Appendix A

Rigged Hilbert Spaces and the Properties of Self-Adjoint Operators in them']

A in quantum mechanics, as well as in analysis, arises from completing the space (/J of 'sufficiently good' (for example, smooth and decrea­ sing in infinity) functions with respect to a norm defined by a scalar product. The RHS theory assumes that (/J is the nuclear'] space in which the convergence is defined by a countable system of norms (the rtj)-convergence). If the scalar product (qJ , 1/1) is introduced in (/J continuous relative to the rtj)-convergence, then (/J will not be complete relative to the new convergence in the norm IIqJlI = = J(qJ , qJ) (the r£-convergence). However, (/J can be completed with respect to this new r£-convergence to the Hilbert space £ . The space £ * ofall antilinear functionals on £ is isomorphic to £ itself. The functionals from £ * are also continuous linear functionals on the space (/J c £ . The space (/J* of all linear functionals on (/J turns out to be broader and to include the Hilbert space £. The set of the three spaces embedded densely into each other

(A.t) is called the rigged Hilbert space (or the Gel'fand triad). Henceforth in the Appendix we shall use the following notation: small letters g, h, ... denote the vectors from the Hilbert space £ ; the Greek letters qJ, 1/1 , ... denote the vectors from (/J; the capital letters F, R, ... denote the vectors from (/J* . The notation for the scalar product in the sense of RHS (A1) should be understood as follows: (h Ig) is the conventional scalar product in £; h, g E £ ; (qJ IF) = (F I q» is the value of the functional FE (/J* on the vector qJ E (/J (the horizontal bar indicates the complex conjugation). Generally, the quantity (F IG) (where F, G E (/J* , but neither F no G belong to (/J) is not defined. In particular, the norm of the Gamow state in the RHS formalism is not defined. Let a linear operator A be given in £ with the definition domain D(A) :::> (/J such that the operator A*, adjoint of A, is defined in the domain D(A*) :::> (/J .

1) Since Appendix A pertains mainly to Chapter 4, the references here are those from that chapter. 2) For the definition see [22]. APPENDIX A 345 Here, A and A* do not map outside the space cP (i.e. cP is invariant with respect to A and A*):

ACP c cP , A*CP c cP . It should be reminded that the operator A* adjoint of A in the Hilbert space is called such an operator that

(A*h Ig) = (hi Ag) (A2) at arbitrary 9 E D(A) and hE D(A*) . If A = A* provided D(A) = D(A*), the operator is called self-adjoint. Throughout the space cP* the extension Aofthe operator A is defined by the relation

(AF I rp) = (F I A*rp) or

(rp IAF) = (A*rp IF) (A3) which must be satisfied for all rp E D(A*) and F E cP* . The extension A+ of the adjoint operator A* may be defined analogously:

(A+F I rp) = (F IArp), rp E D(A), FE CP*. (A4) For these extensions it is possible to formulate the generalized problems for the eigenvalues:

AR(A) = AR(A) , (A5a)

A +L(Ii) = iiL(Ii) (A5b) where the complex number A is called the right eigenvalue of the operator A corresponding to the right generalized eigenvector (GEV) R(A) E cP* ; Ii and L(Ii) are the left eigenvalue and the left GEV of the operator A, respectively. Using the Dirac notation, we can write equations (A5) in a more common manner, whence the meaning of the terms 'right' and 'left' becomes clear:

A IR(A) = A IR(A) , (A6a)

(L(Ii)1 A = Ii (L(Ii)1 . (A6b) However, it should be remembered that the meaning of these equations must conform to the definitions (A3) and (A4):

(A*rp I R(A) = A(rp IR(A), rp E D(A*) ; (A7a)

(L(Ii) I Arp) = Ii (L(Ii) Irp), rp E D(A) . (A7b) 346 APPENDIX A IfR(A.) is the right GEV of the operator A belonging to the eigenvalue A., then it proves to be the left GEV ofthe operator A* corresponding to the eigenvalue A. Indeed, we have

or

i.e.

