The Cauchy Problem for a Forced Harmonic Oscillator

ENSENANZA˜ REVISTA MEXICANA DE FISICA´ 55 (2) 196–215 DICIEMBRE 2009

The Cauchy problem for a forced harmonic oscillator

R.M. Lopez and S.K. Suslov School of Mathematical and Statistical Sciences and Mathematical, Computational, and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287–1804, U.S.A. e-mail:[email protected], [email protected] Recibido el 14 de agosto de 2009; aceptado el 20 de agosto de 2009 We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schrodinger¨ equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg–Weyl group N (3) in a certain special case ﬁrst, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric ﬁelds is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.

Keywords: The Cauchy initial value problem; the Schrodinger¨ equation; forced harmonic oscillator; Landau levels; the hypergeometric functions; the Hermite polynomials; the Charlier polynomials; Green functions; Fourier transform and its generalizations; the Heisenberg– Weyl group N(3). En el presente trabajo construimos una solucion expl´ıcita unidimensional a la ecuacion´ de Schrodinger¨ con condiciones iniciales de Cauchy y con un operador Hamiltoniano dependiente del tiempo para el oscilador armonico´ forzado. La correspondiente funcion´ de Green (propa- gador) se deriva con aplicaciones de la transformada de Fourier generalizada y con una relacion´ a las representaciones del grupo N(3) de Heisenberg–Weyl, para un caso especial primero y despues´ se extiende al caso general. Estudiamos por medio de un producto una extencion´ de tres parametros´ a la integral clasica´ de Fourier. Consideramos, como una aplicacion, el movimiento de una part´ıcula giratoria en un campo electrico´ y en un campo magnetico´ perpendicularmente uniforme; evaluamos en terminos´ de polinomios de Charlier una transicion´ de amplitud entre los niveles de Landau. Ademas´ resolvemos una ecuacion´ similar a la de difusion´ con valores iniciales.

Descriptores: Problema de valor inicial de Cauchy; ecuacion´ de Schrodinger;¨ osilador armonico´ forzado; niveles de Landau; funciones hipergeometricas; polinomios de Hermite; polinomios de Charlier; funciones de Green; transformada de Fourier y sus generalizaciones; el grupo Heisenberg–Weyl. PACS: 45.20.-d; 02.30.-f; 02.30.Nw

1. Introduction where δ (t) is a complex valued function of time t and the symbol ∗ denotes complex conjugation. This operator is Her- The time-dependent Schrodinger¨ equation for the one- mitian, namely, H† (t) = H (t) . It corresponds to the case dimensional harmonic oscillator has the form of the forced harmonic oscillator which is of interest in many ∂ψ i} = Hψ, (1) advanced problems. Examples include polyatomic molecules ∂t in varying external fields, crystals through which an electron where the Hamiltonian is is passing and exciting the oscillator modes, and other in- µ ¶ }ω ∂2 }ω ¡ ¢ teractions of the modes with external fields. It has particular H = − + x2 = aa† + a†a . (2) 2 ∂x2 2 applications in quantum electrodynamics because the electro- magnetic field can be represented as a set of forced harmonic † Here a and a are the creation and annihilation operators, re- oscillators [9,20,23,34,35,45]. Extensively used propagator spectively, given by techniques were originally introduced by Richard Feynman µ ¶ µ ¶ 1 ∂ 1 ∂ in Refs. 16 to 19. a† = √ x − , a = √ x + ; (3) 2 ∂x 2 ∂x On this note we construct an exact solution of the time- see [21] for another definition. They satisfy the familiar com- dependent Schrodinger¨ equation mutation relation ∂ψ i} = H (t) ψ (6) £ ¤ ∂t a, a† = aa† − a†a = 1. (4) A natural modification of the Hamiltonian operator (2) is with the Hamiltonian of the form (5), subject to the initial as follows: condition ψ (x, t)| = ψ (x) , }ω ¡ ¢ t=0 0 (7) H → H (t) = aa† + a†a 2 ψ (x) ¡ ∗ †¢ where 0 is an arbitrary square integrable complex-valued + } δ (t) a + δ (t) a , (5) function from L2 (−∞, ∞) . We shall start with a particular THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 197 choice of the time-dependent function δ (t) given by (22) be- 2. The simple harmonic oscillator in one di- low, which is later extended to the general case. The explicit mension form of Eq. (6) is given by (25) and (77) below, and an extension to similar diffusion-type equations is also discussed. The time-dependent Hamiltonian operator (5) has the following structure: H (t) = H0 + H1 (t) , (8) This paper is organized as follows. In Sec. 2 we re- where }ω ¡ † † ¢ mind the reader about the textbook solution of the station- H0 = aa + a a (9) 2 ary Schrodinger¨ equation for the one-dimensional simple har- is the Hamiltonian of the harmonic oscillator and monic oscillator. In Sec. 3 we consider the eigenfunction ¡ ∗ †¢ expansion for the time-dependent Schrodinger¨ equation (6) H1 (t) = } δ (t) a + δ (t) a (10) and find its particular solutions in terms of the Charlier polynomials for a certain forced harmonic oscillator. The series is the time-dependent “perturbation”, which corresponds to solution to the corresponding initial value problem is ob- an external time-dependent force that does not depend on tained in Sec. 4. It is further transformed into an integral the coordinate x (dipole interaction) and a similar velocity- form in Sec. 7 after discussing two relevant technical tools, dependent term (see Refs. 20, 23, and 34 for more details). namely, the representations of the Heisenberg–Weyl group The solution to the stationary Schrodinger¨ equation for N(3) and the generalized Fourier transform in Sec. 5 and 6, the one-dimensional harmonic oscillator µ ¶ respectively. An important special case of the Cauchy ini- }ω ∂2 H Ψ = EΨ,H = − + x2 (11) tial value problem for the simple harmonic oscillator is out- 0 0 2 ∂x2 lined in Sec. 8 and a three-parameter generalization of the Fourier transform is introduced in Sec. 9 as a by-product. is a standard textbook problem in quantum mechanics (see In Secs. 10 and 11 we solve the initial value problem for Refs. 13, 21, 28, 34, 35, 39, 45, and 52 for example). The the general forced harmonic oscillator in terms of the corre- orthonormal wave functions are given by sponding Green function (or Feynman’s propagator) and the 1 −x2/2 eigenfunction expansion, respectively, by a different method Ψ = Ψn (x) = p √ e Hn (x) (12) 2nn! π that uses all technical tools developed before in the special case. An extension to the case of time-dependent frequency with is given in Sec. 12. Then in Sec. 13, we outline important ∞ ( Z 1, n = m, special and limiting cases of the Feynman propagators. Fi- ∗ Ψn (x)Ψm (x) dx = δnm = (13) nally in Sec. 14, the motion of a charged particle with a spin 0, n 6= m, in uniform magnetic and electric fields that are perpendicular −∞ to each other is considered as an application; we evaluate a where Hn (x) are the Hermite polynomials, a family of the transition amplitude between Landau levels under the influ- (very) classical orthogonal polynomials (see Refs. 1, 2, 4, ence of the perpendicular electric field in terms of Charlier 12, 14, 38, 39, 41, and 49). The corresponding oscillator dis- polynomials and find the corresponding propagator in three crete energy levels are dimensions. Solutions to similar diffution-type equations are µ ¶ 1 discussed in Sec. 15. E = E = }ω n + (n = 0, 1, 2, ... ) . (14) n 2 The actions of the creation and annihilation operators (3) on the oscillator wave functions (12) are given by The Cauchy initial value problem for a forced harmonic √ √ oscillator was originally considered by Feynman in his path † a Ψn = n Ψn−1, a Ψn = n + 1 Ψn+1. (15) integrals approach to the nonrelativistic quantum mechanics [16,17,20]. Since then this problem and its special and limit- These “ladder” equations follow from the differentiation for- ing cases were discussed by many authors [7,23,25,29,34,50] mulas the simple harmonic oscillator; [3, 10, 24, 37, 43] the particle d H (x) = 2nH (x) = 2xH (x) − H (x) , (16) in a constant external field; see also references therein. It is dx n n−1 n n+1 worth noting that an exact solution to the n-dimensional time- dependent Schrodinger¨ equation for a certain modified oscil- which are valid for the Hermite polynomials. lator is found in Ref. 30. These simple exactly solvable mod- els may be of interest in a general treatment of the non-linear 3. Eigenfunction expansion for the time- time-dependent Schrodinger¨ equation (see Refs. 26, 27, 36, dependent Schrodinger¨ equation 44, 46, 53, and references therein). They also provide explicit solutions which can be useful for testing numerical methods In the spirit of Dirac’s time-dependent perturbation theory in of solving the time-dependent Schrodinger¨ equation. quantum mechanics (see Refs. 13, 21, 28, 35, and 45), we are

