Optical potentials using resonance states in supersymmetric quantum mechanics

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1 Home Search Coections Journas About Contact us My IOPscience Optica potentias using resonance states in supersymmetric quantum mechanics This artice has been downoaded from IOPscience. Pease scro down to see the fu text artice J. Phys.: Conf. Ser ( View the tabe of contents for this issue, or go to the journa homepage for more Downoad detais: IP Address: The artice was downoaded on 06/12/2012 at 16:45 Pease note that terms and conditions appy.

2 V Internationa Symposium on Quantum Theory and Symmetries Journa of Physics: Conference Series 128 (2008) IOP Pubishing Optica potentias using resonance states in Supersymmetric Quantum Mechanics Nicoás Fernández-García and Oscar Rosas-Ortiz Departamento de Física, Cinvestav, AP , México DF, Mexico E-mai: and Abstract. Compex potentias are constructed as Darboux-deformations of short range, radia nonsinguar potentias. They behave as optica devices which both refracts and absorbs ight waves. The deformation preserves the initia spectrum of energies and it is impemented by means of a Gamow-Siegert function (resonance state). As straightforward exampe, the method is appied to the radia square we. Anaytica derivations of the invoved resonances show that they are quantized whie the corresponding wave-functions are shown to behave as bounded states under the broken of parity symmetry of the reated one-dimensiona probem. 1. Introduction Soutions of the Schrödinger equation associated to compex eigenvaues ɛ = E α and satisfying purey outgoing conditions are known as Gamow-Siegert functions [1, 2]. These soutions represent a specia case of scattering states for which the capture of the incident wave produces deays in the scattered wave. The time of capture can be connected with the ifetime of a decaying system (resonance state) which is composed by the scatterer and the incident wave. Then, it is usua to take Re(ɛ) as the binding energy of the composite whie Im(ɛ) corresponds to the inverse of its ifetime. The Gamow-Siegert functions are not admissibe as physica soutions into the mathematica structure of Quantum Mechanics since, in contrast with conventiona scattering wave-functions, they are not finite at r. Thus, such a kind of functions is acceptabe in Quantum Mechanics ony as a convenient mode to sove scattering equations. However, because of the resonance states reevance, some approaches extend the formaism of quantum theory so that they can be defined in a precise form [3 9]. In this paper the Gamow-Siegert functions are anayzed not to precisey represent decaying systems but to be used as the cornerstone of compex Darboux-transformations in Supersymmetric Quantum Mechanics (Susy-QM). The unphysica behaviour of these soutions pays a reevant roe in the transformation: Whie the Gamow-Siegert function u diverges at r, the imit of its ogarithmic derivative β = u /u at r is a compex constant. This ast function is used to deform the initia potentia into a compex function. It is worth noticing that compex eigenvaues ɛ have been used as factorization constants in Susy-QM (see for instance [10 18] and the discussion on atypica modes in [19]). However, as far as we know, unti the recent resuts reported in [20 22] the connection between Susy-QM and resonance states has been missing. Our interest in the present work is two-fod. First, we want to show that appropriate approximations ead to anaytica expressions for the rea and imaginary parts c 2008 IOP Pubishing Ltd 1

