÞ 2 Γ ψ k ðxÞ¼ − AkðEk Þ and k is a sufficiently small contour enclosing only B ∂ i bound E E¼Ek 1 Ek . We note that the free resolvent kernel or Green’s Z bound i¼ bound ∞ function R0ðx; yjEÞ does not contain any pole on the tEk × dt e bound Ktðai;xÞ: ð6:6Þ real axis below m [the spectrum of the free part is 0 σðH0Þ¼½m; ∞Þ]. Therefore, all the poles on the real axis smaller than m must come only from the poles of This explicit result of the bound state wave function for the inverse principal matrix. Since it has been shown N Dirac delta potentials is the linear combination of the Φ† bound state wave functions for each single Dirac delta that the principal matrix is a symmetric [ ijðEÞ¼ Φ ðE Þ] holomorphic (analytic) family in Appendix center located ai. In the single center case, we have only ij one bound state energy, namely E . Since the principal A, its eigenvalues and its eigenprojections are also B matrix is just a single function in this case, A ¼ 1 so that holomorphic on the real axis [63]. i we obtain As a result of Hermiticity of the principal matrix on the Z real E axis and its analytical continuation to the complex E ∞ tEB plane, we can apply the spectral theorem to the principal ψ BðxÞ¼N dt Ktðx; 0Þ e ; ð6:7Þ matrix 0 where N is the normalization constant given by XN σ Φ ðEÞ¼ ω ðEÞ½PσðEÞ : ð6:2Þ Z Z ij ij ∞ ∞ 2 −1=2 σ 1 ¼ tEB N ¼ dx dt Ktðx; 0Þ e : ð6:8Þ −∞ 0 P σ σ σ Here σðEÞij ¼ðAi ðEÞÞ Aj ðEÞ and Ai ðEÞ are the pro- jection operator and the normalized eigenvector corre- The wave function (6.7) is nothing but the same formula σ obtained recently in [23]. This can be seen by first sponding to the eigenvalue ω ðEÞ, respectively. Similarly, pffiffiffiffiffiffiffiffiffiffiffi 0 0 −t P2þm2 0 we can write the spectral resolution of the inverse expressing the heat kernel as Ktðx; Þ¼hR je j i principal matrix, ∞ dp and inserting the completeness relation −∞ 2π jpihpj¼1 in front of the exponential X 1 Φ−1 P Z Z ½ ðEÞ ij ¼ σ ½ σðEÞ ij: ð6:3Þ ∞ ∞ pffiffiffiffiffiffiffiffiffiffiffi ω ðEÞ dp − 2 2 σ ψ N ipx t p þm tEB BðxÞ¼ dt 2π e e e Z0 −∞ ∞ ipx The residue of the resolvent at the simple pole E ¼ N dp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie k k ¼ : ð6:9Þ ω −∞ 2π 2 2 Ebound (assuming that only the kth eigenvalue flows to p þ m − EB k its zero at E ¼ Ebound)isgivenby There is an overall minus sign difference between our ResðRðx; yjEÞ; Ek Þ result (6.9) and the one given in [23],whichis bound physically irrelevant. The above improper integral is ∂ωk −1 k ðEÞ discussed in great detail in [23] by using the contour ¼ R0ðx; aijEboundÞ ∂E k integration for three different regimes of bound states, E¼Ebound k k namely weak, strong, and ultrastrong bound states. We × ½P ðE Þ R0ða ;yjE Þ; ð6:4Þ k bound ij j bound are not going to discuss the details of these various cases since they have already been studied in [23].We ∂ωkðEÞ where ∂ j k can be found from Eq. (3.6). will consider the general behavior of the bound state E E¼Ebound Combining all these results yields wave functions in the next sections. For consistency, let us consider the nonrelativistic limit of the bound state wave function (6.6) associated with N ψ k ðxÞðψ k ðyÞÞ B B delta centers. To find the wave function in this limit, we first 1 ∂ωk E −1 rewrite the wave function formula (6.6) in the same way as 2π k − ð Þ ¼ ð iÞR0ðx; aijEboundÞ in Eq. (6.9), 2πi ∂E k E¼Ebound Z k k k k k XN − × ðAi ðEboundÞÞ Aj ðEboundÞR0ðaj;yjEboundÞ: ð6:5Þ ∞ dp eipðx aiÞ ψ k N k k k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi BðxÞ¼ Ai ðEboundÞ ; −∞ 2π 2 2 − k i¼1 p þ m E Then, it is straightforward to read off the bound state bound wave function from the equation above, ð6:10Þ 045004-12 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) N k XN k Δ k where is the normalization constant. Note that the mAi nr ð EboundÞ ψ k ðxÞ ∼ N k ð Þ integral appearing in the wave function (6.10) is exactly B nr ð−2mΔEk Þ1=2 the same integral as in the principal matrix. Using (2.29), i¼1 bound k − Δ k ≪ 1 × exp½−ð−2mΔEk Þ1=2jx − a j ; ð6:15Þ the nonrelativistic limit jEbound mj=m ¼j Eboundj=m bound i of the above integral becomes where N k is the normalization constant and Ak is the m nr iðnrÞ − −2 Δ k 1=2 − 1 2 exp½ ð m EboundÞ jx aij ; kth eigenvector of the nonrelativistic principal matrix (6.14) ð−2mΔEk Þ = bound ωk Δ k associated with the kth eigenvalue nr. Here Ebound must ð6:11Þ ωk Δ k 0 be the solution of nrð EboundÞ¼ . Hence, we show that the nonrelativistic limit of the bound state wave function for Δ k where we ignored the higher order terms in j EBj=m. N centers (6.10) is actually the linear combination of the Similarly, we can find the nonrelativistic limit of the bound state wave function for single nonrelativistic Dirac − ≪ 1 i − ≪ 1 principal matrix (jE mj=m and jEB mj=m ) delta centers. and obtain 8 < m − m VII. POINTWISE BOUND ON THE BOUND −2 Δ i 1=2 −2 Δ 1=2 if i ¼ j ð m EBÞ ð m EÞ STATE WAVE FUNCTION AND EXPECTATION Φ ðEÞ∼ ij : m 1=2 VALUE OF THE FREE HAMILTONIAN − 1 2 exp − −2mΔE a −a if i ≠ j: ð−2mΔEÞ = ½ ð Þ j i jj The exponential decay of the bound state wave functions ð6:12Þ of the Schrödinger operators are known as the consequence of regularity theorems. Basically, square-integrable solutions Let us now go back to the nonrelativistic problem. We do −∇2 ψ ψ not need renormalization in this case, and it is straightfor- of ð þ VÞ ¼ E obey pointwise bounds of the form ward to calculate the resolvent formula for N dirac delta centers jψðrÞj ≤ Ce−ar; ð7:1Þ −1 −1 RϵðEÞ¼ðH0 − EÞ þðH0 − EÞ if the potential energy V is continuous and bounded below XN −∇2 Φ−1 − −1 and E is in the discrete spectrum of þ V (see [42] for × jaii½ ðEÞ ijhajj ðH0 EÞ ; ð6:13Þ the review of the subject). We shall prove that it is still 1 i;j¼ possible to get exponential pointwise bounds for the bound P2 state wave function of our semirelativistic problem. where H0 ¼ 2m and 8 It is easy to find an upper bound for the wave function < 1 − m 1=2 if i ¼ j (6.6) by applying Cauchy-Schwarz inequality λi ð−2mEÞ Φ ðEÞ¼ ij : m 1=2 − 1 2 exp − −2mE a − a if i ≠ j: Z ð−2mEÞ = ½ ð Þ j i jj XN ∞ ψ k ≤ N k k k tEk j BðxÞj j j Ai ðEboundÞ dt e bound Ktðai;xÞ 0 ð6:14Þ i¼1 Z XN ∞ 2 1=2 k tEk Since the bound state energy to the ith center in the ≤ jN j dt e bound Ktðai;xÞ 2 Δ i − λ 2 1 0 nonrelativistic case is given by EB ¼ m i = such that i¼ 1 2 Z 1 λ − −2 Δ i = N = i ¼ m=ð m EBÞ , we show that the nonrelativ- X ∞ k tEk istic limit of the principal matrix (6.12) is equal to the ≤ jN j dt e bound Ktðai;xÞ; ð7:2Þ 0 nonrelativistic principal matrix (6.14). Because of this i¼1 k result, the nonrelativistic limit of the eigenvectors Ai of P N k k 2 1 the principal matrix is equal to the eigenvector of the where i¼1 jAi ðEboundÞj ¼ . Thanks to the upper bound nonrelativistic principal matrix (6.14). This guarantees that of the Bessel function K1ðxÞ given in Eq. (2.16), the wave the nonrelativistic limit of the bound state wave