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 198 R.M. LOPEZ AND S.K. SUSLOV looking for a solution to the initial value problem in (6)–(7) In the next section we will obtain the function cnm (t) in as an infinite series terms of the Charlier polynomials; see Eq. (33) below. In X∞ Sec. 5 we establish a relation with the representations of the ψ = ψ (x, t) = cn (t)Ψn (x) , (17) Heisenberg–Weyl group N(3); see Eq. (46). A generaliza- n=0 tion to an arbitrary function δ (t) will be given later. where Ψn (x) are the oscillator wave functions (12) which 4. Solution of the Cauchy problem depend only on the space coordinate x and cn (t) are the yet unknown time-dependent coefficients. Substituting this We can now construct the exact solution to the origi- form of solution into the Schrodinger¨ equation (6) with the nal Cauchy problem in (6)–(7) for the time-dependent help of the orthogonality property (13) and the “ladder” rela- Schrodinger¨ equation with the Hamiltonian of the tions (15), we obtain the following linear infinite system: form (8)–(10) and (22). More explicitly, we will solve the µ ¶ √ following partial differential equation: dcn(t) 1 µ ¶ i = ω n + cn (t) + δ (t) n + 1 cn+1 (t) 2 dt 2 ∂ψ ω ∂ ψ 2 i = − 2 + x ψ ∗ √ ∂t 2 ∂x + δ (t) n cn−1 (t)(n = 0, 1, 2, ... ) (18) µ ¶ p ∂ψ + 2µ (cos (ω−1) t) xψ+i (sin (ω−1) t) (25) of the first-order ordinary differential equations with ∂x c (t) ≡ 0. The initial conditions are −1 subject to the initial condition Z∞ ∗ ψ (x, t)|t=0 = ψ0 (x)(−∞ < x < ∞) . (26) cn(0) = Ψn(x)ψ0(x) dx (19) −∞ By (17), (19), and (23) our solution has the form X∞ X∞ due to the initial data (7) and the orthogonality property (13). ψ (x, t) = Ψn (x) cnm (t) Now we specify the exact form of the function δ (t) in or- n=0 m=0 der to find a particular solution to the system (18) in terms of Z∞ the so-called Charlier polynomials that belong to the classical × Ψ (y) ψ (y) dy, (27) orthogonal polynomials of a discrete variable (see Refs. 11, m 0 12, 14, 38, and 39). One can easily verify that the following −∞ Ansatz where ³ ´ r n/2 n−m m! n µ c (t) = −µ1/2 e−i(ω(n+1/2)−(n+µ))t cn (t) = (−1) √ nm n! n! m ∞ k −i(ω(n+1/2)−(n+µ))t ¡ −iλt ¢ µ X µ × e e pn (λ) (20) × e−iktcµ (k) cµ (k) e−µ m! n m k! k=0 gives the three-term recurrence relation µ(n+m)/2 = (−1)n−m e−µ √ e−i(ω(n+1/2)−(n+µ))t λpn(λ)= − µpn+1(λ)+(n + µ)pn(λ)−npn−1(λ) (21) n!m! ¡ ¢ µ ∞ −it k for the Charlier polynomials pn (λ) = cn (λ) (see Refs. 38 X µe × cµ (k) cµ (k) , (28) and 39 for example), when we choose n m k! k=0 √ i(ω−1)t ∗ √ −i(ω−1)t δ (t) = µ e , δ (t) = µ e (22) in view of the superposition principle and the orthogonality property with the real parameter µ such that 0 < µ < 1. Thus with ∞ p (λ) = cµ (λ) , Eq. (20) yields a particular solution of the X µk m! n n cµ (k) cµ (k) e−µ = δ (0 < µ < 1) (29) system (18) for any value of the spectral parameter λ. n m k! µm nm k=0 By the superposition principle the solution to this linear of the Charlier polynomials (see Ref. 38 for example). system of ordinary differential equations, which satisfies the The right-hand side of (28) can be transformed into a sin- initial condition (19), can be constructed as a linear combina- gle sum with the help of the following generating relation for tion X∞ the Charlier polynomials: c (t) = c (t) c (0) , (23) n nm m X∞ k (µ1µ2s) m=0 cµ1 (k) cµ2 (k) = eµ1µ2s(1 − µ s)m k! n m 1 where cnm (t) is a “Green” function, or a particular solution k=0 µ ¶ that satisfies the simplest initial conditions s × (1−µ s)n F −n, −m; . (30) 2 2 0 (1−µ s) (1−µ s) cnm (0) = δnm. (24) 1 2

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 199

(See Refs. 31 to 33, and 22 for more information and Ref. 6 where for the definition of the generalized hypergeometric series.) Z∞ −it Choosing µ1 = µ2 = µ and s = e /µ we obtain ∗ i(γ+βx) Tmn(α, β, γ) = Ψm(x)e Ψn(x + α)dx n+m (−i) −i((ω−1)n+(n+m)/2)t −2µ sin2(t/2) −∞ cnm(t) = √ e e 2n+mn!m! m−n i i(γ−αβ/2) −ν/2 ³ p ń+m = √ e e × e−i(µ sin t+(ω/2−µ)t) 2 2µ sin (t/2) m!n! ³ ´m³ ń µ ¶ iα + β iα − β ν 1 × √ √ cm(n) (37) 2 2 ×2 F0 −n, −m; − 2 . (31) 4µ sin (t/2) ¡ ¢ with ν = α2 + β2 /2 [38]. A similar integral The hypergeometric series representation for the Charlier polynomials is Z∞ µ ¶ −x2 1 Hm(x + y)Hn(x + z)e dx cµ (x) = F −n, −x; − (32) n 2 0 µ −∞ √ n n−m n−m (see [38] for example). Thus = π 2 m! z Lm (−2yz) (38)

−i(µ sin t+(ω/2−µ)t) −i((ω−1)n+(n+m)/2)t cnm (t) = e e is evaluated in Ref. 15 in terms of the Laguerre polynomials α n+m Lm (ξ) , whose relation with the Charlier polynomials is (−i) −β2/4 n+m β2/2 × √ e β c (m) (33) −n n+m n µ x−n 2 n!m! cn (x) = (−µ) n!Ln (µ) , (39) √ with β = β (t) = 2 2µ sin (t/2) and, as a result, by sub- see Ref. 38. Its special case y = z in the form of stitution of this expression into the series (27), we obtain the eigenfunction expansion solution to the original Cauchy Z∞ −(x−y)2 problem (25)–(26). We shall be able to ﬁnd an integral form Hm (x) Hn (x) e dx of this solution in Sec. 7 after discussing representations of −∞ the Heisenberg–Weyl group N(3) and a generalization of the √ ¡ ¢ = π 2nm! yn−m Ln−m −2y2 (40) Fourier transform in the next two sections. This complete m solution to the particular initial value problem (25)–(26) will is of particular interest in this paper. suggest a correct form of the Green function (propagator) for The unitary relation the general forced harmonic oscillator in Secs. 10 and 11. X∞ ∗ Tmn (α, β, γ) Tm0n (α, β, γ) = δmm0 (41) 5. Relation with the Heisenberg–Weyl group n=0 N(3) holds due to the orthogonality property of the Charlier poly- Let N(3) be the three-dimensional group of the upper trian- nomials (29). gular real matrices of the form The relevant special case of these matrix elements is   1 α γ im+n Tmn (0, β, 0) = tmn (β) = √ 0 1 β = (α, β, γ) . (34) 2m+nm!n! 0 0 1 −β2/4 m+n β2/2 × e β cm (n) , (42) The map

i(γ+βx) which explicitly acts on the oscillator wave functions as fol- T (α, β, γ)Ψ(x) = e Ψ(x + α) (35) lows: X∞ iβx deﬁnes a unitary representation of the Heisenberg–Weyl e Ψn (x) = tmn (β)Ψm (x) . (43) group N (3) in the space of square integrable functions m=0 Ψ ∈ L2 (−∞, ∞) (see Ref. 51, 38, and 47 for more details). Relations (30), (32) and (42) imply ∞ The set {Ψn (x)} of the wave functions of the har- n=0 ∞ monic oscillator (12) forms a complete orthonormal sys- X t (β ) t (β ) sk tem in L2 (−∞, ∞) . The matrix elements of the representa- mk 1 nk 2 k=0 tion (35) with respect to this basis are related to the Charlier m+n polynomials as follows: i − β2+β2+2β β s /4 = √ e ( 1 2 1 2 ) m+n X∞ 2 m!n! T (α, β, γ)Ψn (x) = Tmn (α, β, γ)Ψm (x) , (36) m n λ × (β1 + β2s) (β2 + β1s) cm (n) (44) m=0