3 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) of the resonance energies in the case of radia short range potentias. Second, we want to enarge the number of appications of the Darboux transformations in constructing new exacty sovabe modes in Quantum Mechanics. In Section 2 the main aspects of soving the Schrödinger equation for radia, nonsinguar short range potentias are reviewed. The reevance of the scattering ampitude S(k) in the anaysis of both the bounded and scattering wave-functions is ceary stated. In Section 3 the anaytica properties of S(k) as a function of the compex kinetic parameter k are studied. It is shown that poes of the scattering ampitude which ive in the ower k pane are connected with the Gamow-Siegert functions whie the poes on the positive imaginary axis ead to bounded physica energies. The genera aspects of the compex Darboux-transformations are incuded in Section 4. A the previous deveopments are then appied to the square radia we and new anaytica expressions for the invoved resonance energies are reported in Section 5. We concude the paper with some concuding remarks in Section Short range radia potentias revisited 2.1. Genera considerations Let us consider the Hamitonian of one partice in the externa, sphericay symmetric fied U( r) = U(r). The Schrödinger equation reduces to the eigenvaue probem: ) Hψ( r) ( 2 2m + U(r) ψ( r) = Eψ( r). (1) We are interested in potentias which decrease more rapidy than 1/r at arge r. Indeed, U(r) is a short-range potentia (i.e., there exists a Dom(U) := D U such that U(r) = 0 at r > a) which is nonsinguar (i.e., U(r) satisfies r 2 U(r) 0 at r 0). The wave function ψ( r) must be singe-vaued and continuous everywhere in D U. Since the operators H, L 2, L 3 and P commute with each other, in spherica coordinates the Schrödinger equation (1) has soutions of the form ψ m ( r) = R (r)y m (θ, ϕ) (2) where θ and ϕ are respectivey the poar and the azimutha anges, R (r) is a function depending on r and Y m (θ, ϕ) stands for the spherica harmonics. For simpicity in cacuations we sha use the function u (r) = rr (r). The introduction of (2) into equation (1) yieds: ] [ d2 ( + 1) + dr2 r 2 + v(r) u (r) := [ d2 dr 2 + V (r) ] u (r) = k 2 u (r) (3) with v = 2mU/ 2 and k 2 = E 2mE/ 2 the dimensioness expressions for the potentia and the kinetic parameter respectivey. As usua, the effective potentia V is defined as the potentia v pus the centrifuga barrier ( + 1)/r 2. Physica soutions ψ( r) of (1) shoud be constructed with functions u fufiing the foowing boundary conditions: u (r) = 0 r 0 u (r) r < r Furthermore, the function u and its derivative u have to be continuous for r > 0 since (3) incudes second order derivatives. As we sha see, the study of the invoved matching conditions is simpified by introducing a function β as foows β (r) := u (r) u (r) = d dr n u (r). (5) (4) 2

4 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) In genera we wi assume that the energy spectrum is composed by negative (bounded) and positive (scattering) energies. For the sake of notation we sha use the symbo φ (r) for the soutions of (3) which satisfy the boundary conditions (4); the symbo u (r) wi stand for genera mathematica soutions of equation (3) Bases of soutions For arbitrary and r > a, the appropriate basis of soutions can be written in the form u (+) (r) = ikr h (1) (kr), u ( ) (r) = ikr h (2) (kr) (6) where h (1) ν and h (2) ν are the spherica Hänke functions of order ν = (see e.g. [23]). For arge vaues of kr in the region r > a, the functions (6) behave as foows u (±) (r) exp [ ±i ( kr π 2 )]. (7) Thereby, u (+) represents a diverging wave and describes partices moving with speed ϑ = k in a directions from the origin. In turn, the function u ( ) represents a converging wave and describes partices moving with speed ϑ towards the origin. It is worth noticing that the Schrödinger equation (3) is invariant under the change k k. Hence, if u (r, k) is soution of (3) for E = k 2, then u (r, k) is aso a soution for the same energy. However, these ast functions must differ at most in a constant factor because the soutions of (3) have to be singe-vaued. In contrast, up to a goba phase, they interchange their roes at r, as indicated by the reationship u (±) (r, k) = e iπ u ( ) (r, k), r. (8) On the other hand, the basis of soutions at r < a may be written u (1) (r) = Λ kr j (kr), u (2) (r) = Λ kr n (kr), Λ := 2 π Γ( + 1/2) (9) where j and n are respectivey the spherica Besse and Neumann functions of order. For very sma vaues of r we have u (1) (r 0) (kr) +1, u (2) (r 0) 1 (kr). (10) Bounded states. Imaginary vaues of the kinetic parameter k = ±i E ±iκ correspond to negative vaues of the energy E = k 2 = κ 2. As usua, we sha consider k to be in the upper compex k pane I +. In this way u ( ) (r, κ) does not satisfy the boundary condition at r and u (2) (r, κ) does not fufi the boundary condition at r = 0. Hence, the physica soutions shoud be constructed with u (1) (r, κ) and u (+) (r, κ). We write: ζ (κ) u (1) (r, κ) g (r, κ) r < R φ (r, κ) = (11) ξ (κ) u (+) (r, κ) r R where the intermediary function g (r, κ) is equa to 1 at r = 0 and depends on the potentia. The coefficients ζ (κ) and ξ (κ) are compex numbers whie the matching condition β (a, κ) = κ 3