function is function is pointwise bounded on the real line Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ mt 1 1 ψ k ≤ N k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k − − 2 2 j BðxÞj j j dt þ exp tEbound m ðx aiÞ þ t 0 π ðx − a Þ2 þ t2 m ðx − a Þ2 þ t2 2 i i m 1 1 m ≤ N k − ffiffiffi − j j 2 þ exp p jx aij ; ð7:3Þ π − pmffiffi − k − 2 2 jx aijð 2 EboundÞ mjx aij 045004-13 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) 2 2 2 where we have used ðx − aiÞ þ t ≥ ðx − aiÞ for the locations of Dirac delta centers ai. This singular behavior expressionsqffiffiffiffiffiffiffiffiffiffi in front of the exponential and the inequality of the bound state wave function is expected due to the aþb a2þb2 2 ≤ 2 in the exponent (for all a, b). This shows that small t asymptotic expansion of the Bessel function (2.17). the bound state wave functions for Salpeter Hamiltonians Nevertheless, the bound state wave function can be shown with point interactions are also pointwise exponentially to be square integrable from its explicit expression using bounded. Note that this upper bound blows up at the the semigroup property of the heat kernel (2.13) Z Z Z Z ∞ ∞ XN ∞ ∞ 2 2 k ψ k N k k k ðt1þt2ÞEbound dx j BðxÞj ¼j j dx Ai ðAj Þ dt1 dt2 Kt1 ðai;xÞ Kt2 ðx; ajÞ e −∞ −∞ 0 0 i;j¼1 Z XN ∞ 2 k N k k k tEbound ¼j j Ai ðAj Þ dt t Kt1 ðai;ajÞ e : ð7:4Þ 0 i;j¼1 In the second line we have made the change of variables t ¼ t1 þ t2 and u ¼ t1 − t2 and then integrated with respect to the variable u. From the explicit expression of the heat kernel (2.14) and the upper bound of the Bessel function (2.16), the above expression is finite so that the bound state wave function is square integrable, ψ k ∈ 2 R B L ð Þ: ð7:5Þ To understand heuristically why our problem can be considered as a self-adjoint extension of the free Hamiltonian, which is also suggested by the Krein formula, let us calculate the expectation value of the kinetic energy for the bound state, Z Z Z ∞ ∞ XN ∞ XN pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k k 2 2 ψ k ψ k N k t1Ebound k t2Ebound k − h BjH0j Bi¼j j dx dt1 e ðAi Þ Kt1 ðai;xÞ dt2 e Aj P þ m Kt2 ðaj;xÞ ; −∞ 0 0 i¼1 j¼1 ð7:6Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k p 2 2 where we have suppressed the energy dependence of Ai for P þ m so the self-adjoint extension of the free simplicity. Using the heat equation (2.3) with its initial Hamiltonian extends the domain of it such that the states condition, and integration by parts for the t2 integral, we see ψ k corresponding to the eigenfunctions BðxÞ are included. that the above expression includes the following term: Z VIII. NONDEGENERACY OF THE ∞ 2 k GROUND STATE k t1Ebound jAi j dt1 e Kt1 ðai;aiÞ: ð7:7Þ 0 The rigorous proof of nondegeneracy and positivity of the ground state in standard quantum mechanics is given This integral is clearly divergent due to the small t in [42], which includes neither the singular potentials nor asymptotic expansion of the Bessel function (2.17). Hence the relativistic cases. Therefore, it is necessary to check we show that the expectation value of the free Hamiltonian whether a similar conclusion can be drawn for our problem. is divergent, The proof here is essentially the same as the one for the nonrelativistic case given in the previous work [29] based ψ k ψ k → ∞ h BjH0j Bi : ð7:8Þ on utilizing the Perron-Frobenius theorem [57]. It states that if A is an N × N matrix and A>0 (i.e., Aij > 0), then The self-adjoint extension of the semirelativistic kinetic the following statements are true: energy operator in the context of a single point interaction (a) The spectral radius ρðAÞ is strictly positive. (Recall was rigorously studied in [25]. We may here heuristically that ρðAÞ¼maxfjωj∶ω is an eigenvalue of Ag); deduce that the extension of the problem to the finitely (b) The spectral radius ρðAÞ is an eigenvalue of the many point interactions can also be considered as a self- matrix A; adjoint extension of the free part since we have proved that (c) There is an x ∈ CN with x>0 and Ax ¼ ρðAÞx; ψ k ρ the bound state wave function BðxÞ that we have found (d) The spectral radius ðAÞ is an algebraically (and hence does not belong to the domain of the free Hamiltonian geometrically) simple eigenvalue of A; 045004-14 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) (e) jωj < ρðAÞ for every eigenvalue ω ≠ ρðAÞ, that is, transmission coefficients for finitely many centers using ρðAÞ is the unique eigenvalue of maximum modulus. the semirelativistic version of the Lippmann-Schwinger The first step is to find a positive “equivalent” matrix to equation [64] the principal matrix (2.24). Let us subtract the maximum of the diagonal part, and reversing the overall sign, jk i¼jki − R0ðEk i0ÞVjk i; ð9:1Þ 0 Φ ðEÞ¼−ðΦðEÞ − ð1 þ εÞImax ΦiiðEÞÞ > 0; ð8:1Þ ’ E where R0ðEÞ is the free resolvent or Greenffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis operator, V p 2 2 represents the interaction, and Ek ¼ k þ m , the energy ε 0 ∈ ∞ Φ where > and E ½Egr; Þ. Since ii is a decreasing of the incoming particles. The notation Ek þ i0 denotes the function of E,maxEΦiiðEÞ¼ΦiiðEgrÞ. Note that the results limit of Green’s function as ε ↓ 0. Following the similar obtained by both Φ and Φ0 are physically equivalent. arguments developed in Sec. II, we can write the regular- First of all, adding a diagonal term to the principal matrix ized semirelativistic Lippmann-Schwinger equation by the Φ does not change its eigenvectors, whereas the eigenval- heat kernel ues are shifted by a constant amount. Nevertheless, this shift is equivalent to a constant shift in the bound state XN ϵ λ ϵ 0 ϵ ϵ spectrum, which is physically unobservable (we can shift jk ð Þi¼jkiþ ið ÞR0ðEk i Þjajihajjk i: ð9:2Þ the spectrum without altering its physics). Hence, this j¼1 transformed matrix Φ0 and Φ have the same common eigenvectors so it guarantees that there exist a strictly Let us consider the outgoingp boundaryffiffiffiffiffiffiffiffiffiffi conditions and Φ ϵ λ ϵ ϵ positive eigenvector Ai for the principal matrix and rescale the ket vectors jfi i¼ ið Þjai i so we have 0 min ρðΦ Þ¼−ω ðEÞþð1 þ εÞΦiiðEgrÞ. min þ ϵ 0 ϵ ϵ þ ϵ For a given E, there is a unique ω ðEÞ, and since we are jk ð Þi¼jkiþR0ðEk þ i Þjfi ihfi jk ð Þi looking for the zeros of the eigenvalues ωðEÞ¼0, the XN E E 0 ϵ ϵ þ ϵ minimum goes to zero at ¼ gr. This means that the þ R0ðEk þ i Þjfjihfjjk ð Þi; ð9:3Þ positive eigenvector Ai corresponds to the ground state j≠i energy. Hence, we prove that the ground state energy is ϵ unique and the associated eigenvector Ai is strictly positive. where we have isolated the j ¼ ith term. By acting on hfi j Because of the positivity property of the heat kernel, it is from the left, we can write the resulting expression in the easy to see that the ground state wave function is strictly following form: positive from Eq. (6.6), ϵ ϵ ϵ þ Z ð1 − hf jR0ðE þ i0Þjf iÞhf jk ðϵÞi ∞ XN i k i i tEgr