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 200 R.M. LOPEZ AND S.K. SUSLOV with where Ψn(z) are the oscillator wave functions deﬁned by (12) and |r| ≤ 1, r 6= ±1 (see Refs. 14, 41, 49, and 52 for ¡ 2 2 ¡ −1¢¢ λ = β1 + β2 + β1β2 s + s /2, example). Using the orthogonality property (13) one gets which is an extension of the addition formula Z∞ n X∞ r Ψn (x) = Kr (x, y)Ψn (y) dy, |r| < 1. (49) tmk (β1) tnk (β2) = tmn (β1 + β2) (45) −∞ k=0 Thus the wave functions Ψn are also eigenfunctions of an for the matrix elements. integral operator corresponding to the eigenvalues rn. In order to obtain functions cnm (t) in terms of the matrix We denote elements tmn (β) of the representations of the Heisenberg– Weyl group, we compare (33) and (42). The result is ei(π/2−τ)/2 Kτ (x, y) = Keiτ (x, y) = √ 2π sin τ n+m −i(µ sin t+(ω/2−µ)t) cnm (t) = (−1) e Ã ¡ ¢ ! 2xy − x2 + y2 cos τ −i((ω−1)n+(n+m)/2)t × exp i (50) × e tmn (β) , (46) 2 sin τ √ where β = 2 2µ sin (t/2) . Our solution (27) takes the form with 0 < τ < π and use the fact that the oscillator wave ∞ X∞ X∞ Z functions are the eigenfunctions of the generalized Fourier ψ (x, t) = Ψn (x) cnm (t) Ψm (y) ψ0 (y) dy transform n=0 m=0 −∞ Z∞ inτ X∞ e Ψn (x) = Kτ (x, y)Ψn (y) dy (51) −i(µ sin t+(ω/2−µ)t) n −it(ω−1/2)n = e (−1) e −∞ n=0 inτ ∞ corresponding to the eigenvalues e . (See Refs. 5, 42, and X∞ Z m −imt/2 48 for more details on the generalized Fourier transform, its × Ψn (x) tmn (β) (−1) e m=0 inversion formula and their extensions. It is worth noting that −∞ the classical Fourier transform corresponds to the particular × Ψm (y) ψ0 (y) dy. (47) value τ = π/2 [52]. Its three-parameter extension will be discussed in Sec. 9.) In the next section, we will discuss a generalization of the Fourier transformation, which will allow us to transform this multiple series into a single integral form in Sec. 7; see 7. An integral form of solution Eqs. (55) and (60)–(62) below. Now let us transform the series (47) into a single integral form. With the help of the inversion formula for the gen- 6. The generalized Fourier transform eralized Fourier transform (see (51) with τ → −τ) and the symmetry property The Mehler generating function, or the Poisson kernel for n Hermite polynomials, is given by Hn (−x) = (−1) Hn (x) X∞ n 1 we get Kr (x, y) = r Ψn (x)Ψn (y) = p π (1 − r2) n=0 Z∞ (−1)me−imt/2Ψ (y) = K (−y, z)Ψ (z)dz. (52) Ã ¡ ¢ ¡ ¢! m −t/2 m 2 2 2 4xyr − x + y 1 + r −∞ × exp , (48) 2 (1 − r2) Then by (43) and Fubuni’s theorem,

  ∞ ∞ ∞ X∞ Z X∞ Z Z m −imt/2 tmn(β) (−1) e Ψm (y) ψ0 (y) dy= tmn(β)  K−t/2(−y, z)Ψm(z)dz ψ0(y)dy m=0 −∞ m=0 −∞ −∞ ∞ Ã ! ∞ Z Z X∞ Z Z ¡ iβz ¢ = K−t/2 (−y, z) tmn (β)Ψm(z) ψ0(y)dydz = K−t/2 (−y, z) e Ψn(z) ψ0 (y) dydz. (53) −∞ m=0 −∞

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 201

Now the series (47) takes the form ∞ X∞ Z Z −i(µ sin t+(ω/2−µ)t) n −it(ω−1/2)n ¡ iβz ¢ ψ (x, t) = e (−1) e Ψn (x) K−t/2 (−y, z) e Ψn (z) ψ0 (y) dydz n=0 −∞ ∞ Ã ! Z Z X∞ −i(µ sin t+(ω/2−µ)t) iβz −it(ω−1/2)n = e K−t/2(−y, z)e e Ψn (−x)Ψn (z) ψ0 (y) dydz −∞ n=0   Z∞ Z∞ −i(µ sin t+(ω/2−µ)t) iβz = e  K−t/2 (−y, z) e Kt(1/2−ω) (−x, z) dz ψ0 (y) dy (54) −∞ −∞ in view of the generating relations (48) and (50). Thus Z∞ −i(µ sin t+(ω/2−µ)t) ψ (x, t) = e Gt (x, y) ψ0 (y) dy, (55) −∞ where we deﬁne the kernel as Z∞ iβz Gt (x, y) := Kt(1/2−ω) (−x, z) e K−t/2 (−y, z) dz. (56) −∞ This can be evaluated with the help of the familiar elementary integrals Z∞ Z∞ r 2 √ 2 πi 2 e−x dx = π, ei(az +2bz), dz = e−ib /a (57) a −∞ −∞

(see Refs. 9, 15, and 40 also). Denoting τ1 = t (ω − 1/2) and τ2 = t/2, from (50) we get i(ωt−π)/2 e 2 2 2 iβz i(x cot τ1+y cot τ2)/2 i(β+x/ sin τ1+y/ sin τ2)z i(cot τ1+cot τ2)z /2 K−τ1 (−x, z) e K−τ2 (−y, z) = √ e e e (58) 2π sin τ1 sin τ2 and ∞ Z i(ωt−π)/2 e 2 2 iβz i(x cot τ1+y cot τ2)/2 K−τ1 (−x, z) e K−τ2 (−y, z) dz = √ e 2π sin τ1 sin τ2 −∞ Z∞ 2 × ei((β+x/ sin τ1+y/ sin τ2)z+(cot τ1+cot τ2)z /2)dz. (59) −∞ As a result Ã ! i(ωt−π/2)/2 2 e 2 2 sin τ sin τ (β + x/ sin τ + y/ sin τ ) i(x cot τ1+y cot τ2)/2 1 2 1 2 Gt (x, y) = √ e exp (60) 2π sin ωt 2i sin ωt √ with τ1 = t (ω − 1/2) , τ2 = t/2 and β = β(t) = 2 2µ sin(t/2). Thus the explicit form of this kernel is given by Ã¡ ¢ ¡ ¢ ! i(ωt−π/2)/2 2 2 2 2 µ 2 ¶ e x + y sin ωt − x − y sin (ω − 1) t ikt (x, y) Gt (x, y) = √ exp exp , (61) 2π sin ωt 2i (cos ωt − cos (ω − 1) t) sin ωt (cos ωt − cos (ω − 1) t) where

kt (x, y) = (x + y) sin (ωt/2) cos ((ω − 1) t/2) p − (x − y) cos (ωt/2) sin ((ω − 1) t/2) − 2µ sin (t/2) (cos ωt − cos (ω − 1) t) . (62) The last expression can be transformed into a somewhat more convenient form µ ¶ µ ¶ sin ((ω − 1/2) t) sin (t/2) β2 (x sin (t/2) + y sin ((ω − 1/2) t)) β G (x, y) = K∗ (x, y) exp exp (63) t ωt 2i sin ωt i sin ωt

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 202 R.M. LOPEZ AND S.K. SUSLOV √ with β = 2 2µ sin (t/2) in terms of the kernel of the has the following explicit solution: generalized Fourier transform (50). Our formulas (55) and (60)–(63) provide an integral form to the solution of the X∞ −iω(n+1/2)t Cauchy initial value problem (25)–(26) in terms of a Green ψ (x, t) = e Ψn (x) function. n=0 By choosing ψ (x) = δ (x − x ) , where δ(x) is the Z∞ 0 0 1 Dirac delta function, we formally obtain × Ψn (y) ψ0 (y) dy = p 2πi sin (ωt) −i(µ sin t+(ω/2−µ)t) −∞ ψ (x, t) =G(x, x0, t)=e Gt(x, x0), (64) Z∞ Ã ¡ ¢ ! x2 + y2 cos (ωt) − 2xy which is the fundamental solution to the time-dependent × exp i ψ0(y)dy. (72) 2 sin (ωt) Schrodinger¨ equation (25). One can show that −∞

lim ψ (x, t) = ψ0 (x) (65) t→0+ The last relation is valid when 0 < t < π/ω. Analytic continuation in a larger domain is discussed in Ref. 29 and 50. by methods of Refs. 5, 42, and 52. The details are left to the Equation (72) gives the time evolution operator (66) for reader. the simple harmonic oscillator in terms of the generalized The time evolution operator for the time-dependent Fourier transform. This result and its extension to a general Schrodinger¨ equation (6) can formally be written as forced harmonic oscillator without the velocity-dependent    Zt term in the Hamiltonian are well-known (see Refs. 7, 17, i 0 0 20, 23, 25, 29, 34, 50, and references therein; further gener- U (t, t0) = T exp − H (t ) dt  , (66) } alizations are given in Secs. 10–12; more special cases will t0 be discussed in Sec. 13). where T is the time ordering operator which orders operators with larger times to the left [9], [21]. Namely, this unitary operator takes a state at time t to a state at time t, so that 0 9. Three parameter generalization of the

ψ (x, t) = U (t, t0) ψ (x, t0) (67) Fourier transform and The properties of the time evolution operator in (67)–(70) 0 0 U (t, t0) = U (t, t ) U (t , t0) , (68) suggest the following extension of the classical Fourier in-

−1 † tegral: U (t, t0) = U (t, t0) = U (t0, t) . (69) Z∞ We have constructed this time evolution operator explic- f (x) = Lt (x, y) g (y) dy, (73) itly as the following integral operator −∞ U (t, t ) ψ (x, t ) = e−i(µ sin(t−t0)+(ω/2−µ)(t−t0)) 0 0 where the kernel given by Z∞

× Gt−t (x, y) ψ (y, t0) dy (70) 0 Lt (x, y) = Kωt (x, y) −∞ µ ¶ sin ((ω − 1/2) t) sin3 (t/2) × exp i ε2 with the kernel given by (60)–(63), for the particular form of 2 sin ωt the time-dependent Hamiltonian in (8)–(10) and (22). The ³ (x sin(t/2)+y sin((ω − 1/2)t)) sin(t/2) ´ Green function (propagator) for the general forced harmonic × exp i ε (74) oscillator is constructed in Sec. 10; see Eqs. (79)–(84). sin ωt

depends on the three free parameters t, ω and ε. If ε = 0 8. The Cauchy problem for the simple har- and ωt = τ we arrive at the kernel of the generalized Fourier monic oscillator transform (50). The formal inversion formula is given by

In an important special case µ = 0, the initial value problem Z∞ µ ¶ g (y) = L∗ (x, y) f (x) dx. (75) ∂ψ ω ∂2ψ t i = − + x2ψ , ∂t 2 ∂x2 −∞