For the self-adjoint operator A = A*, the extensions A and A+ coincide with each other, so any GEV is simultaneously the right vector with the eigenva­ lue A. and the left vector with the eigenvalue A . For the real eigenvalues, in particular, the right and left GEV of the self-adjoint operator are the same. Nevertheless, we retain the differencein the notation (R and L) bearing in mind the continuation of GEV to the complex plane. Furthermore, the right GEV will usually appear in the position of a ket, and the left GEV in the position ofa bra. Generally, the self-adjoint operators in the Hilbert space, e.g. such as the Hamiltonians of physical systems, have not only a discrete, but also a continu­ ous, spectrum; only the eigenvectors of the discrete spectrum belong to Jt1, however. Ifwe use the RHS, we may also treat the (generalized)eigenvectors of the continuous spectrum belonging to (/)*. Moreover, a self-adjoint operator in the RHS has a complete GEV system. The following (Gel'fand-Maurin) nuclear holds [18, 23]: for any self-adjoint operator A in the complex separable Hilbert space Jt1 with the definition domain D(A) there exists a nucle­ ar space (/) E D(A) densely and continuously imbedded in Jt1 which is invariant with respect to A such that the system of generalized eigenvectors of the operator A corresponding to the real eigenvalues from the spectrum a(A) is complete, i.e. for all rp, If/ E cP we get

(rp 11f/) = f dp(A.) ~ (rp IRi(A.) (Li(A.) 11f/) . (A8) utA) In fact, this is the conventional spectral expansion of unity expressed in terms of RHS. The sum in (A8) allows for the multiplicity of the spectrum (for example, in the case of a one-particle Hamiltonian, i is a set of quantum numbers of the angular momentum I proper and of its projection m) and the Stieltjes integral taken over the spectrum of the operator A. The physical Hamiltonians have usually a discrete and absolute continuous spectrum APPENDIX A 347

A.. Now, having introduced the Lebesgue measure dPaAA.) = h(A. )dA. , we can write (for simplicity we omit the multiplicity indices

(9' I '1') = L ( 9' l.fj) (jj I '1') + j

+ f(9' I R(A.) (L(A.) I '1') dA. (A9) If where .fj E Jf are the proper eigenvectors ofthe discrete spectrum; R(A.) = L(A.) are the GEV of the continuous spectrum. Going over from (A8) to (A9), we change their normalization in such a way that the weight function h(A.) disappe­ ars. Itshould be remembered again that R(A.) in (A9)coincides with L(A.) because A. E A. is real. Apart from the GEV entering the expansion (A9) and corresponding to the real eigenvalue A. E u(A), there exist other GEV with eigenvalues located both inside and outside the spectrum. In fact, the GEV of the continuous spectrum entering (A9) form an analytic GEV family, i.e. a vector-valued function with the value in (/J*, analytic in the region Q (which includes A.) such that at every A. E Q R(A.) is a GEV and A. is the corresponding eigenvalue. The analyticity of R(A.) means that the function (9' I R(A.) is analytic at any 9' E (/J • Let us examine an isolated singular point A.oof the function R(A.) . From the conventional Laurent expansion of the function (9' IR(A.) the expansion for the GEV arises:

(AIO) n= -00 whose coefficients are ~ C = f R(A.) dA. (All) 1 n 21ti (A. - A. t + c o where C is a simple closed and positively orientated curve inside the analyticity region Q of the function R(A.). The integral (All) defines the vector C; E (/J* • The properties of the Laurent series coefficients C; in the expansion of GEV R(A.) of the operator A are similar to the properties of the same Laurent series coefficients in the expansion of the resolvent near its singular points, thereby resulting in

(AI2)

This follows immediately from (All) and from the weak continuity ofA in (/J*. Thus, if Cn _ 1 = 0, then C; is GEV of the operator A corresponding to A.o. In particular, if A.o is a first-order pole, the first non-zero coefficient is C-1 and 348 APPENDIX A

C_2 = o. Therefore, C_1 is a GEV corresponding to the eigenvalue AO' Thus, the first-order-pole residue of the GEV R(A) belonging to continuous spectrum and continued to the complex plane Ais also the GEV of the operator A. This residue in the resonance pole is a Gamow state. Consider the simple example of the Schrodinger equation for spinless partic­ les with central local potential V(r) which behaves 'properly' at the origin and at infinity: V(r) '" (O(r- 3/2+8 ), e > 0; V(r) '" '" (O(r- 3-"), b > O. Any r~O r~oo 2 function 'II(r) E L (1Ii) can be expanded in spherical functions:

(A13) and the scalar product is defined as

00

('III 17J) = r fdr Vilm(r)l7Jlm(r) = r('IIlm I I7Jlm)' lm lm o

The functions 'IIlm(r) belong to the Hilbert space L2(R) which is denoted as .Tflm. A formal expansion of the Hamiltonians Hand Hocorresponds to the expansi­ on (A13), namely, the self-adjoint operators HI and HI act in each .Tflm

