Dynamical formulation of low-energy scattering in one dimension
aa r X i v : . [ qu a n t - ph ] F e b Dynamical formulation of low-energy scatteringin one dimension
Farhang Loran ∗ and Ali Mostafazadeh † ∗ Department of Physics, Isfahan University of Technology,Isfahan 84156-83111, Iran † Departments of Mathematics and Physics, Ko¸c University,34450 Sarıyer, Istanbul, Turkey
Abstract
The transfer matrix M of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical formulation of stationary scattering. Weexplore the utility of this formulation in the study of the low-energy behavior of the scatteringdata. In particular, for the exponentially decaying potentials, we devise a simple iterativescheme for computing terms of arbitrary order in the series expansion of M in powers of thewavenumber. The coefficients of this series are determined in terms of a pair of solutions ofthe zero-energy stationary Schr¨odinger equation. We introduce a transfer matrix for the latterequation, express it in terms of the time-evolution operator for an effective two-level quantumsystem, and use it to obtain a perturbative series expansion for the solutions of the zero-energy stationary Schr¨odinger equation. Our approach allows for identifying the zero-energyresonances for scattering potentials in both full line and half-line with zeros of the entries ofthe zero-energy transfer matrix of the potential or its trivial extension to the full line.Keywords: Low-energy scattering, complex potential, zero-energy resonance, non-unitary quan-tum dynamics, Dyson series A real or complex-valued potential v : R → C is called a short-range potential, if for | x | → ∞ , | v ( x ) | tends to zero faster than the Coulomb potential, 1 / | x | , [1]. A characteristic property ofthe short-range potentials v ( x ) is that given any solution ψ ( x ; k ) of the corresponding Schr¨odingerequation, − ∂ x ψ ( x ; k ) + v ( x ) ψ ( x ; k ) = k ψ ( x ; k ) , (1) ∗ E-mail address: [email protected] † E-mail address: [email protected] A ± ( k ) and B ± ( k ) such that ψ ( x ; k ) → A ± ( k ) e ikx + B ± ( k ) e − ikx for x → ±∞ . (2)A 2 × M ( k ) that relates A ± ( k ) and B ± ( k ) according to, (cid:20) A + ( k ) B + ( k ) (cid:21) = M ( k ) (cid:20) A − ( k ) B − ( k ) (cid:21) , (3)is called the transfer matrix of the potential v , [2]. It is uniquely determined by the potential, if wedemand that it is independent of A − and B − , [3].Among the solutions of (1) are the so-called left/right-incident scattering solutions ψ l/t whichfulfill, ψ l ( x ; k ) → (cid:26) e ikx + R l ( k ) e − ikx for x → −∞ ,T ( k ) e ikx for x → + ∞ , (4) ψ r ( x ; k ) → (cid:26) T ( k ) e − ikx for x → −∞ ,e − ikx + R r ( k ) e ikx for x → + ∞ . (5)The coefficient functions R l/r ( k ) and T ( k ) are the left/right reflection and transmission amplitudesof the potential v . They are related to the entries M ij ( k ) of M ( k ) according to [2] R l ( k ) = − M ( k ) M ( k ) , R r ( k ) = M ( k ) M ( k ) , T ( k ) = 1 M ( k ) . (6)Dividing ψ l/r ( x ; k ) by T ( k ), we obtain the Jost solutions ψ ± ( x ; k ) of (1); ψ + ( x ; k ) := T ( k ) − ψ l ( x ; k ) → (cid:26) M ( k ) e ikx − M ( k ) e − ikx for x → −∞ ,e ikx for x → + ∞ , (7) ψ − ( x ; k ) := T ( k ) − ψ l ( x ; k ) → (cid:26) e − ikx for x → −∞ ,M ( k ) e − ikx + M ( k ) e ikx for x → + ∞ . (8)Solving the scattering problem for a short-range potential v means the determination of itsreflection and transmission amplitudes. In view of (6), this can be achieved by computing thetransfer matrix of the potential [4, 5, 6, 7, 8, 9, 10]. Because k has the interpretation of theenergy