N ψ grðxÞ¼N dt e AiðEgrÞKtðai;xÞ > 0; ð8:2Þ X 0 − ϵ 0 ϵ ϵ þ ϵ ϵ i¼1 hfi jR0ðEk þ i Þjfjihfjjk ð Þi ¼ hfi jki; ð9:4Þ j≠i where N > 0. Despite the singular character of the interaction, we prove that the ground state is still non- or it can be written as a matrix equation degenerate. This may seem to be inconsistent with the result discussed in Sec. III for the case where there are twin XN symmetrically located delta centers. We have shown that as ϵ 0 ϵ þ ϵ ϵ 1 2 … Tijð ;Ek þ i Þhfjjk ð Þi¼hfi jki;i¼ ; ; ;N; the distance between the centers goes to infinity, we have j¼1 degeneracy in the bound states. However, this is not ð9:5Þ contradicting with our proof above since this degeneracy occurs due to the vanishing of the off-diagonal terms in the principal matrix so that the positivity hypothesis of the where Perron-Frobenius theorem breaks down. As long as the ϵ ϵ 1 − hf jR0ðE þ i0Þjf i if i ¼ j; distance between the centers is finite, the ground state is ϵ 0 i k i Tijð ;Ek þ i Þ¼ − ϵ 0 ϵ ≠ always nondegenerate. hfi jR0ðEk þ i Þjfji if i j: IX. THE SCATTERING PROBLEM ð9:6Þ FOR N CENTERS Hence, the solution to Eq. (9.5) is given by The reflection and transmission coefficients of the problem for a single center case has recently been inves- XN ϵ þ ϵ −1 ϵ 0 ϵ tigated in [23] by constructing even and odd parity hfi jk ð Þi ¼ ½T ð ;Ek þ i Þ ijhfjjki: ð9:7Þ scattering solutions. Here we calculate the reflection and j¼1 045004-15 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) Substituting this result into the formula (9.3) that we have obtained for the scattering solution, and acting on the position bra vector hxj from the left yields XN þ ϵ ψ þ ϵ ikx 0 ϵ −1 ϵ 0 ϵ hxjk ð Þi ¼ k ð ;xÞ¼e þ hxjR0ðEk þ i Þjfi i½T ð ;Ek þ i Þ ijhfjjki i;j¼1 XN ikx 0 ϵ Φ−1 ϵ 0 ϵ ¼ e þ hxjR0ðEk þ i Þjai i½ ð ;Ek þ i Þ ijhajjki; ð9:8Þ i;j¼1 where ( 1 − ϵ 0 ϵ λ ϵ hai jR0ðEk þ i Þjai i if i ¼ j; Φ ðϵ;E þ i0Þ¼ ið Þ ð9:9Þ ij k − ϵ 0 ϵ ≠ hai jR0ðEk þ i Þjaji if i j: If we insert the choice (2.19) and take the limit as ϵ → 0, we obtain XN −1 ψ þ ikx 0 Φ 0 ikaj k ðxÞ¼e þ R0ðx; aijEk þ i Þ½ ðEk þ i Þ ije ; ð9:10Þ i;j¼1 where the principal matrix ΦðEk þ i0Þ ≡ limε→0þ ΦðEk þ iεÞ is 8 1 > − − ffiffiffiffiffiffiffiffiffiffiiEk <> λ i p 2 2 if i ¼ j ðEk;EBÞ E −m Φ 0 ffiffiffiffiffiffiffiffiffiffik ffiffiffiffiffiffiffiffiffiffi ijðEk þ i Þ¼ p R p 2 2 ð9:11Þ > 2− 2 − 1 μ −m > ffiffiffiffiffiffiffiffiffiffiiEk i E m jai ajj ∞ −μjai−ajj : − p e k − dμ e 2 2 2 i ≠ j: 2− 2 π m μ þE −m if Ek m k λ i The function ðEk;EBÞ is defined as 2 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 3 2 − 2 1 6 E B Ek m C Ei π Ei 7 ¼ −4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik arctanh@ A þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB þ arcsin B 5; ð9:12Þ λ i 2 i 2 2 ðEk;EBÞ π 2 − 2 Ek π m − ðE Þ m Ek m B and called the energy dependent running coupling constant A simple asymptotic analysis applied to the above originally introduced in [23] for a single center. integral shows that it is exponentially damped for The diagonal term of the principal matrix (9.11) is large values of x (x ≫ ai) so that we may ignore actually nothing but the analytic continuation of the it compared to the first oscillating term for the out- formula (2.28). For the scattering problem, we need to going scattering problem. Putting this into Eq. (9.10), determine the asymptotic behavior of the scattering sol- we get x x ≫ a ution for large values of , namely i. For this reason, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 XN p 2 2 let us first express the resolvent kernel R0ðx; aijEk þ i Þ in i k þ m the following way: ψ þðxÞ ∼ eikx þ k k i;j¼1 0 − −1 R0ðx; aijEk þ i Þ ikjx aij Φ 0 ikaj × e ½ ðEk þ i Þ ije : ð9:14Þ ¼hxjR0ðEk þ i0Þjaii Z − This is an explicit and exact solution to the ∞ dp eipðx ajÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi semirelativistic Lippmann-Schwinger equation, and it −∞ 2π 2 2 − 0 p þ m ðEk þ i Þ includes the information about the reflection and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ∞ 2 2 transmission coefficients so that we can immediately i k þ m − 1 −μ − μ − m ikjx aij μ jx aij ikx ¼ e þ π d e μ2 2 : find them by simply reading the factors in front of e k m þ k ikx for x 045004-16 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN 2 2 solutions. In our method, the derivation for the reflection 2 i k þ m RðkÞ¼jrðkÞj ¼ and transmission coefficients is much simpler and more 1 k i;j¼ general. The scattering phase shift δðkÞ can simply be 2 −1 computed from the S-matrix SðkÞ¼rðkÞþtðkÞ¼ × Φ E i0 eikðaiþajÞ ; ð ð k þ ÞÞij expð2iδÞ. Further physical questions have been discussed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN 2 2 in [23]. 2 i k þ m 2 1 2 TðkÞ¼jtðkÞj ¼ 1 þ N ¼ case (EB ¼ EB ¼ EB): 1 k We can always choose our coordinate system such that i;j¼ 2 two Dirac delta centers are located symmetrically with −1 − Φ 0 ikð aiþajÞ − × ð ðEk þ i ÞÞije : ð9:15Þ respect to the origin, so that a1 ¼ a and a2 ¼ a. Since we cannot analytically evaluate the integrals in the off- diagonal part of the principal matrix (9.11), we compute the Here R k represents the reflection coefficient and ð Þ reflection and transmission coefficients numerically with T k the transmission coefficient. It is important to ð Þ the help of Mathematica, and their graphical representa- notice the notational difference that the same letters tions are depicted in Fig. 5. have been used for the scattering amplitudes in [23]. Let us address some issues about the behavior of the Here, we prefer to stick to a more traditional notation. reflection and transmission coefficients. The general pat- Although the above solution is exact, it is difficult to tern of these coefficients as functions of k=m is very similar calculate the inverse of the principal matrix for any to the one in the nonrelativistic version of the same problem number of Dirac delta centers located arbitrarily on the [45,46]. All maxima of the transmission coefficient in line. Moreover, the off-diagonal part of the principal Fig. 5 indicate perfect transmissions. If we plot the trans- matrix (9.11) even includes an integral term that mission coefficient near one of those peaks, say at k=m ∼ 4, cannot be evaluated analytically. For this purpose, in a higher resolution, we can see that the peak has the we shall first consider the simplest possible cases. form, as shown in Fig. 6. This is why these peaks are N ¼ 1 case: First, we consider the case where we have a single center sometimes interpreted as resonances in [45]. However, one (N ¼ 1). We can assume that the Dirac delta potential is must be careful about this terminology since these do not located at the origin without loss of generality. In this case, have to correspond to decaying states [48]. For this reason, the principal matrix (9.11) is simply a function. Hence, the we prefer to call them perfect transmission energies. reflection and transmission