5 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) corresponds to a transcendenta equation which is fufied for a set of discrete vaues of the kinetic parameter κ n, n = 0, 1, 2,... Under these conditions the functions φ (r, κ n ) are finite and the integra + 0 φ (r, κ n ) 2 dr (12) converges. Thereby, φ (r, κ n ) 2 can be identified as the invoved Born s probabiity density Scattering states. Positive energies E = k 2 correspond to rea vaues of k and both soutions (7) remain finite in r a. As a consequence, both of them are physicay acceptabe in this region. To anayze these soutions first et us consider the case = 0 and, for the sake of simpicity, et us drop this subindex from the functions u (r). The genera soution in the free of interaction zone (r > a) may be written as [ ] u out (r) = γ(k) u ( ) (r) S(k)u (+) (r). (13) The constant γ(k) is the ampitude of the converging wave and does not depend on the potentia v(r). The scattering ampitude S(k), in contrast, strongy depends on the potentia and encodes the information of the scattering phenomenon. The matching condition between u out (r) and the soution u in (r) in the interaction zone reads β out (a) = β in (a) (14) and can be satisfied by appropriate vaues of γ and S. Indeed, equation (14) is equivaent to the foowing matrix array ( ) ( ) ( ) uin u (+) γ 1 u ( ) = (15) u in u (+) S r=a r=a the soution of which is given by the system of equations γ = W ( u in, u (+)), S = W ( uin, u ( )) 2ik W ( u in, u (+)), (16) r=a r=a where W (, ) stands for the Wronskian of the invoved functions. To get a better idea of the roes payed by each one of the terms in equation (13) et us consider the free motion (v(r) = 0) in the whoe of D v. From equations (6) and (9) we reaize that a reguar soution at the origin may be written as foows u ( ) u free (r) u ( ) (r) u (+) (r) = 2ikr j 0 (kr) = 2i sin(kr). (17) This term can be introduced into equation (13) to get [ ] u out (r) = γ(k) u free (r) (S(k) 1)u (+) (r). (18) Thus, the tota wave function after scattering u out is given by the wavepacket one woud have if there were no scattering γu free pus a scattered term γ(s 1)u (+). This ast exampe shows the important roe payed by the scattering ampitude in the anaysis of positive energies. On the other hand, the potentia we are deaing with is neither a sink nor a source of partices, then the condition S(k) = 1 (eastic scattering) is true and we have S(k) = e 2iδ(k). (19) 4

6 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) The scattering phase δ(k) represents the difference in phase of the outgoing parts. The situation is cear if one introduces (17) and (19) into equation (18) to get u out (r ) = 2iγ(k) e iδ(k) sin(kr + δ). (20) As we have indicated, the scattering states incude k R since E > 0. However, the function (13) is aso vaid if k is one of the imaginary vaues of the kinetic parameter eading to the bounded energies E n = κ 2 n. Indeed, this case provides further information about the scattering ampitude. For k n = iκ n, the function (13) at r reads u out (r, κ n ) 2iγ(iκ n ) [ e κ nr S(iκ n )e κ nr ]. (21) The first term in square brackets never vanishes (κ n > 0), then the coefficient S must be singuar at the compex point k n = iκ n. In other words, the points k n are nothing but poes of the scattering ampitude. Since these points are in the upper compex k pane I +, we reaize that S(k) is a reguar function in I + except at k n, n = 0, 1, 2,... We sha use these resuts in the next sections. Finay, simiar expressions can be found for arbitrary azimutha quantum numbers. A straightforward cacuation gives ( ) u (r) = γ (k) u ( ) (r) S (k)u (+) (r), r > a (22) the asymptotic behaviour of which reads u (r ) 2iγ (k)e iδ (k) sin ( kr + δ (k) π ). (23) 2 The above discussed properties of the scattering ampitude S(k) are easiy generaized to the case of arbitrary anguar momentum S (k). 3. Gamow-Siegert functions 3.1. Anaytic continuation of the scattering ampitude S (k) In the previous section we reaized that S is a reguar function in I + except at the points k = iκ eading to bound states of the energy. In order to get a we behaved scattering ampitude in the whoe of the compex k pane, this function must be extended to be anaytic in I. With this aim, first et us construct an arbitrary inear combination of the functions (7), it reads [ ] u (r) = ζ (k) u ( ) (r) ξ (k) u (+) (r) ζ (k) u ( ) (r) S (k)u (+) (r). (24) This function is reguar at the origin if u (r = 0) = 0, then we arrive at the foowing reationship S (k) = ξ (k) ζ (k) = u( ) (r) u (+). (25) (r) r=0 Now, after a change of sign in k, the coefficients of the singe-vaued function (24) shoud satisfy e iπ ξ ( k) = αζ (k), e iπ ζ ( k) = αξ (k) (26) where α is a proportionaity factor and we have used equation (8). In this way, the expression (25) eads to S (k) S ( k) = 1. (27) 5