in quantum mechanics, we refer to the k → | v ( x ) | as x → ±∞ , [1].Let L σ ( R ) denote the class of functions f : R → C such that R ∞−∞ (1 + | x | σ ) | f ( x ) | dx < ∞ ,where σ is a nonnegative real number. Suppose that v ∈ L ( R ). Then for each x ∈ R , the Jostsolutions ψ ± are analytic functions of k in the upper complex half-plane C + := { z ∈ C | Im( z ) > } ,and continuous functions in C + \ { } := { z ∈ C | Im( z ) ≥ z = 0 } . In view of (7) and (8),the same applies for (the entries of) the transfer matrix M ( k ). If v is an exponentially decayingfunction as x → ±∞ , i.e., there are positive real numbers C, M , and µ such that for | x | ≥ M , | v ( x ) | ≤ C e − µ | x | , then the Jost solutions and the transfer matrix are analytic functions of k in2 z ∈ C | Im( z ) > − µ & z = 0 } . This in particular implies that 0 is an isolated singularity of M ( k ), R l/r ( k ), and T ( k ). In particular, they admit Laurent series expansions about k = 0. The coefficientsof these series determine the low-energy scattering behavior of the potential. Refs. [11, 12] providea comprehensive study of these coefficients and elaborate on the low-energy scattering propertiesof potentials belonging to L σ ( R ) with σ ≥
2. Refs. [13] and [14] explore the low-energy scatteringbehavior of the potentials belonging to L σ ( R ) with 1 < σ ≤ σ = 1, respectively. See also[1, 15].The treatment of low-energy scattering provided in Refs. [1, 11, 12, 13, 14, 15] relies on varioustechnical results of functional analysis, which are mostly beyond the reach of typical physicists.The purpose of the present article is to offer an alternative and much more accessible approach tolow-energy scattering in one dimension. It is based on a recent formulation of stationary scatteringin which the transfer matrix M ( k ) is expressed in terms of the time-evolution operator for a certaintwo-level quantum system [16, 17].The organization of this article is as follows. In Sec. 2 we provide a brief review of the dynamicalformulation of scattering theory in one dimension. In Sec. 3, we discuss its application in thedetermination of the low-energy series expansion of the transfer matrix of exponentially decayingpotentials. Here we outline an iterative scheme for computing the coefficients of this series in termsof a pair of solutions of the zero-energy Schr¨odinger equation, − φ ′′ ( x ) + v ( x ) φ ( x ) = 0 , (9)and provide a simple characterization of the zero-energy resonances. In Sec. 4, we introduce atransfer matrix M for (9), identify a corresponding two-level quantum system whose evolutionoperator yields M , and use the Dyson series expansion of the former to outline a perturbativesolution of (9). In Sec. 5, we discuss the extension of our results to potential scattering in thehalf-line, [0 , ∞ ), where the scattering problem depends on both the potential and the boundarycondition at x = 0. Here we offer a characterization of zero-energy resonances which turns out to bedifferent for Dirichlet and non-Dirichlet boundary conditions. Finally, in Sec. 6, we offer a summaryof our findings and present our concluding remarks. Let g ( x ; k ) be an invertible 2 × Ψ ( x ; k ) := g ( x ; k ) (cid:20) ψ ( x ; k ) ψ ′ ( x ; k ) (cid:21) , V ( x ; k ) := i (cid:20) v ( x ) − k (cid:21) , (10)where v ( x ) is a real or complex short-range potential. Then, it is an elementary exercise to showthat the time-independent Schr¨odinger equation (1) is equivalent to the time-dependent Schr¨odingerequation, i∂ x Ψ ( x ; k ) = H ( x ; k ) Ψ ( x ; k ) , (11)for the 2 × H ( x ; k ) := g ( x ; k ) V ( x ; k ) g ( x ; k ) − + i [ ∂ x g ( x ; k )] g ( x ; k ) − . (12)3f ψ ( x ; k ) is a solution of (1), so that (2) holds, we can choose g ( x ; k ) such thatlim x →±∞ Ψ ( x ; k ) = (cid:20) A ± B ± (cid:21) . (13)The simplest choice for g ( x ) that ensures (13) is g ( x ; k ) := 12 k (cid:20) ke − ikx − ie − ikx ke ikx ie ikx (cid:21) . (14)Substituting this relation in (10) and (12), we have [16] Ψ ( x ; k ) = 12 (cid:20) e − ikx [ ψ ( x ) − ik − ψ ′ ( x )] e ikx [ ψ ( x ) + ik − ψ ′ ( x )] (cid:21) , (15) H ( x ; k ) = v ( x )2 k (cid:20) e − ikx − e ikx − (cid:21) = v ( x )2 k e − ikx σ K e ikx σ , (16)where K := (cid:20) − − (cid:21) = σ + i σ , (17)and σ j are the Pauli matrices; σ = (cid:20) (cid:21) , σ = (cid:20) − ii (cid:21) , σ = (cid:20) − (cid:21) . (18)The time-dependent Schr¨odinger equation (11) corresponding to the Hamiltonian (16) definesthe dynamics of a two-level quantum system, if we let x play the role of the evolution parameter,i.e., ‘time.’ Note, however, that H ( x ; k ) is neither Hermitian nor diagonalizable. Therefore, itdoes not generate a unitary ‘time-evolution.’ Furthermore, because K = , we have H ( x ; k ) = ,where stands for the 2 × U ( x, x ) be the time-evolution operator for the Hamiltonian (16) with x ∈ R playing therole of the initial ‘time,’ i.e., the 2 × x and x that satisfies i∂ x U ( x, x ; k ) = H ( x ; k ) U ( x, x ; k ) , U ( x , x ; k ) = I , (19)where I labels the 2 × U ( x, x ; k ) := I + ∞ X n =1 ( − i ) n Z xx dx n Z x n x dx n − · · · Z x x dx H ( x n ; k ) H ( x n − ; k ) · · · H ( x ; k )=: T exp (cid:26) − i Z xx d ˜ x H (˜ x ; k ) (cid:27) , (20) If v ( x ) is a real potential, H ( x ; k ) † = σ H ( x ; k ) σ − . This means that H ( x ; k ) is σ -pseudo-Hermitian [18]. If v ( x )is a complex potential, H ( x ; k ) is σ -pseudo-normal, i.e., [ H ( x ; k ) , H ( x ; k ) ♯ ] = , where H ( x ; k ) ♯ := σ − H ( x ; k ) † σ ,[16]. T stands for the time-ordering operation [19]. According to (19), every solution Ψ ( t ; k ) ofthe time-dependent Schr¨odinger equation (11) satisfies Ψ ( x ; k ) = U ( x, x ; k ) Ψ ( x ; k ). We can usethis equation together with (3), (13), and (20) to conclude that [16] M ( k ) = lim x ±→±∞ U ( x + , x − ; k ) = T exp (cid:26) − i Z ∞−∞ dx H ( x ; k ) (cid:27) = I + ∞ X n =1 ( − i ) n Z ∞−∞ dx n Z x n −∞ dx n − · · · Z x −∞ dx H ( x n ; k ) H ( x n − ; k ) · · · H ( x ; k ) . (21)Because the entries of the transfer matrix determine the reflection and transmission amplitudesof the potential, Eq. (21) offers an alternative route for the solution of the scattering problemfor short-range potentials. Refs. [16, 17, 20, 21] explore various implications of this approach topotential scattering in one dimension, while Refs. [22, 23, 24] develop its extensions to potentialscattering in two and three dimensions, electromagnetic scattering by isotropic scatterers, andpotential scattering for long-range potentials.In the present article, we examine the utility of Eq.(21) in the study of the low-energy propertiesof the transfer matrix. This is motivated by the observation that for each x ∈ R , the matrixHamiltonian H ( x ; k ) is an analytic function of k in the punctured complex k -plane, C \ { } , andthat 0 is a simple pole of H ( x ; k ). Consider a finite-range potential v : R → C with support [ x − , x + ], so that U ( x − , x − ; k ) = I , U ( x − , x + ; k ) = M ( k ) , (22)and introduce Γ := (cid:20) −