coefficients become There is actually a small bump around the very small value of k=m, and it can be more clearly observed by 2 ðk2 þ m2Þλ2ðE ;E Þ changing the distance between the centers ma and EB=m. RðkÞ¼ k B ; To see this behavior, we plot the reflection coefficient as a k2 λ2 E ;E k2 m2 þ ð k BÞð þ Þ function of k=m for a particular value of 2ma and E =m in 2 B k Fig. 7. This shows that the reflection coefficient suddenly TðkÞ¼ 2 2 2 2 : ð9:16Þ k þ λ ðEk;EBÞðk þ m Þ vanishes near the zero energy of incoming particles for a certain value of distance between centers (2ma ¼ 0.775 in This is exactly the same result that was derived in [23] Fig. 7) for a given EB=m. The critical value for the distance by constructing the even-parity and odd-parity scattering between the centers is more transparently seen if we plot km km 0 5 10 15 20 25 30 0 5 10 15 20 25 30 1.0 1.0 0.8 0.8 0.6 0.6 k Tk R 0.4 0.4 0.2 0.2 0.0 0.0 FIG. 5. The reflection and transmission coefficients of two symmetric twin Dirac delta centers as a function of k=m for the values EB=m ¼ 1=2, 2ma ¼ 1. 045004-17 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) k m 3.0 3.5 4.0 4.5 5.0 defined as the vanishing reflection coefficient near the 1.0 threshold energy (at the border of the continuum energy spectrum), namely 0.8 RðkÞ → 0; ð9:17Þ 0.6 k as k → 0 for certain values of the parameters in the model. T 0.4 The underlying reason for threshold anomaly is basically the appearance of a bound state very close to the threshold 0.2 energy for some particular choice of the parameters in the model [49]. This anomaly in the nonrelativistic quantum mechanics even exists for the much more general class 0.0 of potentials, and the proof is given in [49]. Here we FIG. 6. The transmission coefficient as a function of k=m observe that this anomaly even exists for the semirelativ- plotted near its first peak k=m ¼ 4 for EB=m ¼ 1=2 and istic case that includes some singular potentials requiring 2ma ¼ 1. renormalization. We recall that the excited state of the system discussed in the reflection coefficient as a function of 2ma for different Sec. III appears near E ¼ m (k ¼ 0) when 2ma ¼ 0.775 small values of k=m, as shown in Fig. 8. It is important to and EB=m ¼ 1=2, as shown in Fig. 4. Hence, we show that notice that the peak around the critical value 2ma ¼ 0.775 the critical value of 2ma observed in Fig. 8 exactly becomes sharper and sharper as k=m decreases. corresponds to the critical case for which the excited state This behavior has also been observed in the nonrelativ- appears. We also realize that the critical value of 2ma istic case and known as a threshold anomaly [49].Itis decreases as we decrease EB=m (see Fig. 9). This is not km km 0 5 10 15 20 25 30 0.0 0.1 0.2 0.3 0.4 0.5 1.0 1.0 0.8 0.8 0.6 0.6 k R Rk 0.4 0.4 0.2 0.2 0.0 0.0 FIG. 7. The reflection coefficient as a function of k=m in different scales for 2ma ¼ 0.775 and EB=m ¼ 1=2. 2ma 2ma 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 1.0 1.0 0.8 0.8 k 1 0.6 m 10 0.6 k 1 k m 1000 Rk R 0.4 0.4 0.2 0.2 0.0 0.0 FIG. 8. The reflection coefficient as a function of 2ma in different scales for different values of k=m. Here we choose EB=m ¼ 1=2. 045004-18 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) 2ma surprising since we physically expect that, as we increase 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.0 EB, the bound state energies of the system must also increase so that the critical value for 2ma must be lowered. Although the reflection and transmission coefficients can 0.8 be obtained numerically for a pair of symmetrical Dirac k 1 m 10 delta centers (in principle for any finite N), we may ask 0.6 whether there is any good approximation, where we have Rk an explicit analytical expression for them and the above 0.4 analysis can be examined analytically. The answer relies on the asymptotic expansion of the integral 0.2 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∞ 2 2 1 −μ − μ − m μ jai ajj d e 2 2 ; ð9:18Þ 0.0 π m μ þ k FIG. 9. The reflection coefficient as a function of 2ma for in the off-diagonal part of the principal matrix (9.11). Let us EB=m ¼ 1=10. first make a change of variable μ ¼ sk so the above integral R pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ∞ −skja −a j s2k2−m2 becomes ds e i j 2 . Now we want to find πk m=k s þ1 an apparent discrepancy near very small values of k=m for the large kja − a j behavior of this integral. Note that −s in i j the fixed values of EB=m and 2ma given below, they are in the exponent has its maximum at s ¼ m=k on the interval complete agreement, as shown in Fig. 10 for larger values ðm=k; ∞Þ. Then, only the vicinity of s ¼ m=k contributes of k=m. In this approximation, the appearance of a thresh- to the full asymptotic expansion of the integral for large old anomaly occurs when 2ma ¼ 0.888; i.e., the asymp- − kjai ajj. Thus, we may approximate the above integral by 2ma R pffiffiffiffiffiffiffiffiffiffiffiffiffi totic approximation overestimates the critical value of . 1 ϵ − − 2 2− 2 skjai ajj s k m ϵ Moreover, the approximation to the reflection coefficient πk m=k ds e 2 1 , where >m=k and replace pffiffiffiffiffiffiffiffiffiffiffiffiffis þ 2 s2k2−m2 approaches its numerically calculated values as ma the function s2þ1 in the integrand by its Taylor or 2 k 2 increases near the region k=m are small ( ka ¼ m ma asymptotic expansion [65]. It is important to emphasize gets bigger). that the full asymptotic expansion of this integral as The phase shift in this particular problem can also be kja − a j → ∞ does not depend on ϵ since all other 2 δ i j calculated numerically from the relation SðkÞ¼e i ðkÞ, and integrations are subdominant compared to the original its graph is illustrated in Fig. 11. We note that δð0Þ¼π=2 integral (9.18). Hence, we find no matter what the values of EB=m for 2ma ¼ 1 are. Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let us consider now the array of Dirac delta potentials 1 ϵ s2k2 − m2 −skjai−ajj equally separated by some fixed distance, namely the ds e 2 πk m=k s þ 1 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi ϵ 1 − − s − m=k 2kmk km ∼ ds e skjai ajj π 2 2 0 5 10 15 20 25 30 m=k k þ m 1.0 Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi Rnumerical ∞ 1 − − s − m=k 2kmk ∼ skjai ajj ds e 2 2 0.8 π m=k k þ m Rapprox 2 2 m =k exp ð−mjai − ajjÞ ffiffiffiffiffiffi 0.6 ¼ p 2 2 3=2 ; ð9:19Þ 2πð1 þ m =k Þ ðmjai − ajj Þ Rk 0.4 where we have used the fact that the contribution to the integral outside of the interval ðm=k; ϵÞ is exponentially small for any ϵ >m=k. Substituting this result into 0.2 Eq. (9.11) and computing the inverse of the principal matrix, we can find an explicit analytic expression for the 0.0 reflection and transmission coefficients (but they are too complicated to write them down explicitly here) as long as FIG. 10. The reflection coefficient Rapprox in the asymptotic approximation and the reflection coefficient Rnumerical obtained we have to keep in mind that these expressions are valid 1 2 1 2 − numerically for the particular values of EB=m ¼ EB=m ¼ = only in the region where kjai ajj is large. 