7 V Internationa Symposium on Quantum Theory and Symmetries Journa of Physics: Conference Series 128 (2008) IOP Pubishing iκ n k α = k i k α = k i k α = k i k α = k i Figure 1. Schematic representation of the poes (disks) and the zeros (circes) of the scattering ampitude S (k) in the compex k pane. Bounded energies E n = κ 2 n correspond to the poes ocated on the positive imaginary axis. Let k i I + be a zero of S. From equation (27) we notice that the scattering ampitude is we behaved in I except at the point k i, for which S ( k i ). Thus, k i I is a poe of S. On the other hand, if the kinetic parameter k is rea, then the compex conjugate of any soution of equation (3) is aso a soution for the same energy. However, the soution is unique so these ast functions differ at most in a goba phase. Thereby, in a simiar form as in the previous case, we get S (k) S (k) = 1 (28) where the bar stands for compex conjugation. This ast equation is a consequence of the eastic scattering we are deaing with since S (k) = 1. However, as k is rea, aso equation (28) has to be extended to the whoe of the compex k pane. The natura condition reads S (k) S (k) = 1. (29) It is cear that S (k i ) diverges because k i I + is a zero of S. In other words, k i I is a poe of S since S (k i ). Moreover, as k i is a poe of S, from equation (29) we aso notice that S ( k i ) = 0. Thus, k i is another zero of the scattering ampitude. In summary, if k α (k i ) is a poe (zero) of S, then k α ( k i ) is another poe (zero) whie k α and k α (k i and k i ) are zeros (poes) of the scattering ampitude. In this way S (k) is a meromorphic function of the compex kinetic parameter k, with poes restricted to the positive imaginary axis (bound states) and to the ower haf-pane I (see Figure 1) Resonance states Let us express one of the poes of the scattering ampitude as k α = α 1 iα 2, with α 1 an arbitrary rea number and α 2 > 0. The function (22), evauated for k α at r reads u (r, k α ) γ (k α )e i(α 1r π/2) [ e α 2r S (k α )e α 2r ]. (30) Observe that the first term in brackets vanishes for arge vaues of α 2 r. Hence, there remains ony the scattering, purey outgoing wave e α 2r. As a consequence, the partice is maximay scattered by the potentia fied v(r) and the wave function is not we behaved since it does not fufi the boundary condition at r. However, as we have seen, this ast unphysica behaviour is not ony natura but necessary to study the eastic scattering process of a partice 6

8 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) by a short range, nonsinguar radia fied interaction. Wave functions behaving ike (30) are known as Gamow-Siegert soutions of the Schrödinger equation (3). We write u (GS) (r) = θ g (r, k α ) u (1) (k α r) r < R γ (k α )S (k α )u (+) (r, k α ) R r where θ is a constant and the intermediary function g (r, k α ) is equa to 1 at r = 0 and depends on the potentia. Besides the matching condition (16), the Gamow-Siegert functions fufi the purey outgoing boundary condition: { } im β (GS) r + (r) + ik α = 0. (32) Our interest is now addressed to the Gamow-Siegert functions not precisey as describing a decaying system but as appropriate mathematica toos to study the construction of compex Darboux-deformed potentias. 4. Compex-Darboux deformations of radia potentias 4.1. Genera considerations In order to throw further ight on the function β (r) we may note that (5) transforms the Schrödinger equation (3) into a Riccati one (31) β (r) + β2 (r) + ɛ = V (r) (33) where the energy k 2 has been changed for the arbitrary number ɛ. Remark that (33) is not invariant under a change in the sign of the function beta: β (r) + β2 (r) + ɛ = V (r) + 2β (r). (34) These ast equations define a Darboux transformation Ṽ(r) Ṽ(r, ɛ ) of the initia potentia V (r). If ɛ is a nontrivia compex number, then the Darboux-deformation is necessariy a compex function Ṽ (r) = V (r) + 2β (r) V (r) 2 d2 dr 2 n ϕ ɛ(r) (35) where the transformation function u (r, ɛ) ϕ ɛ (r) is the genera soution of (3) for the compex eigenvaue ɛ. The soutions y (r) = y (r, ɛ, E) of the non-hermitian Schrödinger equation are easiy obtained where u (r) is eigensoution of equation (3) with eigenvaue E. y (r) + Ṽ(r) y (r) = Ey (r) (36) y (r) W(ϕ ɛ(r), u (r)), (37) ϕ ɛ (r) 7