10 0 (cid:21) , ∆ := (cid:20) (cid:21) . (23)Then it is easy to check that 12 (cid:0) K Γ + K T ∆ (cid:1) = I , (24)where K T stands for the transpose of K .If we multiply both sides of the first equation in (19) from the left by Γ and ∆ and substitute(16) in the resulting equations, we find ∂ x D = v ( x ) (cid:2) − x s ( kx ) D + x c ( kx ) G (cid:3) , (25) ∂ x G = v ( x ) (cid:2) x s ( kx ) G − d ( kx ) D (cid:3) , (26)where D and G abbreviate D ( x, x − ; k ) := i ∆U ( x, x − ; k ) , G ( x, x − ; k ) := k ΓU ( x, x − ; k ) , (27)5nd s ( τ ) := sin 2 τ τ = 1 + ∞ X n =1 s n τ n , s n := ( − n (2 n + 1)! , (28) c ( τ ) := 1 − cos 2 τ τ = 1 + ∞ X n =1 c n τ n , c n := 2( − n (2 n + 2)! , (29) d ( τ ) := 1 + cos 2 τ ∞ X n =1 d n τ n , d n := ( − n n )!] . (30)According to (24) and (27), U ( x, x − ; k ) = 12 (cid:2) K ΓU ( x, x − ; k ) + K T ∆U ( x, x − ; k ) (cid:3) = 12 k (cid:2) K G ( x, x − ; k ) − ik K T D ( x, x − ; k ) (cid:3) . (31)This relation reduces the calculation of U ( x, x − ; k ) to that of D ( x, x − ; k ) and G ( x, x − ; k ).For each ˜ x ∈ [ x − , x + ], we can identify U (˜ x, x − ; k ) with the transfer matrix of the potential˜ v ( x ) := v ( x ) θ (˜ x − x ), where θ ( x ) is the Heaviside step function [16]. Because ˜ v is a finite-rangepotential, either U (˜ x, x − ; k ) is an analytic function of k at 0 or k = 0 is a simple pole of thisfunction. For ˜ x = x ∈ [ x − , x + ], this argument implies the existence of matrix-valued functions U ( m ) ( x, x − ) such that U ( x, x − ; k ) = ∞ X m = − U m ( x, x − ) k m . (32)Substituting this relation in (27), we find D ( x, x − ; k ) = ∞ X m = − D m ( x ) k m , G ( x, x − ; k ) = ∞ X m = − G m ( x ) k m +1 , (33)where D m and G m are matrix-valued functions with vanishing second rows. The latter are alsofunctions of x − , but for brevity we do not make this explicit. In view of (31) – (33), we can expressthe coefficients of the Laurent series expansion (32) of the evolution operator in the form, U ( m ) ( x, x − ) = 12 (cid:2) K G m ( x ) − i K T D m ( x ) (cid:3) . (34)Next, we note that the first relation in (22) together with (32) and (33) imply D m ( x − ) = iδ m ∆ = (cid:26) i ∆ for m = 0 , for m = 0 , G m ( x − ) = δ m Γ = (cid:26) Γ for m = 0 , for m = 0 , (35)where δ ij stands for the Kronecker delta symbol. If we substitute (28) – (30) and (33) in (25) and(26), and match the coefficients of the same powers of k in both sides of the resulting equations, wearrive at the following equations for D m and G m . D − ( x ) = (36) D ′ m +1 ( x ) = − x v ( x ) (cid:2) D m +1 ( x ) − x G m ( x ) + E m ( x ) (cid:3) , (37) G ′ m ( x ) = − v ( x ) (cid:2) D m +1 ( x ) − x G m ( x ) + F m ( x ) (cid:3) . (38) This is generally true for exponentially decaying potentials [11]. m ≥ −
1, a prime stands for the differentiation with respect to x , E − ( x ) := F − ( x ) := E ( x ) := F ( x ) := , (39) E m ( x ) := ⌊ m ⌋ +1 X n =1 s n x n D m +1 − n ( x ) − ⌊ m +12 ⌋ X n =1 c n x n +1 G m − n ( x ) for m ≥ , (40) F m ( x ) := ⌊ m ⌋ +1 X n =1 d n x n D m +1 − n ( x ) − ⌊ m +12 ⌋ X n =1 s n x n +1 G m − n ( x ) for m ≥ , (41)and ⌊ x ⌋ stands for the integer part of x , i.e., the greatest integer that is not greater than x .Next, we multiply both sides of (38) by x and subtract the result from those of (37) to obtain D ′ m +1 ( x ) − x G ′ m ( x ) = x v ( x ) [ F m ( x ) − E m ( x )] . Integrating this equation yields D m +1 ( x ) = Z xx − d ˜ x n ˜ x v (˜ x ) [ F m (˜ x ) − E m (˜ x )] o + ( x∂ x − G m ( x ) + C m , (42)where G m ( x ) is any 2 × G ′ m ( x ) = G m ( x ) , (43)and C m is a 2 × x = x − in (42) and making use of (35). This yields C m = iδ − m ∆ − [ x − G ′ m ( x − ) − G m ( x − )] . (44)If we substitute (42) in (38) and express the result in terms of G m , we discover that − G ′′ m ( x ) + v ( x ) G m ( x ) = v ( x ) S m ( x ) , (45)where S m ( x ) := Z xx − d ˜ x h ˜ x v (˜ x ) (cid:2) F m (˜ x ) − E m (˜ x ) i + F m ( x ) + C m . (46)Because (45) is a second-order differential equation, we can determine G m in a unique manner, ifwe impose a pair of initial conditions. Eqs. (35) and (43) provide one of these conditions, namely G ′ m ( x − ) = δ m Γ . (47)Our analysis of low-energy scattering does not single out a second initial condition on G m . Thisreveals a certain degree of freedom in the choice of this function. We use this freedom to simplifyour calculation of G m and D m by demanding that G m satisfies x − G ′ m ( x − ) − G m ( x − ) = iδ − m ∆ . (48)In view of (44), this implies that C m = , and (42) and (46) become D m +1 ( x ) = Z xx − d ˜ x n ˜ x v (˜ x ) [ F m (˜ x ) − E m (˜ x )] o + ( x∂ x − G m ( x ) , (49) S m ( x ) := Z xx − d ˜ x h ˜ x v (˜ x ) (cid:2) F m (˜ x ) − E m (˜ x ) i + F m ( x ) . (50)7he initial conditions (47) and (48) determine a unique solution of (45). This solution takes aparticularly simple form, if we express it in terms of the solutions, φ and φ , of the zero-energySchr¨odinger equation (9) that are subject to the initial conditions φ ( x − ) − x − φ ′ ( x − ) = 1 , φ ′ ( x − ) = 0 ,φ ( x − ) − x − φ ′ ( x − ) = 0 , φ ′ ( x − ) = ℓ − . (51)Here ℓ is an arbitrary positive real parameter with the dimension of length that we can identifywith a relevant length scale entering the definition of the potential. In terms of φ and φ , thesolution of the initial-value problem given by (45), (47) and (48) takes the form, G m ( x ) = − iδ − m φ ( x ) ∆ + δ m ℓφ ( x ) Γ + Z xx − d ˜ x G ( x, ˜ x ) v (˜ x ) S m (˜ x ) , (52)where G ( x, ˜ x ) is the Green’s function for the operator − ∂ x + v ( x ) given by G ( x, ˜ x ) := ℓ [ φ ( x ) φ (˜ x ) − φ ( x ) φ (˜ x )] φ ( x − ) . (53)In view of (39) – (41) and (50), E m ( x ), F m ( x ), and S m ( x ) only involve G n and D n with labels n < m . This together with (43), (49), and (52) allow us to determine D m and G m iteratively. Inthe following we give the details of this calculation for m ≤ D − ( x ) = . If we set m = − G − ( x ) = − iφ ( x ) ∆ , G ( x ) = ℓφ ( x ) Γ , (54) G − ( x ) = − iφ ′ ( x ) ∆ , G ( x ) = ℓφ ′ ( x ) Γ , (55) D ( x ) = − i [ xφ ′ ( x ) − φ ( x )] ∆ , D ( x ) = ℓ [ xφ ′ ( x ) − φ ( x )] Γ . (56)To determine G , we first use (28) – (30), (40), (41), and (50), to show that s = − / c = − / d = −
1, and E ( x ) = − i x (cid:2) φ ( x ) − xφ ′ ( x ) (cid:3) ∆ , (57) F ( x ) = − i x (cid:2) φ ( x ) − xφ ′ ( x ) (cid:3) ∆ , (58) S ( x ) = i ς ( x ) ∆ , (59)where ς ( x ) := − (cid:26) x (cid:2) φ ( x ) − xφ ′ ( x ) (cid:3) + Z xx − d ˜ x ˜ x v (˜ x ) φ (˜ x ) (cid:27) . (60)Finally, we set m = 1 in (52) and use (43) and (59) to show that G ( x ) = iℓg ( x ) ∆ , (61) We do not state these conditions in the form φ ( x − ) = 1, φ ′ ( x − ) = ℓ − , and φ ( x − ) − ℓ − x − = φ ′ ( x − ) = 0,because we will consider a generalization of our approach to the infinite-range exponentially decaying potentials byconsidering the limit x − → −∞ . We have introduced the length scale ℓ to make sure φ and φ have the same physical dimension. In mathematicalliterature, x is taken to be dimensionless, and ℓ is set to 1, [13]. g ( x ) := ℓ − Z xx − d ˜ x ∂ x G ( x, ˜ x ) v (˜ x ) ς (˜ x ) . (62)Having computed D m and G m for m ≤