HI = __1 ~ + 1(1 + 1) . 2j.! d,z 2j.!,z'

HI = HI + V(r) . Following [20], we shall now construct an example of the nuclear subspace (/J convenient for 'rigging' the Hilbert space of the examined problem. First, we shall costruct the in each partial Hilbert space .Tflm. Examine the space ofthe complex-valued infinitely-differentiable finite functi­ ons 17J(x) on the straight line with the carrier [a, b], i.e. vanishing outside [a, b]. The convergence on this space is usually prescribed using a countable set of norms; for example,

1Il7Jllu = max sup 117J«(x)1 «SU x i.e. the convergence in this space (denoted as Cr([a, b])) means a uniform on [a, b] convergence of I7J together with a convergence of all its derivatives. Such a space is called countably normed and it is a metrizable, but not normalizable, complete nuclear space. Let us examine further an increasing sequence ofsuch spaces, for example !i)n = = Cr([I/n, n]) and treat their union. As a result, we obtain a set ofall infinitely APPENDIX A 349 differentiable functions along the semistraight line (0, 00) with a finite carrier. This set will be denoted Cg:'(O, 00) . Given such a union, we may introduce the topology of the inductive limit; as a result the linear manifold Cg:'(O, 00) turns into a locally-convex nonmetrizable topological space g)(O, 00) which is known in the theory of generalized functions to be the space of the trial Schwartz functions (along a ray) [27]. The convergence of the sequence {qJk} in the space g)(O, 00) means that there exists such a number n that all qJk vanish outside [lin, n] and the sequence {qJk} converges in the space Cg:'([l/n, n]), i.e. converges uniformely on [lin, n] together with all the derivatives. In its capacity of being the inductive limit of nuclear spaces, the space g)(O, 00) is nuclear (but is not countably-normalizedJ- At each value ofI, m such space g)(O, 00) will be denoted g)lm' The operators HI and H, are essentially self-adjoint in g)'m at I =F 0. At I = ° the self-adjointness requires the additional conditions at r > R.

Let us construct now the complete nuclear space f/J imbedded densely in .1f = L 2(f1i ). For this purpose we shall again use the same technique, i.e. we take the direct sum of spaces g)'m at I ~ L:

LI g)L = LL ffi g)'m I=Om=-1 and then construct the union of the increasing sequence of spaces g)L(g)L+ 1 ::l g)L) and introduce the inductive-limit topology in the union. The so c~n­ structed space f/J is a closed subspace ofthe space ofthree-dimensional Schwartz functions g)(1R 3)[27 ]. It comprises only the vectors with a finite number of nonvanishing components rp'm each of which is an infinitely differentiable fun­ ction with a finite carrier. Let us determine now the GEV of the operators H O and H. First, we shall examine H 0. The normalized Ricatti-Bessel functions

~( J x )= X 1/2J,+ 1/2 (X ) which are regular solutions for the free radial Schrodinger equation If? Ji(kr) =

= ~:Ji(kr) ,define the continuous functionals on g)'m' Therefore, the functionals

(klml

00 drJ~(kr)Xlm(r) , (kim IX> = f X E f/J o 350 APPENDIX A

2/2#. are continuous on tP and are the GEV ofthe operator H 0 with eigenvalue k Moreover, they are analytic throughout the k-plane. The completeness conditi­ on is

00 (X I qJ) = f dk L (X Ikim) (kim I qJ) . 1m o Consider now the total Hamiltonian. The radial Schrodinger equation is known [3] to have the regular solution qJI,k(r)

k2 HI qJI k(r) = - qJI k(r) , 2# ' which behaves as (kr)l+l/(21 + 1)!! at r -+ O. Under the adopted assumptions concerning V(r), this regular solution is an entire function throughout the k-plane at any value of r [3]. The regular solution qJI,k(r) is related to the normalized physical solution corresponding to an incoming wave as

"'( +)(r) = f!-.qJI,k(r) , (A 14) I, k ~ ft fi(k) where fi(k) = 1;( -k) is the Jost function which is analytic in the upper halfplane and continuous along the real axis (may be, except for the point k = 0). The functionals IR~:)(k) defined on tP by the relation

00 drilm(r)",f.~)(r) (qJ IRf:)(k) = f (AI5) o are the GEV of the operator Hand forin an analytic family in the upper k-halfplane. The functionals (L~: )(k)l:

00 drl,iif.~)(r)Xlm(r) (Lf:)(k) IX) = f (AI6) o are analytic in the lower k-halfplane. Thus, the formula for continuing the resolvent is