9 V Internationa Symposium on Quantum Theory and Symmetries Journa of Physics: Conference Series 128 (2008) New exacty sovabe optica potentias IOP Pubishing Let the transformation function ϕ ɛ (r) be a Gamow-Siegert function (31). The superpotentia β (GS) (r, k α ) and the new potentia Ṽ(r) behave at the edges of D v as foows β (GS) (r, k α ) = { +1 r r 0 ik α r + Ṽ (r) = { V+1 (r) r 0 V (r) r + (38) It is remarkabe that even for an initia potentia with = 0 the corresponding Darbouxdeformation contains a nontrivia centrifuga term in the interaction zone r < a. Notice aso that the new potentia is mainy a rea function at the edges of D v. Thus, the function Im(Ṽ) is zero at the origin and vanishes at r +. In genera Im(Ṽ) osciates aong the whoe of (0, + ) according with the eve of excitation of the Gamow-Siegert soution u (GS) (r, k α ): The higher the eve of excitation the greater the number of osciations. Such a behaviour impies a series of maxima and minima in Im(Ṽ) which can be anayzed in terms of the optica bench [19]. As regards the physica soutions of the new potentia it is immediate to verify that, in a cases (bounded and scattering states of the initia potentia), the corresponding transformations fufi the boundary condition at r = 0. The anaysis of the behaviour at r is as foows Scattering states. Let u (r) be a scattering state of the initia potentia in r > a. At r +, the corresponding Darboux-Deformation (37) behaves as foows [ ] im y (r) = iγ (k) (k α + k)u ( ) (r) (k α k)s (k)u (+) (r) r + r + where we have used equation (22). As we can see, the Darboux-deformations of u behave as scattering states at r > a. If the scattering state u (r) is a Gamow-Siegert function and k k α, k α, k α, k α, the Darboux-deformation preserves the purey outgoing condition (32). Specia cases are: (i) k = k α the Wronskian in (37) is zero and there is no transformation. (ii) k = k α S (k α ) = 0, then y (r) is an exponentiay decreasing function. (iii) k = k α S ( k α ) = 0 and the coefficient of u ( ) (r) vanishes, then y (r) = 0. (iv) k = k α S ( k α ), then y (r) fufis the purey outgoing condition (32) Bounded states. If I + k = iκ = i E the function u (r, E ) corresponds to one of the bounded states (11). Thereby, from equation (37) we get (39) im y (r, E ) (ik α E ) ξ ( π i E ) e 2 e E r. (40) r + Thus, the Darboux-deformations of bounded wave-functions vanishes at r. However, athough this new functions are square-integrabe, they do not form an orthogona set in the Hibert space spanned by the initia square-integrabe wave-functions [14] (see aso [24] and the puzzes with sef orthogona states [25]). In the next section we are going to sove the Schrödinger equation for the radia square we. We sha get the soutions corresponding to compex energy eigenvaues and purey outgoing boundary conditions. Thus, we sha construct the invoved Gamow-Siegert functions. 8