1, we can use (34), (35), (55), (56), and (61) to showthat U ( − ( x, x − ) = − iφ ′ ( x )2 K , (63) U (0) ( x, x − ) = 12 n ℓφ ′ ( x )( I − σ ) + [ φ ( x ) − xφ ′ ( x )]( I + σ ) o , (64) U (1) ( x, x − ) = iℓ n g ( x ) K + [ φ ( x ) − xφ ′ ( x )] K T o , (65)where we have also benefitted from the identities K Γ = I − σ , K ∆ = K , K T Γ = K T , K T ∆ = I + σ . Substituting (63) – (65) in (32), we have U ( x, x − ; k ) = − iφ ′ ( x )2 k K + 12 n ℓφ ′ ( x )( I − σ ) + [ φ ( x ) − xφ ′ ( x )]( I + σ ) o + ikℓ n g ( x ) K + [ φ ( x ) − xφ ′ ( x )] K T o + O ( k ) . (66)The above iterative procedure for computing the coefficients U ( m ) ( x, x − ) of the low-energy seriesexpansion of U ( x, x − ; k ) reduces their determination to finding the solutions φ and φ of the zero-energy Schr¨odinger equation (9) that fulfill the initial conditions (51). Because v ( x ) vanishes outsidethe interval [ x − , x + ], to each solution φ of (9) there corresponds a pair of complex numbers, a and b , such that φ ( x ) = a + ℓ − b x for x ≤ x − . In particular, φ ( x ) − xφ ′ ( x ) = a and φ ′ ( x ) = ℓ − b for x ≤ x − . In light of this observation, we can establish the equivalence of (51) with the requirementthat for all x ∈ ( −∞ , x − ], φ ( x ) − xφ ′ ( x ) = 1 , φ ′ ( x ) = 0 , φ ( x ) − xφ ′ ( x ) = 0 , φ ′ ( x ) = ℓ − . (67)These are in turn equivalent to the asymptotic boundary conditions,lim x →−∞ [ φ ( x ) − xφ ′ ( x )] = 1 , lim x →−∞ φ ′ ( x ) = 0 . lim x →−∞ [ φ ( x ) − xφ ′ ( x )] = 0 , lim x →−∞ φ ′ ( x ) = ℓ − . (68)Similarly, we can use the fact that v ( x ) = 0 for x / ∈ [ x − , x + ] to infer the existence of complexnumbers a j and b j , with j ∈ { , } , such that φ j ( x ) = a j + ℓ − b j x for x ≥ x + . Again this implies φ j ( x ) − xφ ′ j ( x ) = a j and φ ′ j ( x ) = ℓ − b j for x ≥ x + . Hence, a j = lim x → + ∞ [ φ j ( x ) − xφ ′ j ( x )] , b j = ℓ lim x → + ∞ φ ′ j ( x ) . (69)Because the Wronskian of φ and φ , i.e., W := φ ( x ) φ ′ ( x ) − φ ′ ( x ) φ ( x ), is a constant [25], (67)and (69) imply a b − a b = ℓ lim x → + ∞ [ φ ( x ) φ ′ ( x ) − φ ′ ( x ) φ ( x )] = ℓ lim x →−∞ [ φ ( x ) φ ′ ( x ) − φ ′ ( x ) φ ( x )] = 1 . (70)9t is also not difficult to see from (53) and (60), that their right-hand sides remain the same if wereplace them with their x − → −∞ limit, so that ς ( x ) = − (cid:26) x (cid:2) φ ( x ) − xφ ′ ( x ) (cid:3) + Z x −∞ d ˜ x ˜ x v (˜ x ) φ (˜ x ) (cid:27) , (71) G ( x, ˜ x ) = ℓ [ φ ( x ) φ (˜ x ) − φ ( x ) φ (˜ x )]lim x − →−∞ φ ( x − ) . (72)Furthermore, because according to (62), for all x ∈ [ x − , + ∞ ), g ( x ) = g ( x + ) =: g , we have g = lim x → + ∞ g ( x ) = ℓ − Z ∞−∞ d ˜ x ∂ x G ( x, ˜ x ) v (˜ x ) ς (˜ x ) . (73)In view of these observations and Eqs. (22), (66), and (69), we can express the low-energy expansionof the transfer matrix as follows. M ( k ) = − i b kℓ K + 12 (cid:2) b ( I − σ ) + a ( I + σ ) (cid:3) + ikℓ (cid:0) g K + a K T (cid:1) + O ( k ) . (74)Next, consider an exponentially decaying potential v : R → C that has an infinite range. Then v ∈ L ( R ), and we can use the results of Ref. [13] to conclude that the zero-energy Schr¨odingerequation for this potential has a pair of global solutions φ and φ satisfying the asymptotic boundaryconditions (68), and that for these solutions the limits in (69) exist, i.e., there are a j and b j fulfilling(69). This implies that φ j ( x ) can grow at most linearly as x → ±∞ . Furthermore, we can view v ( x ) as the x ± → ±∞ limit of the potentials w x − ,x + : R → C that are defined, for all x ± ∈ R with x + > x − , by w x − ,x + ( x ) := (cid:26) v ( x ) for x ∈ [ x − , x + ] , x / ∈ [ x − , x + ] . Because v ( x ) decays exponentially as x → ±∞ , the terms D m ( x ) and G m ( x ) in the series expansionof D ( x + , x − ; k ) and G ( x + , x − ; k ) for w x − ,x + tend to finite values as x ± → ±∞ . This in turn impliesthat we can use our