2 1 1 2 and ma ¼ . Notice that they slightly differ only near the region In particular, for twin (EB ¼ EB) symmetrically oriented when k=m is zero for a fixed value of 2ma. This is expected since Dirac delta centers, we can compare the predictions of our the asymptotic approximation becomes better and better as 2ka approximation with the numerical results. Although there is takes larger values. 045004-19 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) k m 0.0 0.5 1.0 1.5 2.0 which is of the order Oð1Þ. The above limit (9.20) is the 2.0 principal matrix for the nonrelativistic version of the same problem, and it can be directly seen from Eq. (6.14). Then, we obtain the nonrelativistic limit of 1.5 the scattering solution (9.14) EB 0.99 m XN k 1.0 im − −1 ψ þðxÞ ∼ eikx þ eikjx aij½Φ ðE þ i0Þ eikaj ; k k k ij EB 0 i;j¼1 0.5 ð9:22Þ EB 0.99 m where ΦijðEk þ i0Þ is given by (9.20). Then, we can obtain 0.0 the reflection and transmission coefficients from this solution, which is consistent with the standard results in FIG. 11. Phase shift δðkÞ as a function of k=m for three different the literature (see [46] for the two-center case). values of EB=m and for 2ma ¼ 1. X. THE BOUND STATES AND THE SCATTERING semirelativistic Kronig-Penney model. In this case, the PROBLEM IN THE MASSLESS CASE transmission coefficients in Fig. 12 indicate the formation of the band gaps in the spectrum as we increase the number We first consider the bound state problem in the massless of centers. The nonrelativistic version of the problem by case m ¼ 0. In this case, we have only ultrastrong bound studying the transmission coefficient has been given states since they must occur in the negative E axis. Using in [66]. the explicit expression of the heat kernel (2.25), the To discuss the nonrelativistic limit of the reflection and principal matrix is transmission coefficients, we study the nonrelativistic limit − 8 E m ≪ 1 1 ( m ) of the scattering solution of the semirelativistic > i <>π logðE=EBÞ if i ¼ j Lippmann-Schwinger equation (9.14). The nonrelativistic Φ 0 Φ ðEÞ¼ 1 2cos E a −a Ci −E a −a limit of the principal matrix ijðEk þ i Þ is ij > 2π ð ð ð i jÞÞ ð j i jjÞ :> ( þsinðEjai −ajjÞðπ þ2SiðEjai −ajjÞÞÞ if i ≠ j; 1 − im λ k if i ¼ j Φ E i0 → i : ð10:1Þ ijð k þ Þ − ð9 20Þ − im ikjai ajj ≠ k e if i j; where Ei < 0 and E is real and negative (for bound states). −λ i → λ B where we have used the fact that ðE; EBÞ i in the Here Ci and Si are the sine integral and the cosine integral nonrelativistic limit, which is shown for a single center in functions defined by their integral representations [53] [23]. Here we have ignored the second integral term in the Z Z off-diagonal part of the principal matrix since ∞ x − cos t sin t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CiðxÞ¼ dt ;SiðxÞ¼ dt : ð10:2Þ Z x t 0 t 1 ∞ μ2 − 2 −μ − m dμ e jai ajj π μ2 − 1 2 m þðEk mÞðEk þ mÞ For simplicity, we assume that EB ¼ EB ¼ EB and Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − − 1 ∞ μ2 − 2 a1 ¼ a2 ¼ a (twin symmetrically located centers). −μ − m ¼ dμ e jai ajj The bound state energies can be found from the transcen- π μ2 þ ηðη þ 2Þm2 Zm dental equation det ΦðEÞ¼0 or the zeros of the eigenval- 1 ∞ 1 −μja −a j ues of the principal matrix (10.1) as emphasized earlier. ≤ dμ e i j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ K0ðmja − a jÞ; π 2 2 i j m μ − m In contrast to the complications in the massive case, the eigenvalues can be explicitly calculated in this case and ð9:21Þ given by 2 logð E Þþ2 cos ð2aEÞCið−2aEÞþπ sin ð2aEÞþ2 sin ð2aEÞSið2aEÞ ω1 EB ðEÞ¼ 2π ; 2 logð E Þ − 2 cos ð2aEÞCið−2aEÞ − π sin ð2aEÞ − 2 sin ð2aEÞSið2aEÞ ω2 EB ðEÞ¼ 2π : ð10:3Þ 045004-20 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) 2.0 2.0 1.5 1.5 Tvs.k m T vs. k m 1.0 1.0 0.5 0.5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2.0 2.0 1.5 1.5 T vs. k m T vs. k m 1.0 1.0 0.5 0.5 51015202530 5 1015202530 FIG. 12. The transmission coefficient as a function of k=m for different values N ¼ 1, 2, 4, 8, respectively. Here we choose that all i ’ 1 10 − 2 EB s are the same, EB=m ¼ = , and mjai ajj¼ . Let us analyze the behavior of bound states for this case by exactly the same except for the fact that the form of the heat plotting them as a function of E for different values of aEB. kernel is given by (2.25)]. In Fig. 13, one can apparently notice that the eigenvalues of Figure 13 also illustrates that we may have one or two the principal matrix become degenerate as we increase ultrastrong bound states depending on the choice of the jaEBj. values of aEB. It is not difficult from Eq. (10.3) to show that 1 1 This can be analytically justified from the following fact: limE→−∞ω ¼ ∞ and limE→0þ ω ¼ −∞ for all a and EB. Since the eigenvalues are decreasing functions, ω1 must 2 E E have exactly one zero. On the other hand, limE→−∞ω ¼ ∞ lim 2 cos 2ajEBj Ci −2ajEBj 2 γþlogð−2aEBÞ ajEBj→∞ jE j jE j ω − γ ≈ B B and limE→0þ ¼ π for all a and EB. Here E 0.5772 is Euler’s constant. This means that ω2 could cross þ π sin 2ajE j B jE j the E axis only when B 2 2 E 2 E 0 1 þ sin ajEBj Si ajEBj ¼ ð10:4Þ ajE j > : ð10:5Þ jEBj jEBj B 2eγ 1 2 for all finite E=jEBj. Hence we conclude that ω → ω as The second bound state appears only if the condition → ∞ 1 ω1 ajEBj for all E=jEBj. As a result of this, the bound ajEBj > 2eγ is fulfilled. The point E at which has a states become degenerate. simple zero is the ground state energy. It is also important to realize from Fig. 13 that the Alternatively, this critical value can also be estimated eigenvalues ω1 and ω2 are decreasing functions of E,as analytically by working out the characteristic equation proved in Sec. III [the proof for the massless case would be det ΦðEÞ¼0, whose solutions are the bound state energies, 045004-21 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) aE aE 10 8 6 4 2 0 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.4 0.8 0.2 0.6 aE 5 B 0.0 E E 0.4 2 2 0.2 , , 0.2 E E 1 1 0.4 0.0 0.6 aEB 12 0.2 0.8 0.4 1.0 aE aE 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8 0.6 0.4 0.2 0.0 0.4 0.4 0.2 0.2 0.0 E E 2 2 0.2 , , 0.0 E E 1 1 0.4 0.2 1 0.6 aEB 1 aEB 2 0.8 0.4 1.0 FIG. 13. The flow of the eigenvalues of the principal matrix as a function of aE in the massless case for different values of aEB. E 2 cos ð2aEÞCið−2aEÞþsin ð2aEÞðπ þ 2Sið2aEÞÞ log ¼ : ð10:6Þ EB 2 The principal matrix ΦðEk þ i0Þ in the scattering problem turns out to be 8 < 1 k i þ π logðjEi jÞ if i ¼ j Φ ðE þ i0Þ¼ B ð10:7Þ ij k : 1 2π ð2 cos ðkðai − ajÞÞCið−kjai − ajjÞ þ sin ðkjai − ajjÞðπ þ 2Siðkjai − ajjÞÞÞ if i ≠ j: 1 2 − − Then, the scattering solution to the semirelativistic For EB ¼ EB ¼ EB and a1 ¼ a2 ¼ a, the behavior of Lippmann-Schwinger equation is the