10 V Internationa Symposium on Quantum Theory and Symmetries Journa of Physics: Conference Series 128 (2008) Gamow-Siegert states of the radia square we Let us consider the potentia v(r) = { v0 r < a 0 a r with a and v 0 positive rea numbers. The reguar soution of the Schrödinger equation (3) for this potentia may be written u (r) = [ iγ (k) kr h (2) θqr j (qr) ] (kr) + S (k)h (1) (kr) r < a a r where the interaction parameter q is defined by q 2 = v 0 + k 2. For simpicity, we sha anayze the s-wave soutions ( = 0) for which the matching conditions (16) ead to S(q) = IOP Pubishing [ ] ik sin(qa) + q cos(qa) e 2ika, γ(k) = θeika [ik sin(qa) q cos(qa)]. (43) ik sin(qa) q cos(qa) 2ik Let us take θ = 2ik. The scattering ampitude S(k) has zeros and poes which respectivey correspond to the roots of the transcendenta equations (41) (42) iq k = tan(qa), iq k = tan(qa). (44) Indeed, equations (44) correspond to the even and odd quantization conditions for negative energies if k = iκ I +, just as it has been discussed in the previous sections. However, we are ooking for points k α in the compex k pane such that Re(k α ) 0 and Im(k α ) 0. Thereby, if k α is a poe of S, the converging wave in (42) vanishes. Then the Gamow-Siegert function reads u (GS) =0 (r) = { 2ik α sin[q(k α )r] 2ik α sin[q(k α )a] e ikα(r a) r < a a r In order to sove the second equation in (44) et us rewrite the scattering ampitude as [ e 2iqa ] (k + q) + (q k) S(k) = e 2iqa e 2ika. (46) (k q) (q + k) In this way, the poes of S are aso the soutions of the transcendenta equation (45) e 2iqa = k 2 q 2 + 2Re(kq) k q 2. (47) If the interaction parameter q is rea, then sin(2qa) = 0 and we obtain the foowing quantization rue for the energy q n = πn 2a E n = ( nπ ) 2 v0, n = 0, 1,... (48) 2a On the other hand, et k and q be the compex numbers k = K 1 + ik 2 and q = Q 1 + iq 2. Then q 2 = v 0 + ɛ = (v 0 + ɛ 1 ) + iɛ 2, with ɛ = k 2. Hence we have Q 2 1 Q 2 2 = v 0 + ɛ 1, 2Q 1 Q 2 = ɛ 2 (49) 9

11 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) and K1 2 K2 2 = ɛ 1, 2K 1 K 2 = ɛ 2. (50) The imaginary part of equation (47) is sin(2q 1 a) = 0 and we obtain the foowing quantization rue Q 1,n = πn, n = 0, 1,... (51) 2a Therefore, if Q 2 << 1, the first equation in (49) shows that equation (51) eads to ɛ 1 E n. The second equation in (49), on the other hand, gives a vaue of ɛ 2 which is not necessariy equa to zero. In this way, we sha study the poes of S for which the compex points ɛ are such that Re(ɛ) E n and Im(ɛ) Γ 2, with 0 < Γ << 1. The main probem is then to approximate the adequate vaue of Γ as connected with the discreteness of Q 1 in (51). With this aim, for a given finite vaue of a we take Q 2 a << 1. A straightforward cacuation shows that the second equation in (44) uncoupes into the system Q 1 = v 0 sin(q 1 a) cosh(q 2 a), Q 2 = v 0 cos(q 1 a) sinh(q 2 a). (52) In turn, the second one of these ast equations reduces to cos(q 1 a) 1 a v 0 = 1 η. (53) Now, et δ << 1 be a correction of Q 1 around the quantized vaues (51), that is Q 1 a Q 1,n a+δ. Then, equation (53) eads to δ n = sin(πn/2), n odd. (54) η Thus, ony odd vaues of n are aowed into the approximation we are deaing with. Now, to ensure sma vaues of Q 2 we propose Q 2 a = + m=1 λ m/η m. From (49) we have ɛ 1 v ( aq1,n η ) 2 2aQ 1,n sin(πn/2) η λ2 1 η 4 2λ 1λ 2 η 5 (55) In order to cance the term incuding η 4 we take λ 1 = ±1 (the appropriate sign wi be fixed beow). Higher exponents in the power of 1/η are dropped by taking λ m>1 = 0. In order to get ɛ 1 E n, after introducing (51) into (55), we reaize that the condition η >> 1 aows us to drop the term incuding η 3. Thus, at the first order in the approximation of Q 2 a we get ɛ 1 E n. It is aso necessary to have regard to the positiveness of energy ɛ 1, which is ensured whenever n exceeds a minimum vaue. Let us take n := n inf +m, m Z +, where n inf is the ceiing function of 2η/π, i.e. n inf = 2η π = (even). We finay arrive at 2a v0 π. Since n is odd, the integer m is even (odd) if n inf is odd ( [(ninf ] { + m)π 2 0, 2, 4,... ninf Re(ɛ) 2a odd 1) v 0, m = v 0 1, 3, 5,... n inf even On the other hand, it is worth noticing that the introduction of Q 2 a = ±1/η and (54) into the second equation in (49) eads to the concusion that Q 2 a = +1/η has to be dropped. The expression for the imaginary part of the compex energy ɛ is cacuated as foows. First, et us rewrite equations (52) in the form K 1 = ± v 0 sin(q 1 a) sinh(q 2 a), K 2 = ± v 0 cos(q 1 a) cosh(q 2 a). (57) (56) 10