ψ (x, t)|t=0 = ψ0 (x)(−∞ < x < ∞) (71) The details are left to the reader. Note that, in terms of a

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 203 distribution, provided a (0) = b (0) = c (0) = 0. The case g (t) ≡ 0 is Z∞ discussed in Refs. 17, 20, and 34, but the answers for b (t) L∗ (x, y) L (x, z) dx and c (t) are given in different forms; we shall elaborate on t t this later. −∞ Indeed, the previously found solution (63)–(64) in the 2 2 = ei cot(ωt)(y −z )/2 eiε(z−y) sin((ω−1/2)t) sin(t/2)/ sin ωt special case of the forced oscillator (25) suggests to look for a general Green function in the form (80), namely, Z∞ 1 × eix(z−y)/ sin ωt dx = δ(y − z), (76) ψ = u eiS, (85) 2π sin ωt −∞ where u = G0 (x, y, t) is the fundamental solution of the which gives the corresponding orthogonality property of the Schodinger¨ equation for the simple harmonic oscillator (71) L-kernel. These results admit further generalizations with the and S = a (t) x + b (t) y + c (t) . Its substitution into (77) help of the time evolution operators found in Secs. 10 and 12. gives µ ¶ da db dc ³ ω ´ 10. The general forced harmonic oscillator x + y + u = ag + xf − a2 u dt dt dt 2 Our solution to the initial value problem (25)–(26) obtained ∂u in the previous sections admits a generalization. The Cauchy + i (aω − g) , (86) ∂x problem for the general forced harmonic oscillator µ ¶ where by (81) ∂ψ ω ∂2ψ ∂ψ i = − +x2ψ −f (t) xψ+ig (t) , (77) ∂t 2 ∂x2 ∂x ∂u x cos ωt − y = i u. (87) ∂x sin ωt where f (t) and g (t) are two arbitrary real valued functions of time only (such that the integrals in (82)–(84) below con- As a result verge and a (0) = 0), with the initial data da db dc ω x + y + = ag + xf − a2 ψ (x, t)|t=0 = ψ0 (x)(−∞ < x < ∞) , (78) dt dt dt 2 x cos ωt − y has the following explicit solution: − (aω − g) , (88) sin ωt Z∞ and equating the coefﬁcients of x, y and 1, we obtain the fol- ψ (x, t) = G (x, y, t) ψ (y) dy. (79) 0 lowing system of ordinary differential equations −∞ d Here the Green function (or Feynman’s propagator [17], [20], (sin ωt a (t)) = f (t) sin ωt + g (t) cos ωt, (89) [34]) is given by dt d ωa (t) − g (t) i(a(t)x+b(t)y+c(t)) b (t) = , (90) G (x, y, t) = G0 (x, y, t) e (80) dt sin ωt with d ω c (t) = g (t) a (t) − a2 (t) , (91) 1 dt 2 G0 (x, y, t)=√ 2πi sin ωt whose solutions are (82)–(84), respectively, if the integrals Ã ¡ ¢ ! converge. This method is equivalent to solving the quantum x2+y2 cos ωt−2xy × exp i (81) mechanical Hamilton–Jacobi equation for the general forced 2 sin ωt harmonic oscillator [34]. and Equation (83) can be rewritten as Zt Zt Zt 1 g (s) a (t) = (f (s) sin ωs + g (s) cos ωs) ds, (82) b (t) = − (sin ωs a (s)) d cot ωs − ds sin ωt sin ωs 0 0 0 Zt ωa (s) − g (s) and, integrating by parts, b (t) = ds, (83) sin ωs 0 b (t)=− cos ωt a (t) Zt Zt ³ ω ´ c (t) = g (s) a (s) − a2 (s) ds (84) + (f (s) cos ωs−g (s) sin ωs) ds (92) 2 0 0

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 204 R.M. LOPEZ AND S.K. SUSLOV by (89). With the help of (82) and the addition formulas for trigonometric functions we finally arrive at Zt ×(−f(s) cos ωs + g(s) sin ωs)ds. (94) 1 b (t) = − (f (s) sin ω (s − t) sin ωt This can be transformed into the form given in Refs. 17 0 and 20 when g (t) ≡ 0. The details are left to the reader. +g (s) cos ω (s − t)) ds, (93) Evaluation of elementary integrals results in (63) again in which is equivalent to the form obtain in Refs. 17, 20, and 34, the special case (25). The simple case f (t) = 2 cos ωt and when g (t) ≡ 0. g (t) ≡ 0 gives In a similar fashion, sin ωt Zt Zt a (t) = , b (t) = t, 1 2 ω c (t) = g (s) a (s) ds + (sin ωs a (s)) d cot ωs 1 1 2 c (t) = sin 2ωt − t. (95) 0 0 8ω2 4ω and as a result Zt The corresponding propagator in (80) does satisfy the 1 Schrodinger¨ equation (77), which can be verified by a direct c (t) = sin ωt cos ωt a2(t) + sin ωs a (s) 2 differentiation with the help of a computer algebra system. 0 The details are left to the reader. A case of the forced modified oscillator is discussed in Ref. 30.

11. Eigenfunction expansion for the general forced harmonic oscillator

Separation of the x and y variables in Feynman’s propagator (80) –(84) with the help of the Mehler generating function (48) written as X∞ −iω(k+1/2)t G0 (x, y, t) = e Ψk (x)Ψk (y) (96) k=0 gives X∞ i(ax+by+c) i(c−ωt/2) −iωkt ¡ iax ¢ ¡ iby ¢ G (x, y, t) = G0 (x, y, t) e = e e e Ψk (x) e Ψk (y) k=0 Ã !Ã ! X∞ X∞ X∞ i(c−ωt/2) −iωkt = e e tnk (a)Ψn (x) tmk (b)Ψm (y) k=0 n=0 m=0 Ã ! X∞ X∞ X∞ i(c−ωt/2) −iωkt = e Ψn (x)Ψm (y) e tnk (a) tmk (b) (97) n=0 m=0 k=0 by (43). The last series can be summed by using the addition formula (44) in the form

X∞ m+n −iωkt i i(ab sin ωt)/2 −χ2/4 n m χ2/2 e tnk (a) tmk (b) = √ e e (a + bz) (b + az) c (n) , (98) m+n m k=0 2 m!n! with z = e−iωt and χ2 = a2 + b2 + 2ab cos ωt. As a result we arrive at the following eigenfunction expansion of the forced harmonic oscillator propagator:

X∞ X∞ n+m i(c−(ωt−ab sin ωt)/2) −χ2/4 i n m χ2/2 G (x, y, t) = e e Ψn (x)Ψm (y) √ (a + bz) (b + az) c (n) (99) n+m m n=0 m=0 2 n!m! in terms of the Charlier polynomials. The special case g (t) ≡ 0 is discussed in Ref. 20 but the connection with the Charlier polynomials is not emphasized. The solution (79) takes the form

∞ X∞ X∞ Z ψ (x, t) = Ψn (x) cnm (t) Ψm (y) ψ0 (y) dy, (100) n=0 m=0 −∞

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 205 where n+m i(c−(ωt−ab sin ωt)/2) −χ2/4 i n m χ2/2 cnm (t) = e e √ (a + bz) (b + az) c (n) (101) 2n+mn!m! m with z = e−iωt and χ2 = a2 + b2 + 2ab cos ωt. Functions a = a (t) , b = b (t) and c = c (t) here are given by the integrals (82)–(84), respectively, and limt→0+ cnm (t) = δnm. If ψ0 (x) = ψ (x, t)|t=0 = Ψm (x) , Eq. (100) becomes X∞ ψ (x, t) = cnm (t)Ψn (x) . (102) n=0

Thus function cnm (t) gives explicitly the quantum mechanical amplitude that the oscillator initially in state m is found at time t in state n [20]. An application to the motion of a charged particle with a spin in uniform perpendicular magnetic and electric ﬁelds is considered in Sec. 14. As a by-product we found the fundamental solution cnm (t) √of the system (18) in terms of the Charlier polynomials for an arbitrary complex valued function δ (t) = (−f (t) + ig (t)) / 2. The explicit solution of the corresponding initial value problem in given by (23).

12. Time-dependent frequency and the system (89)–(91) becomes An extension of the Schodinger¨ equation to the case of the forced harmonic oscillator with the time-dependent fre- d (sin τ a (τ)) = f (τ) sin τ + g (τ) cos τ, (109) quency is as follows: dτ 1 1 µ ¶ d a (τ) − g (τ) ∂ψ ω(t) ∂2ψ ∂ψ b (τ) = 1 , (110) i = − +x2ψ −f(t)xψ+ig(t) , (103) dτ sin τ ∂t 2 ∂x2 ∂x d 1 2 c (τ) = g1 (τ) a (τ) − a (τ) . (111) where ω (t) > 0, f (t) and g (t) are arbitrary real valued dτ 2 functions of time only. It can be easily solved by the sub- Thus stitution Zt Zt 1 dτ a (τ)= (f (s) sin τ (s) +g (s) cos τ (s)) ds, (112) τ = τ (t) = ω (s) ds, = ω (t) , (104) sin τ dt 0 0 Zt ω (s) a (τ (s)) −g (s) which transforms this equation into a familiar form b (τ)= ds, (113) µ ¶ sin τ (s) ∂ψ 1 ∂2ψ ∂ψ 0 i = − +x2ψ −f (τ) xψ+ig (τ) , (105) 2 1 1 Zt µ ¶ ∂τ 2 ∂x ∂x 1 c (τ)= g(s)a (τ (s)) − ω (s) a2 (τ (s)) ds, (114) see the original Eq. (77) with respect to the new time variable 2 0 τ, where ω = 1 and which is an extension of equations (82)–(83) to the case of f (t) f (t) the forced harmonic oscillator with the time-dependent fre- f1 (τ) = , g1 (τ) = . (106) ω (t) ω (t) quency. The solution to the Cauchy initial-value problem is given by Therefore by (80)–(81) the propagator has the form Z∞ G (x, y, τ) = G (x, y, τ) ei(a(τ)x+b(τ)y+c(τ)) (107) 0 ψ (x, t) = G (x, y, τ) ψ0 (y) dy (115) −∞ with 1 with G0 (x, y, τ) = √ 2πi sin τ Zt Ã ¡ ¢ ! x2 + y2 cos τ − 2xy τ = ω (s) ds. × exp i (108) 2 sin τ 0 The details are left to the reader.