iterative calculation of the coefficients of the low-energy series expansion ofthe transfer matrix for v ( x ) provided that for every function f ( x, x − ; k ) appearing in the samecalculation for w x − ,x + ( x ) we replace f ( x + , x − ; k ) with lim x ± →±∞ f ( x + , x − ; k ). In particular, (74)holds for exponentially decaying potentials provided that we define a j , b j , and g using (69) and(73).To obtain the low-energy expansion of the reflection and transmission amplitudes, we substitutethe Laurent series for the entries of the transfer matrix in (6) and express the result as power seriesin k . In view of (70) and (74), for situations where b = 0, this gives R l ( k ) = − − i b kℓ b + 2( b + 1)( kℓ ) b + O ( k ) , (75) R r ( k ) = − − i a ℓk b + 2( a + 1)( kℓ ) b + O ( k ) , (76) T ( k ) = − ikℓ b + 2( a + b )( kℓ ) b + 2 i ( a + b + a b − b g + 1)( kℓ ) b + O ( k ) . (77)10f b = 0, (70) implies that b = 0 and a = b − . Furthermore, the term of order ( kℓ ) − on theright-hand side of (74) vanishes. These together with (6) lead to the following low-energy seriesexpansions for the reflection and transmission amplitudes when b = 0. R l ( k ) = b − b + 1 + 2 i b ( b g − a ) kℓ ( b + 1) + O ( k ) , (78) R r ( k ) = − b − b + 1 + 2 i b ( g − a b ) kℓ ( b + 1) + O ( k ) , (79) T ( k ) = 2 b b + 1 + 2 i b ( a + g ) kℓ ( b + 1) + O ( k ) . (80)A couple of remarks are in order:1. The coefficients of various powers of kℓ that enter the above series expansions for the transfermatrix and the reflection and transmission amplitudes depend on the length scale ℓ in such away that the terms of these series are ℓ -independent. In view of (68), (69), and (71) – (73), itis easy to see that a and b are independent of ℓ ; a and g are proportional to ℓ − , and b isproportional to ℓ . These imply the ℓ -independence of the terms appearing on the right-handsides of (74) and (75) – (80).2. The reflection and transmission amplitudes display different low-energy properties for b = 0and b = 0. In particular, unlike for b = 0, for kℓ →
0, the left and right reflection amplitudesdiffer by a sign and the transmission amplitude tends to a nonzero value for b = 0. We cantrace this strange behavior to the fact that b = 0 marks the presence of a zero-energyresonance [1, 26, 27, 28, 29]. By definition, this means that the Wronskian W ( k ) of the Jostsolutions ψ ± ( x ; k ) tend to zero as k →
0, [1]. This follows from (7), (8), and (74), for theyimply W ( k ) = − ikM ( k ) and lim k → [ − ikM ( k )] = − b /ℓ . We state this result as thefollowing characterization theorem for the zero-energy resonances of exponentially decayingpotentials. Theorem 1:
Let v : R → C be an exponentially decaying potential and φ be the solutionof the zero-energy Schr¨odinger equation (9) that satisfies lim x →−∞ [ φ ( x ) − xφ ′ ( x )] = 1 andlim x →−∞ φ ′ ( x ) = 0. Then v has a zero-energy resonance if and only if lim x → + ∞ φ ′ ( x ) = 0.An exactly solvable example that we can use to check the utility of our approach to low-energyscattering is the barrier potential, v ( x ) := (cid:26) z for x ∈ [ a, a + L ] , x / ∈ [ a, a + L ] , (81)where z is a possibly complex coupling constant, a and L are real parameters, and L >