12 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) Since ɛ 2 = Γ/2, from equations (50) we reaize that the sign of K 1 must be the opposite one of K 2. The introduction of (53) into the second equation in (57) eads to K 2 ± 1 ( ) a 2η 2 (58) where we have used Q 2 a 1/η. If η >> 1 we get ak 2 ±1. Then, this ast resut together with the rea part of the factorization constant ɛ gives K 2 1 = ɛ 1 + K 2 2 ɛ a 2. (59) Thus, for a 2 ɛ 1 > 1 we obtain K 1 ± ɛ 1. Finay, the introduction of (58) and (59) into the second equation of (50) produces Im(ɛ) = Γ 2 2 K 1 K 2 2 a Re(ɛ). (60) In Figure 2 we show the absoute vaue of the scattering ampitude cose to one of its poes k α and the corresponding zero k α. Some of the first resonances are reported in Tabe 1 for different vaues of the potentia strength v 0 and the cutoff a. Re k S k Im k Figure 2. The absoute vaue of the scattering ampitude S(k) cose to the poe k α (and the zero k α ) which corresponds to the first resonance E α = k 2 α of the square we reported in Tabe 1 for n inf = 64. The imits of our approach give the number k α = i Compex Darboux-deformations of the radia square we Figure 3 shows the goba behaviour of a typica Gamow-Siegert function associated with the radia square we. Notice the exponentia growing of the ampitude for r > a. This kind of soutions are used in (35) to compex Darboux-deform the square we as it is shown in Figure 4. The cardiod-ike behaviour seems to be a profie of the compex potentias derived by means of Darboux-deformations (compare with [20] and [22]). In this case, the compex potentia shows concentric cardiod curves for vaues of r inside the interaction zone, one of them is shown at the eft of Figure 4. The rea and imaginary parts of Ṽ(r) are then characterized by osciations and changes of sign depending on the position in r < a. As a consequence, these compex potentias behave as an optica device which both refracts and absorbs ight waves (see detais in [22] and 11

13 V Internationa Symposium on Quantum Theory and Symmetries Journa of Physics: Conference Series 128 (2008) IOP Pubishing Figure 3. The rea (eft) and imaginary (right) parts of the Gamow-Siegert function (45) associated with the first resonance state reported in Tabe 1 for n inf = 64. discussions on the optica bench in [19]). The presence of this kind of osciations is aso noted at distances sighty greater than the cutoff r = a. Hence, the compex Darboux-deformations (35) are short range potentias which enarge the initia interaction zone as it is shown in the right part of Figure Figure 4. The Argand-Wesse diagram of the compex Darboux-deformed square we with v 0 = 100 and a = 10. Left: Detai of the cardiod-ike behaviour of the new potentia between r = 9.8 (disk) and r = 10 (circe). Right: The disk is evauated at r = 10.1 and the circe at r = 13. As compementary information: Ṽ 0 (r = 10) = i and Ṽ0(r = 10.1) = i Tabe 1. The first four resonance energies for the radia square we of intensity v 0 = 100 and cutoff a = 10 (n inf = 64) and for the radia square we of intensity v 0 = 1000 and cutoff a = 10 (n inf = 202). Notice that, in each case, the even vaues of m obtained for the reated one-dimensiona probem are missing (see e.g. Tabe A.1 of reference [22]). n inf = 64 n inf = 202 m = i i m = i i m = i i m = i i Concuding remarks We have studied the eastic scattering process of a partice by short range, nonsinguar radia fied interactions. The poes of the invoved scattering ampitude S(k) pay a reevant roe 12