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 206 R.M. LOPEZ AND S.K. SUSLOV

13. Some special cases Here r ¨ mω The time-dependent Schrodinger equation for the forced har- G0 (x, y, t, t0) = monic oscillator is usually written in the form 2πi} sin ω (t − t0) Ã £ ∂Ψ imω 2 2 i} = HΨ (116) × exp (x + y ) ∂t 2} sin ω(t − t0) ! with the following Hamiltonian ¤ × cos ω(t − t0) − 2xy , (124) p2 mω2 } ∂ H= + x2−F (t) x−G (t) p, p= , (117) Zt Ã 2m 2 i ∂x mω sin ω(s − t0) a (t, t0) = F (s) } sin ω (t − t0) mω where } is the Planck constant, m is the mass of the particle, t0 ω is the classical oscillation frequency, F (t) is a uniform in ! space external force depending on time, function G(t) rep- + G(s) cos ω(s − t0) ds, (125) resents a similar velocity-dependent term, and p is the linear momentum operator. The initial value problem is b (t, t0) = −a (t0, t) (126) ∂Ψ }2 ∂2Ψ mω2 i} = − + x2Ψ and ∂t 2m ∂x2 2 ∂Ψ Zt µ ¶ −F (t) xΨ+i}G (t) (118) } ∂x c (t, t ) = G (s) a (s, t ) − a2 (s, t ) ds. (127) 0 0 2m 0 with t0 The simple harmonic oscillator propagator, when Ψ(x, t)| = Ψ (x, t0) . (119) t=t0 F = G = 0, is given by Eq. (124); see Refs. 7, 17, 20, 23, 25, 29, 34, 50, and references therein for more details. In the Among important special cases are: the free particle, when limit ω → 0 we obtain ω = F = G = 0; a particle in a constant external ﬁeld, r Ã ! where ω = G = 0 and F = constant; the simple harmonic m im(x − y)2 oscillator with F = G = 0. In this section for the beneﬁts G0(x, y, t, t0)= exp (128) 2πi}(t−t0) 2}(t − t0) of the reader we provide explicit forms for the corresponding propagators by taking certain limits in the general solution. as the free particle propagator [20]. The usual change of the space variable r mω Ψ(x, t) = ψ (ξ, t) , ξ = x (120) } reduces Eq. (118) to the form (77) with respect to ξ with r F (t) mω f (t) = √ , g (t) = G (t) . (121) }ωm }

The time evolution operator is

Z∞

Ψ(x, t) = G (x, y, t, t0)Ψ(y, t0) dy (122) −∞ with the propagator of the form

G (x, y, t, t0) = G0 (x, y, t, t0)

i(a(t,t0)x+b(t,t0)y+c(t,t0)) × e . (123) FIGURE 1. Magnetic and electric ﬁelds in R3.

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 207

For a particle in a constant external ﬁeld ω = G = 0 and wave function in the Schrodinger¨ equation (131) can be taken F = constant. The corresponding propagator is given by as an ordinary coordinate function Ψ = Ψ (r, t, σ) . Ã ! r 2 The Hamiltonian (133) does not contain the coordinates m im (x − y) x and z explicitly. Therefore the operators pb and pb also G (x, y, t, t0) = exp x z 2πi} (t − t0) 2} (t − t0) commute with the Hamiltonian and the x and z components

µ 2 ¶ of the linear momentum are conserved. The corresponding iF (x + y) iF 3 × exp (t − t0) − (t − t0) . (129) eigenvalues px and pz take all values from −∞ to ∞; see [28] 2} 24}m for more details. In this paper we consider the simplest case This case was studied in detail in Refs. 3, 10, 20, 24, 37, when the magnetic ﬁeld H is a constant and the electric force and 43. We have corrected a typo in Ref. 20. F is a function of time t (see Fig. 1); a more general case will be discussed elsewhere. Then the substitution 14. Motion in uniform perpendicular mag- Ψ(r, t) = ei(xpx+zpz −S(t,t0))/} ψ (y, t) , (134) netic and electric ﬁelds

14.1. Solution of a particular initial value problem where

A particle with a spin s has also an intrinsic magnetic mo- µ ¶ Zt p2 µσ cp mentum µ with the operator S(t, t )= z − H (t − t ) + x F (τ) dτ, (135) 0 2m s 0 eH µb = µbs/s, (130) t0 where bs is the spin operator and µ is a constant character- results in the one-dimensional time-dependent Schrodinger¨ izing the particle, which is usually called the magnitude of equation of the harmonic oscillator driven by an external the magnetic momentum. For the motion of a charged parti- force in the y-direction cle in uniform magnetic H and electric E fields, which are perpendicular to each other (Fig. 1), the corresponding three- ∂ψ }2 ∂2ψ i} = − dimensional time-dependent Schrodinger¨ equation ∂t 2m ∂y2 ∂Ψ mω2 i} = HbΨ (131) + H (y − y )2 ψ − F (t)(y − y ) ψ (136) ∂t 2 0 0 has the Hamiltonian of the form [28] with µ ¶2 b 1 eH H = pbx + y |e| H cp 2m c ω = , y = − x . (137) H mc 0 eH 1 2 1 2 µ + pby + pbz − sbzH − yF, (132) 2m 2m s The Cauchy initial-value problem subject to special data where pb = −i}∇ is the linear momentum operator, functions H and F/e are the magnitudes of the uniform magnetic and i(xpx+zpz )/} Ψ(r, t)|t=t = e ψ (y, t0) electric fields in z and y directions, respectively. The corre- 0 i(xpx+zpz )/} sponding vector potential A = −yH ex is defined up to a = e ϕ (y − y0) (138) gauge transformation. Here we follow the original choice of Ref. 28 (see a remark at the end of this section). has the following solution: Since (132) does not contain the other components of the spin, the operator sbz commutes with the Hamiltonian Hb and Ψ(r, t) = Ψ (r, t, p , p ) = ei(xpx+zpz −S(t,t0))/} the z-component of the spin is conserved. Thus the operator x z ∞ sbz can be replaced by its eigenvalue sz = σ in the Hamilto- Z nian × G (y − y0, η, t, t0) ϕ (η) dη, (139) µ ¶ 1 eH 2 1 −∞ Hb = pb + y + pb2 2m x c 2m y where the propagator takes the form 1 µσ + pb2 − H − yF (133) 2m z s G (y, η, t, t0) = G1 (y, η, t − t0) with σ = −s, −s + 1, ... , s − 1, s. Then the spin depen- i(a(t,t0)y+b(t,t0)η+c(t,t0)) dence of the wave function becomes unimportant and the × e (140)

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 208 R.M. LOPEZ AND S.K. SUSLOV with The ﬁrst term here gives the discrete energy values corre- r sponding to motion in a plane perpendicular to the ﬁeld. They mωH G1 (y, η, t) = are called Landau levels. The expression (149) does not con- 2πi} sin ωH t µ ¶ tain the quantity px, which takes all real values. Therefore imωH ¡¡ 2 2¢ ¢ the total energy levels are continuously degenerate. For an × exp y + η cos ωH t − 2yη , (141) 2} sin ωH t electron, µ/s = − |e| }/mc, and formula (149) becomes

1 µ ¶ 2 a (t, t0) = 1 pz } sin ω (t − t ) En = En (pz, σ) = }ωH n + + σ + . (150) H 0 2 2m Zt In this case, there is an additional degeneracy: the levels with × F (τ) sin ωH (τ − t0) dτ, (142) n, σ = 1/2 and n + 1, σ = −1/2 coincide: t0

b (t, t0) = −a (t0, t) (143) En (pz, 1/2) = En+1 (pz, −1/2) . and The three-dimensional wave functions corresponding to the Zt energy levels (149) are given by } c (t, t ) = − a2 (τ, t ) dτ. (144) 0 2m 0 Ψn (r, t, σ) = Ψn (r, t, px, pz, σ) t0

−iEn(pz ,σ)(t−t0)/} i(xpx+zpz )/} See Eqs. (122)–(127) with G ≡ 0. Function c (t, t0) can be = e e χn (y) . (151) written in several different forms. They are the eigenfunctions of the following set of commut- 14.2. Landau levels ing operators pbx, pbz, sbz, and Hb with F ≡ 0 :