reflection coefficient as a function of ka is shown below for particular values of ajEBj.Thisisalsoa XN − −1 typical behavior of the reflection coefficient in the ψ þ x ∼ eikx ieikjx aij Φ E i0 eikaj ; 10:8 k ð Þ þ ½ ð k þ Þ ij ð Þ nonrelativistic case [46,47]. The particle is fully trans- i;j 1 ¼ mitted at some certain energies that can easily be seen where ΦðE þ i0Þ is given by Eq. (10.7). Hence, we can from Fig. 14. Also the above graph is plotted for k 1 2 analytically find the reflection and transmission coefficient ajEBj¼ = . In contrast to the massive and the non- relativistic problems, the reflection coefficient is always XN 2 zero for small values of ka no matter what value ajEBj −1 Φ 0 ikðaiþajÞ RðkÞ¼ i½ ðEk þ i Þ ije ; is. In the massive and the nonrelativistic cases, the i;j¼1 reflection coefficient is always unity for very small XN 2 values of k=m. Nevertheless, an anomalous behavior is −1 − 1 Φ 0 ikð aiþajÞ 1 TðkÞ¼ þ i½ ðEk þ i Þ ije : ð10:9Þ also observed in this case when ajEBj¼2eγ (this critical i;j¼1 value corresponds to the condition for the appearance of 045004-22 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) ka 0 5 10 15 20 25 30 XI. THE RENORMALIZATION GROUP 1.0 EQUATIONS AND THE BETA FUNCTION FOR N CENTERS 0.8 One possible way for the renormalization scheme to determine how the coupling constant changes with the 0.6 energy scale is to define the following renormalized k λR R coupling constant i ðMÞ in terms of the bare coupling 0.4 constants λiðϵÞ: Z ∞ −M t 0.2 1 1 e i − dt ; 11:1 λR ¼ λ ϵ π ð Þ i ðMiÞ ið Þ ϵ t 0.0 where Mi is the renormalization scale. Then, the renor- FIG. 14. The reflection coefficient as a function of ka for a pair malized principal matrix in terms of the renormalized of symmetrically located centers in the massless case for coupling constant is a E 1=2 j Bj¼ . 8 R − < 1 − ∞ tE − e Mit λRðM Þ 0 dt ðKtðai;aiÞe πt Þ if i ¼ j ΦR i i ijðEÞ¼ R : − ∞ tE ≠ a new bound state near E ¼ 0). This can easily be seen 0 dt Ktðai;ajÞe if i j; by plotting the reflection coefficient as a function ajE j B ð11:2Þ near k ¼ 0 (ka ¼ 0.01). Around ajEBj in Fig. 15,the reflection coefficient suddenly drops to zero at this and the bound state energy is determined from the critical value of ajE j for small values of ka.Note B condition det ΦR ðEÞ¼0 which gives the relation that there is a curious sudden change near ajE j¼0. ij B λR However, our model is not properly defined when the between i ðMiÞ and Mi. Here the integral in the diagonal centers coincide as long as E is nonzero (recall that part of the matrix Φ is convergent due to the short time B ∼ 1 a ≠ 0 as defined in Sec. II). More interesting, the asymptotic expansion of the Bessel function K1ðmtÞ πt reflection coefficient always vanishes as k → 0 in con- as t → 0. Explicit dependence on Mi cancels out the trast to the nonrelativistic and massive case. Never- implicit dependence on Mi through the renormalized λR theless, the threshold anomaly still occurs very close to coupling constant i ðMiÞ. Physics is determined by the threshold energy. the value of the renormalized coupling constant at an The massless problem is a simple quantum mechanical arbitrary value of the renormalization point Mi. However, λR model where we have an explicit example of dimensional the above choice of i ðMiÞ may not be physically transmutation. Initially, the problem has no intrinsic energy appropriate since we have to deal with more than one scale, but we obtain an energy scale through the renorm- renormalized coupling constant with the same type of alization procedure. interaction, which essentially differ from each other by arbitrary constants. These constants can be determined by deciding the excited energy levels. We instead prefer a a EB 0.0 0.2 0.4 0.6 0.8 1.0 1.0 single renormalized coupling constant by redefining it without altering the physics of the problem. This could be performed in the following way. 0.8 Instead of using the bound state energy to fix the flow, we may fix the relative strengths of individual delta 0.6 i interactions. We know that EB is the bound state energy k for the individual ith Dirac delta center so that it corre- R ΦR i 0 0.4 sponds to the solution iiðEBÞ¼ . Without loss of ΦR 1 0 generality, let us assume that 11ðEBÞ¼ . This allows 0.2 us to choose the renormalized coupling constant as Z 1 1 ∞ e−Mt 0.0 ¼ − dt ; ð11:3Þ λRðMÞ λ1ðϵÞ ϵ πt FIG. 15. The reflection coefficient as a function of ajEBj for a pair of symmetrically located centers in the massless case at some scale M. Once the renormalized coupling constant (ka ¼ 0.01). is fixed under this condition, we must also impose 045004-23 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) ΦR 2 0 ≠ 1 iiðEBÞ¼ for i with this choice at the same scale M. Then, the renormalized principal matrix is This is always possible if we add a constant term to the ΦR definition of a renormalized coupling constant. Let us ijðEÞ consider the i ¼ 2 case 8 R < 1 ∞ tE e−Mt λ − 0 dt Ktðai;aiÞe − π − Σi if i ¼ j ¼ RðRMÞ t 1 : ∞ R 2 − tE ≠ Φ22ðE Þ¼ 0 dt Ktðai;ajÞe if i j: B λ ðMÞ RZ ∞ −Mt ð11:7Þ e tE2 þ dt − Ktða2;a2Þe B − Σ2 0 πt Z To find the beta function, we need the renormalization ∞ tE1 tE2 group equation, given by ¼ dt ðKtða1;a1Þe B − Ktða2;a2Þe B Þ − Σ2 0 dΦR ðM; λ ðMÞ;E;m;ja − a jÞ ¼ 0; ð11:4Þ M ij R i j dM ∂ ∂ ΦR 1 0 where we have used Eq. (11.3) and 11ðEBÞ¼ . This ¼ M þ βðλRÞ ∂M ∂λR means that there always exists a constant Σi depending i Σ 0 Σ ≠ 0 ≠ 1 ΦR λ − only on EB with 1 ¼ and i for i such that the × ijðM; RðMÞ;E;m;jai ajjÞ ΦR Ei 0 condition iið BÞ¼ can be fulfilled. Hence, the renor- ¼ 0; ð11:8Þ malized coupling constant becomes Z where the beta function is 1 1 ∞ −Mt − e Σ λ ¼ λ ϵ dt π þ i; ð11:5Þ ∂λ RðMÞ ið Þ ϵ t βðλ Þ¼M R : ð11:9Þ R ∂M Σ and the choice of i refers to the relative strengths of delta The renormalization group equation essentially tells us i interactions in this new renormalization scheme. If all EB that physics should be independent of the renormaliza- Σ 0 are the same, then i ¼ . We can explicitly determine the tion scale. It is worth pointing out that the renormaliza- renormalized constant by evaluating the integral and tion condition (11.8) corresponding to the problem in ϵ removing , the two-dimensional nonrelativistic version of the prob- lem has been written in terms of the T matrix in [67]. 