GII(z) = G(z) - 2ft iI IRf:)II(k) (Lf:)(k)1 , 1m

k=~ , Imk

ACCC 219 Baz states 19-20 Analytic continuation Born series 77, 138, 166 by means of Pade approximant 61,226, convergence of 78 228,230 orthogonalized 167 by means of power series 58 Breit-Wigner parametrization 13, 23-25, in complex coupling constant 223, 239, 28-29,94,96,98-99, 169,305 258,259 in coupling constant 15, 219, 225, 226, Charge exchange 305 235-237, 251-254, 262, 264, 266, 268 Collective resonances 22 in energy 154, 219, 231, 235, 266, 278 Complex coordinate rotation method 316 in partial coupling constant 223, 239, 258, Complex phase shift 294 259 Complex potential 237 of contour integrals 64 Complex scaling method 15,261,265,270, of Faddeev equations 179 316 of Lippmann-Schwinger equation 172 Compound states 17-19 of scattering equations 169 Continued Faddeev equation 179 stability of 62, 278 Continued fractions 47, 142-147 Analytic properties in coupling constant Continued Lippmann-Schwinger equation for deformed potential. 242-244 174 in one- or two-dimensional case 225 Contour deformation method 154, in partial coupling constants 223, 224, 239 265 of energy eigenvalue 220-225, 238, 256, Correlation 262 angular 330 of matrix elements with Gamow wave func­ radial 330 tion 235-237 Coupled channel resonance 19 of resonance width 224 of S-matrix 238 Analytic properties in energy Decay law of Gamow wave function 232 exponential 26,30 of Hilbert-Schmidt eigenvalue 78, 79, 85, non exponential 31 138, 154 Decay probability 24, 26 of Jost function 79, 245-248, 250 Deformed potential 241 of matrix elements with Gamow functions Degenerate kernel 57 235-236 Determination of S-matrix 220, 245, 279 of branch points of analytic functions Antibound state (see virtual state) 54-55 Antiresonance state 91, 114, 229, 240, 284 of zeros and poles of analytic functions 53 Argand diagram 94-98, 277 Dibaryon resonances 21, 148 Argand loop 96 Dilatation analytic potentials 317 Auger's transition 304-305 Dilatation method (see Complex scaling) Autoionization state 15-18, 261, 304, 305, Dipole interaction 291 330 DWBA 265 INDEX 353

Effective range expansion Hilbert-Schmidt for polarization potential 302 basis 113 for short range + Coulombic potential 285 eigenvalues determination of 138, 142 for short range potentials 280, 302 eigenvalues for complex potential 240 Efimov effect 202 eigenvalues in three-body case 85, 151-152, Efimov states 21, 342 155, 269 Eigenvalue trajectory 80 eigenvalue trajectory 14, 140, 142, 153, 156 (seealso Hilbert-Schmidt eigenvalue trajecto­ expansion for Lippmann-Schwinger equation ry and S-matrix pole trajectory) 77,134 Elastic scattering resonance 22 for resonances 134 Energy shift operator 162, 187,200,247 for three particles 148 ERE 279 method 14,73 Euler function 52 Inelasticity parameter 96-98, 294 summation method 43 transformation 43 lost function 79, 89, 108, 110, 245, 247-248, Extraction of resonance parameters from expe­ 250, 280 rimental data 15,96,98, 147 for short range + Coulombic interaction 251 Faddeev equations 14, 58, 66, 69, 72-73, 77, lost solution 108-109,294,311 84, 138, 149, 167, 180, 197,255,263,268 for heavy core and two light particles 71-73, Kapur-Peierls 263 basis 113 on nonphysical energy sheet 182 formalism 14,117,127,131,214 Faddeev reduction 69, 71-72 functions 118-119 Faddeev-Yakubovskyequation 56, 165, 167 K-matrix 143, 145-146 Feshbach Kohn variational principle 309 projection technique 14-15, 159,292,306, 334 Level shift 130 resonance 17-19, 22, 246, 264, 304 Levinson theorem for complex potentials 241 resonance amplitude 169 Life time of quasistationary state 16, 26, 90, First energy sheet (see also physical energy 102 sheet) 89,99, 150,219 Lippmann-Schwinger equation 56, 58, 66, 73, Fredholm equation 56 76-78,82, 165-167, 172, 198,229,263 Friedrich's model 99 on nonphysical energy sheet 176, 193 Long-lived states type of 16 Gamow basis 113 Lorentz law 31 completeness of 116 Gamow state (see also Siegert state) 30, Many channel resonances 19,292 102-103, 214, 308 Mercer's theorem 75 Gamow wave function 26, 27, 103-104, Meromorphic function 45, 57, 220, 238, 269 107-110, 112, 114, 134, 169, 186, 187, 193, Method of continued fractions 142 214, 232-233, 264-265, 272 Method of orthogonalized pseudopoten­ normalization of 104, 107, 110, 112, 234 tials 163 Generalized virial theorem 325 Mittag-Leffler theorem 113, 127 GEV 170 Golden rule 189-190 N/D technique 21, 270 Near threshold states 19-21, 126 Hadamard method 59, 61, 131, 230 Neumann series 56-57,74 Half-life period 26 Non-physical energy sheet 89, 99, 103, 152, Hellmann-Feynmann theorem 137, 222-223, 219, 224, 252, 255, 279, 296 231,237 Nuclear quasimolecular resonance 19-20 354 INDEX