In an absence of the external force F ≡ 0, Eq. (136) is for- HbΨn = EnΨn, sbzΨn = σΨn, mally identical to the time-dependent Schrodinger¨ equation pb Ψ = p Ψ , pb Ψ = p Ψ . (152) for a simple harmonic oscillator with the frequency ωH . The x n x n z n z n standard substitution The orthogonality relation in R3 is −iε(t−t )/} ψ (y, t) = e 0 χ (y) (145) Z ∗ 0 0 0 gives the corresponding stationary Schrodinger¨ equation as Ψn (r, t, px, pz, σ)Ψm (r, t, px, pz, σ ) dxdydz follows [28]: R3 µ ¶ 2m 1 2 0 0 χ00 + ε − mω2 (y − y )2 χ = 0, (146) = (2π}) δnm δσσ0 δ (px − px) δ (pz − pz) , (153) }2 2 H 0 which has the square integrable solutions only when where µ ¶ Z∞ 1 1 ε = }ωH n + , n = 0, 1, 2, ... . (147) δ (α) = eiαξdξ (154) 2 2π −∞ The eigenfunctions are 1 is the Dirac delta function. χn (y) = p √ n 2 n!aH π Ã ! µ ¶ 14.3. Transition amplitudes (y − y )2 y − y × exp − 0 H 0 , 2 n In the presence of external force, the quantum mechanical 2aH aH r amplitude of a transition between Landau’s levels under the } influence of the perpendicular electric field can be explicitly aH = , (148) mωH found as a special case of our formulas (100)–(102). Indeed, solution (139) takes the form where Hn (η) are the Hermite polynomials. Thus the total energy levels of a particle in a uniform X∞ −iS(t,t0)/} magnetic field have the form Ψ(r, t, σ) = e Ψn (r, t0, σ) n=0 µ ¶ 2 1 pz µσ E = E (p , σ) = }ω n + + − H ∞ Z∞ n n z H 2 2m s X × cnm (t, t0) χm (η) ψ (η, t0) dη (155) (n = 0, 1, 2, ... ) . (149) m=0 −∞

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 209 in view of the bilinear generating relation (99). If where functions a (px, pz) do not depend on time t and ψ (y, t0) =χm (y) , this equation becomes S (t, t0) is given by (135). Now we replace the special initial data (138) in R3 by the general one Ψ(r, t, σ) = e−iS(t,t0)/} ∞ Ψ(r, t)| = φ (x, y, z) , (160) X t=t0 × cnm (t, t0)Ψn (r, t0, σ) , (156) n=0

which is independent of px (and y0). Letting t → t0 in (159) where coefﬁcients cnm (t, t0) are given by (101) in terms of Charlier polynomials as follows: and using the fundamental property of the Green function,

2 ∞ i(c−(ωH (t−t0)−ab sin ωH (t−t0))/2) −γ /4 Z cnm (t, t0) = e e lim G (y − y0, η, t, t0) ϕ (η) dη=ϕ (y − y0) , (161) n+m + i 2 t→t0 × √ (a + bδ)n (b + aδ)m cγ /2 (n) (157) −∞ 2n+mn!m! m one gets −iωH (t−t0) 2 2 2 with δ=e and γ =a + b + 2ab cos ωH (t − t0) . ∞ Functions a = a (t, t0) , b = b (t, t0) and c = c (t, t0) are ZZ evaluated by the integrals (142)–(144), respectively. The last φ (x, y, z) = a (px, pz) ϕ (y − y0) two formulas (156)–(157) and (135) give us the quantum me- −∞ chanical amplitude that the particle initially in Landau state i(xpx+zpz )/} m is found at time t in state n. For the particle initially in the × e dpxdpz, (162) ground state m = 0, the probability of occupying state n at time t is given by the Poisson distribution where y0 is a function of px in view of (137). Thus n 2 −µ µ 1 |c (t, t )| = e , a (px, pz) ϕ (y − y0) = n0 0 n! (2π})2

1 ¡ 2 2 ¢ ZZ∞ µ = a + b + 2ab cos ωH (t − t0) < 1. (158) 2 × φ (ξ, y, ζ) e−i(ξpx+ζpz )/} dξdζ (163) The details are left to the reader. −∞

14.4. Propagator in three dimensions by the inverse of the Fourier transform. Its substitution into (159) gives Our particular solutions (139) subject to special initial data ZZ∞ (138) have been constructed above as eigenfunctions of the 1 i(xpx+zpz −S(t,t0))/} operators pbx and pbz, whose continuous eigenvalues px and Ψ(r, t) = dpxdpz e (2π})2 pz vary from −∞ to ∞. By the superposition principle, one −∞ 3 can look for a general solution in R as a double Fourier in- Z∞ tegral of the particular solution × dη G (y − y0, η − y0, t, t0) ZZ∞ −∞ Ψ(r, t) = a (p , p )Ψ(r, t, p , p ) dp dp x z x z x z ZZ∞ −∞ × φ (ξ, η, ζ) e−i(ξpx+ζpz )/}dξdζ (164) ZZ∞ −∞ i(xpx+zpz )/} −iS(t,t0)/} = dpxdpz a (px, pz)e e −∞ as a solution of our initial value problem. A familiar integral form of this solution is as follows: ∞ Z Z × G (y − y0, η, t, t0) ϕ (η) dη, (159) Ψ(r, t) = G (r, ρ, t, t0) φ (ξ, η, ζ) dξdηdζ, (165) −∞ R3 where the Green function (propagator) is given as a double Fourier integral

ZZ∞ 1 i((x−ξ)px+(z−ζ)pz )/} −iS(t,t0)/} G (r, ρ, t, t0) = e e G (y − y0, η − y0, t, t0) dpxdpz (166) (2π})2 −∞ with the help of the Fubini theorem.

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 210 R.M. LOPEZ AND S.K. SUSLOV

This integral can be evaluated in terms of elementary functions as follows. Integration over pz gives the free particle propagator of a motion in the direction of magnetic ﬁeld ∞ Ã ! Z µ µ 2 ¶¶ r 2 1 i pz m im (z − ζ) G0 (z − ζ, t − t0) = exp (z − ζ) pz− (t − t0) dpz= exp (167) 2π} } 2m 2πi} (t − t0) 2} (t − t0) −∞ by the integral (57). Thus   µ ¶ Z∞ µ ¶ Zt iµσH 1 i icp G (r, ρ, t, t ) = exp (t − t ) G (z − ζ, t − t ) exp (x − ξ) p exp − x F (τ) dτ 0 }s 0 0 0 2π} } x }eH −∞ t0 µ ¶ iµσH × G (y − y , η − y , t, t ) dp = exp (t − t ) G (z − ζ, t − t ) G (y, η, t − t ) 0 0 0 x }s 0 0 0 1 0    Z∞ Zt 1 ipx c × ei(a(t,t0)y+b(t,t0)η+c(t,t0)) exp  x − ξ − F (τ) dτ 2π} } eH −∞ t0 µ ¶ µ µ 2 ¶ ¶ i (a (t, t0) + b (t, t0)) cpx −i px |e| × exp exp + (y+η) px tan (ωH (t−t0) /2) dpx. (168) eH } mωH e In view of (57), the last integral is given by    Z∞ Zt µ ¶ 1 ip c i (a (t, t ) + b (t, t )) cp exp  x x − ξ − F (τ) dτ exp 0 0 x 2π} } eH eH −∞ t0 r µ µ 2 ¶ ¶ −i px e mωH cot (ωH (t − t0) /2) × exp + (y + η) px tan (ωH (t − t0) /2) dpx = } mωH |e| 4πi} µ ¶ imω cot (ω (t − t ) /2) × exp H H 0 β2 , (169) 4} where e β = x − ξ − (y + η) tan (ω (t − t ) /2) + d (t, t ) (170) |e| H 0 0 with Zt c d (t, t0) = F (τ) (sin ωH (τ − t0) − sin ωH (τ − t) − sin ωH (t − t0)) dτ. (171) eH sin ωH (t − t0) t0 Here we have used (143)–(142). As a result, we arrive at the following factorization of our propagator:

i(a(t,t0)y+b(t,t0)η+c(t,t0)) G (r, ρ, t, t0) = G0 (z − ζ, t − t0) G1 (y, η, t − t0) e G2 (x, ξ, y, η, t, t0) , (172) where G0 (z − ζ, t − t0) is the free particle propagator in (167), G1 (y, η, t − t0) is the simple harmonic oscillator propagator in (141), and µ ¶ r µ ¶ iµσH mω cot (ω (t − t ) /2) imω cot (ω (t − t ) /2) G (x, ξ, y, η, t, t ) = exp (t − t ) H H 0 exp H H 0 β2 (173) 2 0 }s 0 4πi} 4} with β = β (x, ξ, y, η, t, t0) given by (170)–(171). Our propagator can be simpliﬁed to a somewhat more convenient form as follows:

G (r, ρ, t, t0) = G0 (z − ζ, t − t0) GH (x, ξ, y, η, t − t0) GF (x, ξ, y, η, t, t0) . (174)

Here G0 (z, t) is the free particle propagator in the direction of magnetic ﬁeld. The function µ ¶ iµσHt mωH GH (x, ξ, y, η, t) = exp }s 4πi} sin (ωH t/2) µ µ ¶¶ imω ³ ´ e × exp H (x − ξ)2 + (y − η)2 cot (ω t/2) − 2 (x − ξ)(y + η) (175) 4} H |e|