1 Ei 1 2M Using Eq. (11.7) in Eq. (11.8), we can find the beta ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB þ log − 1 λ 2 i 2 π function RðMÞ m − ðE Þ m B 1 − i 2 ð1;0;0;0Þ m EB λ þ 2F1 0; 2; 3=2; β λ − R π 2m ð RÞ¼ π : ð11:10Þ i 0 1 0 0 m − E þ Fð ; ; ; Þ 0; 2; 3=2; B It is important to note that the beta function here is 2 1 2m formally different from the one derived for a single i 0 0 1 0 m − E center case [23] and that our formula (11.10) is much þ Fð ; ; ; Þ 0; 2; 3=2; B þ Σ ; ð11:6Þ 2 1 2m i simpler than the one given in [23]. This difference is due to the choice of the renormalization condition, and the beta function has been expressed in terms of the where F1 is the hypergeometric function [53]. The super- 2 energy-dependent running coupling constant λ E; E in scripts on the hypergeometric functions denote the deriva- ð BÞ there. However, the physics is the same. The negativity ð1;0;0;0Þ 0 2; tive with respect to each variable; e.g., 2F1 ð ; of the beta function (11.10) implies that our model is m−Ei m−Ei B B λ 0 3=2; 2 Þ is the derivative of 2F1ðx; 2; 3=2; 2 Þ with asymptotically free and the zero of it is R ¼ so that it m m λ → 0 → ∞ respect to x evaluated at x ¼ 0. Although this is a rather is an ultraviolet fixed point since R as M . complicated function, we will see that this gives us a This result is consistent with the case when there is only simple formula for the β function. The renormalized one center in [23]. We realize that our convention is coupling constant (11.6) logarithmically vanishes for more convenient and simpler to investigate for more large values of energy M as can easily be seen from than one center. By integrating its expression so that the particle becomes free in this ∂λ ðM¯ Þ λ2 ðM¯ Þ limit. This is a phenomenon that appears in QCD and is βðλ Þ¼M¯ R ¼ − R ð11:11Þ called asymptotic freedom. R ∂M¯ π 045004-24 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) from M¯ ¼ M to M¯ ¼ αM with α > 0, we can find the For the massless case, the beta function is formally the flow equation for the coupling constant same but the renormalized coupling constant is λ 1 1 λ α RðMÞ ¼ logð−M=Ei ÞþΣ : ð11:19Þ Rð MÞ¼ 1 : ð11:12Þ λ ðMÞ π B i 1 þ π λRðMÞ log α R When relative strengths are the same, i.e., Σ ¼ 0,we From the explicit expression of the renormalized prin- i obtain the beta function cipal matrix, we can easily see that π −1 β λ − ΦR λ α α α − ð RÞ¼ M 2 ; ð11:20Þ ijðM; RðMÞ; E; m; jai ajjÞ ðlogð− ÞÞ EB ΦR α−1 λ − ¼ ijð M; RðMÞ;E;m;jai ajjÞ: ð11:13Þ which is exactly the same formula as the one given for one If we take the scale-invariant derivative with respect delta center in [23]. Similar to the single center case, the to α of both sides, we find the renormalization group model has both ultraviolet and infrared fixed points. ΦR λ α α equation for the principal operator ijðM; RðMÞ; E; m; −1 XII. A POSSIBLE EXTENSION OF THE MODEL α jai − ajjÞ, The method we have developed for the model of a d single semirelativistic particle interacting with finitely α ΦR ðM; λ ðMÞ; αE; αm; α−1ja − a jÞ dα ij R i j many pointlike Dirac delta potentials can be applied to ∂ more general types of singular interactions, e.g., Dirac þ M ΦR ðM; λ ðMÞ; αE; αm; α−1ja − a jÞ ¼ 0; ∂M ij R i j delta potentials supported by curves in two dimensions and supported by surfaces in three dimensions. The ð11:14Þ nonrelativistic version of this kind of interactions has been studied from several points of view [68–70].The or renormalization is required only if the codimension is two for the nonrelativistic case, whereas the semirelativistic d ∂ α − β λ ΦR λ α α α−1 − case needs to be renormalized when the codimension ð RÞ ijðM; RðMÞ; E; m; jai ajjÞ dα ∂λR is one. ¼ 0: ð11:15Þ Here we only illustrate how our method of renorm- alization can be performed for the general kind of the If we postulate the following functional form for the singular Dirac delta interactions without going into principal matrix: details of their spectrum. Let us consider a semirela- tivistic particle interacting with finitely many singular ΦR λ α α α−1 − interactions, each of which is supported by arc-length ijðM; RðMÞ; E; m; jai ajjÞ parametrized closed regular curve Γi of finite length Li α ΦR λ α − ¼ fð Þ ijðM; Rð MÞ;E;m;jai ajjÞ; ð11:16Þ in two dimensions. We assume that each curve is not self-intersecting and there is no intersection among the and substitute into Eq. (11.15), we obtain an ordinary curves as well. Then, the semirelativistic Schrödinger differential equation for the function f, equation is Z D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E Xn λ dfðαÞ r 2 2 ψ − i δ r Γ α 0: 11:17 j P þ m j dli ð ; iðsÞÞ ¼ ð Þ Γ dα 1 Li i Z i¼ α 1 This gives the solution fð Þ¼ using the initial condition × dli ψðΓiðsÞÞ ¼ EψðrÞ; ð12:1Þ Γ at α ¼ 1. Therefore, we get i v ΦR λ α α α−1 − where dli ¼j iðsÞjds is the ith integration line element, ijðM; RðMÞ; E; m; jai ajjÞ _ viðsÞ¼ΓiðsÞ is the tangent vector to the curve Γi,ands ΦR λ α − ¼ ijðM; Rð MÞ;m;jai ajjÞ; ð11:18Þ is the arc-length parameter. Here ψðΓiðsÞÞ is the restric- tion of the wave function ψðrÞ to the curve Γi.Notethat which means that there is no anomalous scaling. We can the potential energy term in the above Schrödinger also verify that if the renormalized coupling constant equation has a nonlocal character. evolvesasinEq.(11.12), the scaling relation (11.18) is Similar to the formal definitionR of pointlike Dirac delta ∞ satisfied. function hδa; ϕi ≔ ϕðaÞ¼“ −∞dx δðx − aÞϕðxÞ” for any 045004-25 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) test function ϕ, the Dirac delta function supported by a It is important to notice that as ϵ → 0þ, we obtain the delta Γ closed arc-length parametrized curve i of length Li can be function supported by the curve Γi. Moreover, we have defined formally [71] ZZ Γϵ Γϵ 0 Γ Γ 0 Z Z h i j ji¼ dli dlj Kϵ=2ð iðsÞ; jðs ÞÞ: ð12:5Þ L i Γi×Γj hδΓ; ϕi ≔ dli ϕ ¼ ds jviðsÞj ϕðΓiðsÞÞ Γ 0 ZZi Z We can then write the regularized semirelativistic L 2 i Schrödinger equation ¼ “ d r ϕðrÞ ds jviðsÞj δðr; ΓiðsÞÞ”; R2 0 XN λiðϵÞ ϵ ϵ ð12:2Þ H0 − jΓ ihΓ j jψi¼Ejψi: ð12:6Þ L j i i¼1 i from which we have Following the same line of arguments introduced in Sec. II Z L for pointlike Dirac delta potentials, we obtain the resolvent r Γ i v δ r Γ h j ii¼ ds j iðsÞj ð ; iðsÞÞ: ð12:3Þ after the renormalization of the coupling constant 0 −1 −1 RðEÞ¼ðH0 − EÞ þðH0 − EÞ In analogy with the regularization of point Dirac delta potential with the heat kernel, we introduce XN Γ Φ−1 Γ − −1 Z × j ii½ ðEÞ ijh jj ðH0 EÞ : ð12:7Þ i;j¼1 r Γϵ Γϵ r r Γ h j i i¼ i ð Þ¼ dli Kϵ=2ð ; iðsÞÞ: ð12:4Þ Γ i Here, the principal matrix is defined as 8 RR R 1 0 ∞ tEi tE 0 < Γ Γ dl dl 0 dt ðe B − e ÞK ðΓ ðsÞ; Γ ðs ÞÞ if i ¼ j Li i× i i i t i i Φ ðEÞ¼ RR R ð12:8Þ ij : − ffiffiffiffiffiffiffi1 0 ∞ Γ Γ 0 tE ≠ p Γ ×Γ dli dlj 0 dt Ktð iðsÞ; jðs ÞÞ e if i j: LiLj i j Similarly, we can apply our method to the Dirac delta studied the behavior of the reflection and transmission potentials supported by a regular surface in three dimen- coefficients for the two-center case numerically and sions. This analysis can be even further extended to the approximately and observed the threshold anomaly that curved manifolds; see the nonrelativistic discussion of it also exists in the nonrelativistic problem. We have found in [68,69]. that this anomaly is due to the appearance of the bound state appearing just near the threshold energy. In par- XIII. CONCLUSIONS ticular, we have analytically analyzed the bound state and scattering