Orthogonal projecting method 163, 246 190-191, 198, 200, 214, 224, 226, 247, 249, Orthogonality scattering 252-253, 256, 271, 312, 316, 326, 338 amplitude 169 partial 25, 188-189, 200, 215 wave function 167 total 25, 188-189,200,215 Orthogonalized without S-matrix poles 99 Lippmann-Schwinger equation 166, 168 RHS 169 resolvent 165-166 R-matrix 14, 127, 277 R-function 129 Pade approximants 14,40-58 Rigged Hilbert space 15, 169 applications of 50, 138-139, 155, 167, 195-196,204,226,230-232,234-239,256, Scattering function 279, 285 259, 262, 280-281, 286, 297 Schwarz reflection principle 79, 85, 91, 139 basic properties of 41-44 Second energy sheet 89, 159, 171,306 diagonal 40 Shape resonance 16-17,22,208 of type I 40, 53, 58, 61, 139, 230 Siegert state (see also Gamow state) 102-103, oftype II 40,47,52-54,58,61,138,226,231 307 of type III 40,49, 61, 226, 231, 138 Singlet deuteron 17,284 Pade conjecture 44 S-matrix 220, 245, 321 PCI 331 for complex potential 238 PEBS 240 for long range potential (Coulombic) Penalty function 163, 290 247-248 Penetration factor of potential barrier for short range potential 279 16 multichannel 295 Perturbation series for parametrization 89, 278-280, 286, 297 decay amplitude 191 poles 15,80,89-91,99, 102, 131, 134, 140, orthogonalized Green function 167 221, 226, 247, 278, 281, 286, 297, 316, 321 orthogonalized resolvent 197 pole trajectory 220-221, 225-227, 230 orthogonalized scattering wave function for complex potential 239-240 190 for deformed potential 242-244 Perturbation theory for resonance states 134 for short range + Coulombic interaction Physical energy sheet 89, 150, 152, 219, 279, 251 296 for short range + dipol interaction 254 Plane orthogonalized wave 169 for Woods-Saxon potential 273 Post-collision interaction model 329 Stabilization method 312 Predissociation 305 Stak effect 327 Projection methods 159-216 Stieltjes function 45 QBSEC 17 moment technique 314 Quasi bound states embedded in a continu­ Sturm eigenvalues 82 um 17-18,22,211,246 Sturm-Liouville problem 82 Quasiparticle excitations 22 Summation of poorly convergent or divergent Quasi-stationary states 16 series by means of Pade approximants decay law of 30-31 of type I 50-51 of type II 52 Reduced width 129 Surface quasi stat ionary state 20 Regularization of divergent integrals 35 Regularizing factor 36-39 Three particle Resonance Hilbert Schmidt expansion 148 dynamic manifestation of 27 near threshold long lived states 20-22 phase shift 92-93, 99, 130, 288-289 resonances 147, 255-256 width 23-24, 90, 92-93, 130, 153, 185, 187, Time delay 28-29 INDEX 355

Unitarity relation Virtual states (Antibound states) 16, 20, 38, for Green function 193 80,91 , 114, 159,221,225,229,240,252,254, for r-matrix 192 271-272, 284, 286 for three body transition operator 199 Weinstein-Aronszain determinant 165 Van der Waals complex 20 Wigner condition 27,29 Variational methods for resonance states 15, Wigner-Eisenbud basis 113, 127, 131 307 Virial theorem 325 Zel'dovich regularization 36,38-39, Ill, 113, 121, 237