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 211 is the propagator corresponding to a motion in a plane perpendicular to the magnetic ﬁeld in the absence of an electric ﬁeld (compare our expression with one in Ref. 20, where F = µ = 0, and see a remark below in order to establish an identity of two results). The third factor

iWF (t,t0)/} GF (x, ξ, y, η, t, t0) = e (176) with 1 W (t, t ) = } (a (t, t ) y + b (t, t ) η + c (t, t )) + mω d (t, t ) F 0 0 0 0 4 H 0 µ ¶ e × (d (t, t ) + 2 (x − ξ)) cot (ω (t − t ) /2) − 2 (y + η) (177) 0 H 0 |e| is a contribution from the electric ﬁeld. When F = 0, WF = 0 and GF = 1. and in view of (179) The solution to the Cauchy initial value-problem in R3 µ ¶ µ ¶ subject to the general initial data ief (x, y) ief (ξ, η) GH → exp GH exp − Ψ(r, t)|t=t = Ψ (r, t0) = φ (x, y, z) (178) }c }c 0 Ã has the form mω imω ³£ ¤ Z = H exp H (x − ξ)2 + (y − η)2 Ψ(r, t) = G (r, ρ, t, t0)Ψ(ρ, t0) dξdηdζ, (179) 4πi} sin(ωH t/2) 4}

3 ! R e ´ which gives explicitly the time evolution operator for a mo- × cot(ω t/2) − 2 (xη − ξy) , (183) H |e| tion of a charged particle in uniform perpendicular magnetic and electric ﬁelds with a given projection of the spin sz = σ in the direction of the magnetic ﬁeld. By choosing which is, essentially, Eq. (3-64) on page 64 of [20], where Ψ(r, t0) = δ (r − r0) , where δ (r) is the Dirac delta func- we have corrected a typo. The details are left to the reader. tion in three dimensions, we formally obtain

Ψ(r, t) = G (r, r0, t, t0) (180) as the wave function at time t of the particle initially located 15. Diffusion-type equation at a point r = r0. Then Eq. (179) gives a general solution by the superposition principle. 15.1. Special case Remark. The vector potential of the uniform magnetic ﬁeld in the z-direction is deﬁned up to a gauge transformation [28] A formal substitution√ of t → −it and ψ → u into Eq. (25) with ω = 2κ and 2µ = ε yields the following time- A = −yH e → A0 = A + ∇f x dependent diffusion-type equation: 1 1 1 = − yH ex + xH ey = H × r (181) µ ¶ 2 2 2 ∂u ∂2u = κ − x2u with f (x, y) = xyH/2. The corresponding transformation ∂t ∂x2 of the wave function is given by µ ¶ µ ¶ ∂u ief (x, y) −ε (cosh (2κ−1) t) xu+ (sinh (2κ−1) t) , (184) Ψ → Ψ0 = Ψ exp ∂x }c µ µ ¶¶ imω e where the initial condition is = Ψ exp H 2 xy (182) 4} |e|

u (x, t)|t=0 = u0 (x)(−∞ < x < ∞) . (185)

As in the case of the time-dependent Schrodinger¨ equation, in order to solve this initial value problem, we use the eigenfunction expansion method. Hence the solution is given by

∞ X∞ X∞ Z u (x, t) = Ψn(x) cnm (t) Ψm (y) u0 (y) dy, (186) n=0 m=0 −∞

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 212 R.M. LOPEZ AND S.K. SUSLOV where n+m 2 −t 2 ¡ ¢ n−m ε −(ε /2)(1−e ) −((2κ−1)n+κ−ε /2)t −t m+n cnm(t) = (−1) √ e e 1 − e 2F0 2n+mn!m! Ã ! 2e−t × −n, −m; (187) ε2 (1 − e−t)2 by analytic continuation t → −it with ω = 2κ and µ = ε2/2 < 1 in (28) and (31). One can easily verify that

lim cnm (t) = δnm t→0+ √ and that if 0 < ε < 2 and κ ≥ 1/2, κ > ε2/2,

lim cnm (t) = 0 (m, n = 0, 1, 2, ... ) . t→∞ Thus the limiting distribution is

lim u (x, t) ≡ 0 (−∞ < x < ∞) , (188) t→∞ which is independent of the initial data (185). Relation (47) becomes

∞ X∞ X∞ Z −(ε2/2) sinh t−(κ−ε2/2)t n −t(2κ−1/2)n m −mt/2 u (x, t) =e (−1) e Ψn (x) tmn (β) (−1) e Ψm (y) u0 (y) dy (189) n=0 m=0 −∞ with β = β (t) = −2iε sinh (t/2) . With the help of (49), (43) and the Fubuni theorem we transform

∞ ∞ Ã ! X∞ Z Z Z X∞ −mt/2 tmn(β) e Ψm (−y) u0 (y) dy = Ke−t/2 (−y, z) tmn (β)Ψm (z) u0 (y) dydz m=0 −∞ −∞ m=0 Z Z∞ γz = Ke−t/2 (−y, z)(e Ψn (z)) u0 (y) dydz, (190) −∞ where γ = iβ = 2ε sinh (t/2) . The series (189) becomes

∞ X∞ Z Z −(ε2/2) sinh t−(κ−ε2/2)t −t(2κ−1/2)n γz u (x, t) = e e Ψn (−x) Ke−t/2 (−y, z)(e Ψn (z)) u0 (y) dydz n=0 −∞ ∞ Ã ! Z Z X∞ −(ε2/2) sinh t−(κ−ε2/2)t γz −t(2κ−1/2)n = e Ke−t/2 (−y, z) e e Ψn (−x)Ψn (z) u0 (y) dydz −∞ n=0 ∞ Z µZ ∞ ¶ −(ε2/2) sinh t−(κ−ε2/2)t γz = e Ke−t/2 (−y, z) e Ke−t(2κ−1/2) (−x, z) dz u0 (y) dy (191) −∞ −∞ in view of the generating relation (48). Therefore, the integral form of the solution (186)–(187) is Z∞ −(ε2/2) sinh t−(κ−ε2/2)t u (x, t) = e Ht (x, y) u0 (y) dy, (192) −∞ where by the deﬁnition Z∞ γz Ht (x, y) := Ke−t(2κ−1/2) (−x, z) e Ke−t/2 (−y, z) dz. (193) −∞

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 213

−t(2κ−1/2) −t/2 Denoting r1 = e and r2 = e we obtain by (48) that Z∞ Ã ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ! 1 1 + r2 1 − r2 x2 + 1 − r2 1 + r2 y2 K (−x, z) eγzK (−y, z) dz = p exp − 1 2 1 2 r1 r2 2 2 2 (1 − r2) (1 − r2) π (1 − r1) (1 − r2) 1 2 −∞ ∞ Ã¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ! Z 2 2 2 2 µ 2 2 ¶ 1 − r1 1 − r2 γ − 2r1 1 − r2 x − 2r2 1 − r1 y 1 − r1r2 2 × exp 2 2 z exp − 2 2 z dz (194) (1 − r1) (1 − r2) (1 − r1) (1 − r2) −∞ and the integral can be evaluated with the help of an elementary formula Z∞ r 2 π 2 e−az +2bzdz = eb /a, a > 0. (195) a −∞ As a result, an analog of the heat kernel in (192)–(193) is given by Ã ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢! 1 1 − r2r2 x2 + y2 + r2 − r2 x2 − y2 H (x, y) = p exp − 1 2 1 2 t 2 2 2 (1 − r2) (1 − r2) π (1 − r1r2) 1 2 Ã£¡ ¢ ¡ ¢ ¤ ! 2 2 2 1 − r1 1 − r2 γ − (r1 + r2) (1 − r1r2)(x + y) − (r1 − r2) (1 + r1r2)(x − y) × exp 2 2 2 2 (196) 4 (1 − r1) (1 − r2) (1 − r1r2)

−t(2κ−1/2) −t/2 with r1 = e , r2 = e and γ = 2ε sinh (t/2) , t > 0. The last expression can be simpliﬁed to a somewhat more convenient form Ã¡ ¢ ¡ ¢ ! µ µ ¶ ¶ 2 2 2 r1 + r2 r1 − r2 γ 1 − r1 1 − r2 γ Ht (x, y) = exp − (x + y) + (x − y) exp 2 2 Kr1r2 (x, y) (197) 1 + r1r2 1 − r1r2 2 1 − r1r2 4 in terms of the Mehler kernel (48). One can show that tion

lim u (x, t) = u0 (x) (198) t→0+ by methods of Ref. 52. The details are left to the reader. u (x) = δ (x − x ) A formal substitution of 0 0 into (192) µ ¶ gives ∂u ∂2u ∂u = κ − x2u + f (t) xu − g (t) , (201) ∂t ∂x2 ∂x u (x, t) = H (x, x0, t)

−(ε2/2) sinh t−(κ−ε2/2)t = e Ht (x, x0) (199) as the fundamental solution of the diffusion Eq. (184). In the limit ε → 0 we obtain where f (t) and g (t) are real-valued functions of time, sub- Z∞ −κt ject to the initial condition u (x, t) = e Ke−2κt (x, y) u0 (y) dy (200) −∞ as the exact solution of the corresponding initial value problem in terms of the Mehler kernel (48). This kernel gives also a familiar expression in statistical mechanics for the density matrix for a system consisting of a simple harmonic oscilla- u (x, t)|t=0 = u0 (x)(−∞ < x < ∞) . (202) tor [20].