problem in the massless version of the prob- In conclusion, we have considered in this paper the lem. Finally, we have derived renormalization group one-dimensional spinless Salpeter Hamiltonian with equations and computed the beta function for the model. finitely many Dirac delta potentials. Similar to the We hope that our construction using the heat kernel one-center case, the problem requires renormalization. techniques can be generalized to the many-body version We have constructed the resolvent formula by using heat of the problem so that all the techniques we have kernel regularization and renormalizing the model. We developed here can guide us for more complicated field have discussed the bound state spectrum and proved that theoretical problems. the ground state energy is bounded from below. Then, we have shown that there exists a unique self-adjoint ACKNOWLEDGMENTS operator associated with the resolvent formula. We have obtained an explicit wave function formula for N centers The present work has been fully financed by TUBITAK and illustrated the fact that our problem is actually from Turkey under “the 2221—Visiting Scientist consistent with the self-adjoint extension theory in Fellowship Programme.” We are very grateful to mathematics literature. We have also proved that the TUBITAK for this support. We also acknowledge to ground state is nondegenerate and discussed some new Osman Teoman Turgut for clarifying discussions and his results on the number of bound states. Moreover, we interest in the present research. Finally, we would like to have solved exactly the semirelativistic Lippmann- mention that the present work follows the lines of Projects Schwinger equation and found an explicit expression No. MTM2014-57129-C2-1-P and No. VA057U16 for the reflection and transmission coefficients. We have from Spain. 045004-26 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) APPENDIX A: A PROOF OF THE ANALYTICITY test. Since all its matrix elements of Φ are holomorphic, OF THE PRINCIPAL MATRIX the principal matrix Φ is a matrix-valued holomorphic function on R, and the derivatives of all orders of Φ We first recall the following theorem (theorem 1.1 in with respect to z can be found by differentiating under Chapter 2 of [72]): the sign of integration. Then, its eigenvalues and Assume that the function fðz; tÞ [z is a complex variable eigenfunctions are also infinitely differentiable due to ranging over a domain R and t is a real variable over the corollary of Theorem II. 6. 1 in [63]. ð0; ∞Þ] satisfies: (i) fðz; tÞ is a continuous function of both variables. (ii) For each fixed value of t, fðz; tÞ is a APPENDIX B: A PROOF OF THE EXISTENCE Rholomorphic function of z. (iii) The integral FðzÞ¼ ∞ OF THE SELF-ADJOINT HAMILTONIAN 0 fðz; tÞdt converges uniformly at both limits in any compact set in R. Then, FðzÞ is holomorphic in R and its Equation (5.2) requires the following condition to derivatives of all orders may be found by differentiating complete the second part of the proof: under the integral sign. The above two hypotheses for the matrix elements of the ‖jEkjRðEkÞjfi − jfi‖ → 0; ðB1Þ principal matrix Φ are satisfied since the heat kernel Ktðx; yÞ defined on R × R × ð0; ∞Þ is C1—a continuously differ- as k → ∞, where jfi belongs to some appropriate Hilbert entiable function with respect to the variable t and expo- space and its usual L2 norm is equal to one. Using the nential function etz is an entire function for each fixed value explicit expression of the full resolvent (2.23) and sepa- of t. What is left is to show that all the matrix elements rating the free part, we can find an upper bound to the norm converge uniformly on a compact subset of the chosen above that we are interested in, region R.LetR be the complex plane with RðzÞ 045004-27 FATIH ERMAN, MANUEL GADELLA, and HAYDAR UNCU PHYSICAL REVIEW D 95, 045004 (2017) Z XN To find the upper bound for the norm of the inverse ‖ ‖ ≤ jEk jA jEkj dx R0ðai;xjEkÞR0ðx; aljEkÞ principal matrix, we first decompose the principal matrix 1 R Z i;j;r;l¼ into two positive matrices Φ−1 × dy R0ðaj;yjEkÞR0ðy; arjEkÞj ij ðEkÞ R ! Φ ¼ D − K; ðB9Þ 1=2 × ‖Φ−1 E : B4 rl ð kÞj ð Þ where D and K stand for the on-diagonal and the off- diagonal parts of the principal matrix, respectively. Then, it Φ 1 − −1 Let us first consider the diagonal case l ¼ i and r ¼ j for is easy to see ¼ Dð D KÞ. The principal matrix is 1 − −1 the terms inside the bracket above. invertible if and only if ð D KÞ is invertible. The matrix ð1 − D−1KÞ has an inverse if the matrix norm Z satisfies ‖D−1K‖ < 1. Then, we can write the inverse of XN Φ as a geometric series, jEkj dx R0ðai;xjEkÞR0ðx; aijEkÞ R i;j¼1 Z −1 −1 −1 −1 1=2 Φ ¼ð1 − D KÞ D Φ−1 ‖Φ−1 × dy R0ðaj;yjEkÞR0ðy; ajjEkÞj ij ðEkÞ ji ðEkÞj −1 −1 2 −1 R ¼ð1 þðD KÞþðD KÞ þ ÞD ; ðB10Þ ≤ 2 α α Φ−1 2 1=2 jEkjðN max iðEkÞ max jðEkÞ max j ij ðEkÞj Þ ; 1≤i≤N 1≤j≤N 1≤i;j≤N and the norm has the following upper bound: ðB5Þ ‖Φ−1‖ ¼ ‖ð1 − D−1KÞ−1D−1‖ R ≤ ‖ 1 − −1 −1‖‖ −1‖ where we have defined αiðEkÞ¼ R dyR0ðai;yjEkÞ ð D KÞ D R0ðy; aijEkÞ for simplicity. It is easy to see that αiðEkÞ is 1 ≤ ‖D−1‖: 1 − ‖ −1 ‖ ðB11Þ Z D K dx R0ðai;xjEkÞR0ðx; aljEkÞ Since we are not concerned with the sharp bounds on the R Z Z −1 ∞ ∞ norm of Φ here, we can choose jEkj sufficiently large − −1 ðt1þt2ÞjEkj ‖ ‖ 1 2 ¼ dt1 dt2 Kt1þt2 ðai;alÞe such that D K < = without loss of generality and get Z0 0 ∞ − −1 −1 tjEkj ‖Φ ‖ ≤ 2‖ ‖ ¼ dttKtðai;alÞ e ; ðB6Þ ðEkÞ D ðEkÞ ; ðB12Þ 0 where D−1 ¼ diagðΦ−1; Φ−1; …; Φ−1 Þ and by using the fact that the free resolvent kernel is just the 11 22 NN Laplace transform of the heat kernel. Using the explicit ‖ −1‖ Φ−1 D ¼ max j ii j: ðB13Þ expression of the heat kernel (2.14) and the upper bound of 1≤i≤N the Bessel function (2.16), we get −1 Since D and K are decreasing functions of jEkj, we can 1 m always make ‖D−1K‖ < 1=2 by sufficiently large values of max α ðE Þ < þ : ðB7Þ 1≤ ≤ i k π m m 2 i N ð 2 þjEkjÞ 2πð 2 þjEkjÞ jEkj. By using the lower bound of the Bessel function (4.9), we find We have also −1 π ‖D ðEkÞ‖ < ; ðB14Þ mþjEkj logð m−Ei Þ XN B Φ−1 2 ≤ Φ−1 2 max j ij j max j ij j 1≤i;j≤N 1≤i≤N so that j¼1 " Φ−1 Φ−1 ≤ ρ Φ−2 ¼ max ð ðEkÞ ðEkÞÞii ð ðEkÞÞ 2 1≤i≤N 1 m jE ‖jA‖ < jE j 4π2N2 þ −2 −1 2 k k π m 2π m 2 ≤ ‖Φ ðE Þ‖ ≤ ‖Φ ðE Þjj ; ðB8Þ ð 2 þjEkjÞ ð2 þjEkjÞ k k # X 1 1=2 Φ† Φ ∈ R ρ × : ðB15Þ where we have used ðEkÞ¼ ðEkÞ for Ek and is i 2 mþjEkj log ð m−Ei Þ the spectral radius. B 045004-28 ONE-DIMENSIONAL SEMIRELATIVISTIC HAMILTONIAN … PHYSICAL REVIEW D 95, 045004 (2017) If we take the limit k → ∞, the right hand side H† −E ¼ðR−1ðE ÞÞ† ¼ðR†ðE ÞÞ−1 ¼ðRðEÞÞ−1 ¼ H −E: goes to zero, the same analysis can be found similarly B16 for the off-diagonal terms, and this completes the ð Þ proof. Let us denote this densely defined closed The self-adjointness also requires that the domains of H operator as H. and H must be the same. 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