14 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) in the construction of physica soutions connected with either bounded or scattering states of the energy. It has been shown that poes k α in the ower haf compex k pane ead to the unphysica Gamow-Siegert functions which are necessary to depict the behaviour of the scattering phenomenon. The Gamow-Siegert functions were used to transform the radia short range potentias into compex ones which preserve the initia energy spectrum. These new potentias are opaque in the sense that they simutaneousy emit and absorb fux, just as an optica device which both refracts and absorbs ight waves. The transformation preserves the square-integrabiity of the soutions at the cost of producing non-orthogona sets of wave-functions. It is aso notabe that scattering states are transformed into scattering states whie deformed Gamow-Siegert functions can be either a new Gamow-Siegert function or an exponentiay decreasing function depending on the invoved kinetic parameter k and the poe k α. The method has been appied to the radia square we in the context of s-waves ( = 0). Anaytica expressions were derived for the corresponding compex energies in the ong ifetime imit (i.e., for sma vaues of Im(E α ) = Γ/2) and, as a consequence, such energies fufi a quantization rue. In contrast with the even and odd resonances of short range potentias defined in the whoe of the straight-ine (see e.g. [22]), the resonances of radia short range potentias are abeed by ony odd (positive) integers. This resut enforces the interpretation of Gamow-Siegert functions as representing quasi-bounded states: The bounded spectrum of a potentia which is invariant under the action of the parity operator v(x) = v( x) incudes odd and even functions. The symmetry is broken by adding an impenetrabe wa at the negative part of the straight ine and ony the odd soutions are preserved. As we have shown, the same is true for resonance states. Finay, the resuts reported in this paper are compementary to our previous work [22]. It is remarkabe that expicit derivations of Gamow-Siegert functions are barey reported in the iterature, not even for simpe modes ike those studied here (however see [5, 20, 21, 26 29]). We hope our approach has shed some ight onto the soving of the Schrödinger equation for compex energies and functions fufiing the purey outgoing boundary condition. Acknowedgments. The support of CONACyT project F is acknowedged. The authors are gratefu to M. Gadea and D. Juio for interesting discussions. References [1] Gamov G 1928 Z Phys [2] Siegert AJF 1939 Phys Rev [3] Bohm A, Gadea M and Mainand GB 1989 Am J Phys [4] Bohm A and Gadea M 1989 Lecture Notes in Physics vo 348 (New York: Springer) [5] de a Madrid R and Gadea M 2002 Am J Phys [6] Civitarese O and Gadea M 2004 Phys Rep [7] Aguiar J and Combes JM 1971 Commun Math Phys [8] Basev E and Combes JM 1971 Commun Math Phys [9] Simon B 1972 Commun Math Phys 27 1 [10] Cannata F, Junker G and Trost J 1988 Phys Lett A [11] Andrianov AA, Ioffe MV, Cannata F and Dedonder JP 1999 Int J Mod Phys A [12] Bagchi B, Maik S and Quesne C 2001 Int J Mod Phys A [13] Fernández DJ, Muñoz R and Ramos A 2003 Phys Lett A [14] Rosas-Ortiz O and Muñoz R 2003 J Phys A: Math Gen [15] Muñoz R 2005 Phys Lett A [16] Samsonov BF and Pupasov AM 2006 Phys Lett A [17] Samsonov BF 2006 Phys Lett A [18] Cabrera-Munguia I and Rosas-Ortiz O 2007 this issue [19] Mienik B and Rosas-Ortiz O 2004 J Phys A: Math Gen

15 V Internationa Symposium on Quantum Theory and Symmetries IOP Pubishing Journa of Physics: Conference Series 128 (2008) [20] Rosas-Ortiz O 2007 Rev Mex Fis 53 Supp [21] Fernández-García N 2007 Rev Mex Fis 53 Supp 4 42 [22] Fernández-García N and Rosas-Ortiz O 2008 Ann Phys doi: /j.aop [23] Abramowitz M and Stegun I 1972 Handbook of Mathematica Functions (New York: Dover Pub Inc) [24] Ramírez A and Mienik B 2003 Rev Mex Fis 49 Supp [25] Sokoov AV, Andrianov AA and Cannata F 2006 J Phys A: Math Gen [26] Webber TA, Hammer CL and Zide VS 1982 Am J Phys [27] Sprung DWL and Wu H 1996 Am J Phys [28] Antoniou IE, Gadea M, Hernandez E, Jauregui A, Menikov Y, Mondragón A and Pronko GP 2001 Chaos, So. and Fractas [29] de a Madrid R 2004 J Phys A: Math Gen