15.2. Generalization

A formal substitution of t → −it and ψ → u into (77) with ω = 2κ and f → f, g → −ig yields a diffusion-type equa-

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 214 R.M. LOPEZ AND S.K. SUSLOV

An analog of the expansion (99) is The exact solution is X∞ X∞ ∞ Z H (x, y, t) = cnm (t)Ψn (x)Ψm (y) (212) u (x, t) = H (x, y, t) u0 (y) dy (203) n=0 m=0 −∞ with and the Green function can be found in the form 1 c−κt−(ab/2) sinh(2κt)+λ2/4 cnm (t) = √ e n+m a(t)x+b(t)y+c(t) 2 n!m! H (x, y, t) = H0 (x, y, t) e , (204) µ ¶ 2 × (a + br)n (b + ar)m F −n, −m; . (213) where 2 0 λ2 r r Here r = e−2κt, λ2 = a2 +b2 +2ab cosh (2κt) and functions H0 (x, y, t) = 2 π (1 − r ) a (t) , b (t) and c (t) are given by the integrals (209)–(211), Ã ¡ ¢ ¡ ¢! 4xyr − x2 + y2 1 + r2 respectively. This can be derived by expanding the kernel × exp (205) (204) in the double series in the same fashion as in Sec. 11, 2 (1 − r2) or by the substitution t → −it, a → −ia, b → −ib, and c → −ic in (99). The coefﬁcients c (t) are positive when with r = e−2κt. Indeed, substitution of (204) into (201) gives nm t > 0. the system of equations The solution (203) takes the form d (sinh (2κt) a (t))=f (t) sinh (2κt) ∞ ∞ dt X X u (x, t) = Ψn (x) cnm (t) +g (t) cosh (2κt) , (206) n=0 m=0 d 2κa (t) − g (t) Z∞ b (t) = , (207) dt sinh (2κt) × Ψm (y) u0 (y) dy. (214) d −∞ c (t) = κa2 (t) − g (t) a (t) (208) dt These results can be extended to the case when parameter κ and the solutions are is a function of time in Eq. (201). The details are left to the reader. 1 a (t) = sinh (2κt) Acknowledgments Zt × (f (s) sinh (2κs) + g (s) cosh (2κs)) ds, (209) This paper is written as a part of the summer 2007 program on 0 analysis of the Mathematical and Theoretical Biology Insti- tute (MTBI) at Arizona State University. The MTBI/SUMS Zt 2κa (s) − g (s) undergraduate research program is supported by The Na- b (t) = ds, (210) sinh (2κs) tional Science Foundation (DMS–0502349), The National 0 Security Agency (DOD–H982300710096), The Sloan Foun- Zt dation, and Arizona State University. The authors are grateful ¡ ¢ c (t) = κa2 (s) − g (s) a (s) ds (211) to Professor Carlos Castillo-Chavez´ for support and Ref. 8. We thank Professors George Andrews, George Gasper, Slim 0 Ibrahim, Hunk Kuiper, Mizan Rahman, Svetlana Roudenko, provided a (0) = b (0) = c (0) = 0. and Hal Smith for valuable comments.

1. G.E. Andrews and R.A. Askey, Classical orthogonal polynomi- CBMS–NSF Regional Conferences Series in Applied Mathe- als, in: Polynomesˆ orthogonaux et applications, Lecture Notes matics, (SIAM, Philadelphia, Pennsylvania, 1975). in Math. 1171, (Springer-Verlag, 1985) p. 36. 5. R.A. Askey, M. Rahman, and S.K. Suslov, J. Comp. Appl. 2. G.E. Andrews, R.A. Askey, and R. Roy, Special Functions, Math. 68 (1996) 25. (Cambridge University Press, Cambridge, 1999). 6. W.N. Bailey, Generalized Hypergeometric Series, (Cambridge 3. G.P. Arrighini and N.L. Durante, Am. J. Phys. 64 (1996) 1036. University Press, Cambridge, 1935). 4. R.A. Askey Orthogonal Polynomials and Special Functions, 7. L.A. Beauregard, Am. J. Phys. 34 (1966) 324.

Rev. Mex. F´ıs. 55 (2) (2009) 196–215 THE CAUCHY PROBLEM FOR A FORCED HARMONIC OSCILLATOR 215

8. L.M.A. Bettencourt, A. Cintron-Arias,´ D.I. Kaiser, and 31. J. Meixner, J. London Math. Soc. 9 (1934) 6. C. Castillo-Chavez,´ Phisica A 364 (2006) 513. 32. J. Meixner, Mathematische Zeitscrift 44 (1939) 531. 9. N.N. Bogoliubov and D.V.Shirkov, Introduction to the Theory 33. J. Meixner, Deutsche Math. 6 (1942) 341. of Quantized Fields, third edition, (John Wiley & Sons, New York, Chichester, Brisbane, Toronto, 1980). 34. E. Merzbacher, Quantum Mechanics, third edition, (John Wiley & Sons, New York, 1998). 10. L.S. Brown and Y. Zhang, Am. J. Phys. 62 (1994) 806. 35. A. Messia, Quantum Mechanics, two volumes, (Dover Publica- 11. C.V.L. Charlier, Arkiv for¨ Matematik, Astronomi och Fysik 2 tions, New York, 1999). (1905) 35. 36. V. Naibo and A. Stefanov, Math. Ann. 334 (2006) 325. 12. T.S. Chihara, An Introduction to Orthogonal Polynomials, (Gordon and Breach, New York, 1978). 37. P. Nardone, Am. J. Phys. 61 (1993) 232. 13. A. S. Davydov, Quantum Mechanics, Pergamon Press, (Oxford 38. A.F. Nikiforov, S.K. Suslov, and V.B. Uvarov, Classical Or- and New York, 1965). thogonal Polynomials of a Discrete Variable, (Springer–Verlag, 14. A. Erdelyi,´ Higher Transcendental Functions, Vol. II, Berlin, New York, 1991). A. Erdelyi,´ ed., (McGraw–Hill, 1953). 39. A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathe- 15. A. Erdelyi,´ Tables of Integral Transforms, Vols. I–II, A. Erdelyi,´ matical Physics, (Birkhauser,¨ Basel, Boston, 1988). ed., (McGraw–Hill, 1954). 40. J.D. Paliouras and D.S. Meadows, Complex Variables for Sci- 16. R.P. Feynman, The Principle of Least Action in Quantum Me- entists and Engineers, second edition, (Macmillan Publishing chanics, Ph. D. thesis, Princeton University, 1942; reprinted in: Company, New York and London, 1990). textquotedblleft Feynman’s Thesis – A New Approach to Quan- 41. E.D. Rainville, Special Functions (Chelsea Publishing Com- tum Theory, L.M. Brown, Editor, (World Scientiﬁc Publishers, pany, New York, 1960). Singapore, 2005). pp. 1. 42. M. Rahman and S.K. Suslov, Singular analogue of the Fourier 17. R.P. Feynman, Rev. Mod. Phys. 20 (1948) 367; reprinted in: transformation for the Askey–Wilson polynomials, in: Sym- textquotedblleft Feynman’s Thesis – A New Approach to Quan- metries and Integrability of Difference Equations; D. Levi, L. tum Theory, L.M. Brown, Editor, (World Scientiﬁc Publishers, Vinet, and P. Winternitz, Amer. Math. Soc. 9 (1996) 289. Singapore, 2005) pp. 71. 43. R.W. Robinett, Am. J. Phys. 64 (1996) 803. 18. R.P. Feynman, Phys. Rev. 76 (1949) 749. 44. I. Rodnianski and W. Schlag, Invent. Math. 155 (2004) 451. 19. R.P. Feynman, Phys. Rev. 76 (1949) 769. 45. L.I. Schiff, Quantum Mechanics , third edition, (McGraw-Hill, 20. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path New York, 1968). Integrals, (McGraw–Hill, New York, 1965). 46. W. Schlag, Dispersive estimates for Schrodinger¨ operators: a 21. S. Flugge,¨ Practical Quantum Mechanics, (Springer–Verlag, survay, arXiv: math/0501037v3 [math.AP] 10 Jan 2005. Berlin, 1999). 47. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Or- 22. G. Gasper, J. Math. Anal. Appl. 42 (1973) 438. thogonality, and Oscillatory Integrals , (Princeton University 23. K. Gottfried and T.-M. Yan, Quantum Mechanics: Funda- Press, Princeton, New Jersey, 1993). mentals, second edition, (Springer–Verlag, Berlin, New York, 48. S.K. Suslov, Electronic Transactions on Numerical Analysis 2003). (ETNA) 27 (2007) 140. 24. B.R. Holstein, Am. J. Phys. 65 (1997) 414. 49. G. Szego,˝ Amer. Math. Soc. (Colloq. Publ., Vol. 23, Rhode Is- 25. B.R. Holstein, Am. J. Phys. 67 (1998)583. land, 1939). 26. J. Howland, Indiana Univ. Math. J. 28 (1979) 471. 50. N.S. Thomber, and E.F. Taylor, Am. J. Phys. 66 (1998) 1022. 27. D.R. Jafaev, Teoret. Mat. Fiz. 45 (1980) 224. 51. N. Ya. Vilenkin, Special Functions and the Theory of Group 28. L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Nonrel- Representations, (American Mathematical Society, Providence, ativistic Theory, (Pergamon Press, Oxford, 1977). 1968). 29. V.P. Maslov and M.V. Fedoriuk, Semiclassical Approximation 52. N. Wiener, The Fourier Integral and Certain of Its Applica- in Quantum Mechanics, (Reidel, Dordrecht, Boston, 1981). tions, (Cambridge University Press, Cambridge, 1933); (Dover 30. M. Meiler, R. Cordero–Soto, and S.K. Suslov, J. Math. Phys. edition published in 1948). 49 (2008) 072102. 53. K. Yajima, J. Math. Soc. Japan 29 (1977) 729.

Rev. Mex. F´ıs. 55 (2) (2009) 196–215