0. This is afinite-range potential with support [ a, a + L ], so x − = a and x + = a + L .The standard approach for the determination of the transfer matrix of the barrier potential (81)involves the solution of the corresponding time-independent Schr¨odinger equation (1). This is quitestraightforward, because this potential is piecewise constant. One obtains the general solution of(1) in [ a, a + L ] and matches this solution and its derivative at the boundary points, x = a and11 = a + L , to linear combinations of the plane waves, A ± ( k ) e ikx + B ± ( k ) e − ikx , which solve theSchr¨odinger equation (1) outside [ a, a + L ]. This allows for expressing A + ( k ) and B + ( k ) in terms of A − ( k ) and B − ( k ) and determines the transfer matrix M ( k ) via (3). There is an alternative approachfor calculating M ( k ) which makes use of the dynamical formulation of scattering [2]. Both give thefollowing expressions for the entries of the transfer matrix of (81). M ( k ) = M ( − k ) = e − ikL " cosh( kL p z /k − − i (cid:16) z k − (cid:17) sinh( kL p z /k − p z /k − = − iL z s k + c − L z s − iL h c − (3 + L z ) s i k + O ( k ) , (82) M ( k ) = M ( − k ) = − i z e − ik (2 a + L ) sinh( kL p z /k − k p z /k − , = − iL z s k − L (2 a + L ) z s iL n c + [(2 a + L ) z − s o k + O ( k ) , (83)where c := cosh( L √ z ) , s := sinh( L √ z ) L √ z . (84)The barrier potential (81) involves two length scales, namely L and | z | − / . According to (82)and (83), k enters the expression for M ij ( k ) through kL and z /k . This shows that a reasonablechoice for the characteristic length scale ℓ is the largest of L and | z | − / . This is also supported bythe fact that in the k → kL ≪ k / | z | ≪
1, hold.We can solve the zero-energy Schr¨odinger equation (9) for the barrier potential and identify thesolutions satisfying the initial conditions (68). For x ∈ [ a, a + L ], they are given by φ ( x ) = cosh[ √ z ( x − a )] , φ ( x ) = 1 ℓ (cid:26) a cosh[ √ z ( x − a )] + sinh[ √ z ( x − a )] √ z (cid:27) . Because φ j ( x ) − xφ ′ j ( x ) and φ ′ j ( x ) have the same values for x ≥ a + L , we can use these relationstogether with (69) and (71) – (73) to show that a = c − L ( a + L ) z s , b = ℓL z s , (85) a = − Lℓ n c + [ a ( a + L ) z − s o , b = c + aL z s , (86) g = − L ℓ n c − [(2 a + 2 aL + L ) z + 1] s o . (87)Substituting these equations in (74) reproduces (82) and (83). This confirms the validity of theanalysis leading to (74).Let us also note that according to (85), we have b = 0, and the barrier potential develops azero-energy resonance if and only if s = 0. This happens whenever z = − ( πn/L ) for some positiveinteger n . For these choices of the coupling constant, (81) describes a real potential well, and itstransmission and reflection coefficients, | T ( k ) | and | R l/r ( k ) | , respectively tend to 1 and 0, for k →
0, i.e., the potential is effectively reflectionless for low-energy incident waves. This behavior12isappears once we perturb the coupling constant. For example, suppose that z = − ( πn/L ) (1 + ǫ )for a real number ǫ such that | ǫ | <
1. Then for ǫ = 0, | T ( k ) | → | R l/r ( k ) | → k →
0, i.e.,the potential begins acting as a perfect mirror for low-energy incident waves as soon as ǫ becomesnonzero, while it is reflectionless for these waves for ǫ = 0. The coefficients of the low-energy expansions of the reflection and transmission amplitudes that wehave derived in Sec. 3 involve the constants a j , b j , and g which depend on the solutions φ and φ of the zero-energy Schr¨odinger equation (9). In this section we pursue the approach of Sec. 2to introduce a transfer matrix for the zero-energy Schr¨odinger equation and express it in termsof the evolution operator for an effective two-level quantum system. This leads to a Dyson seriesexpansion for this transfer matrix which we can use to obtain series expansions for a j , b j , and g .The zero-energy time-independent Schr¨odinger equation (9) is equivalent to the time-dependentSchr¨odinger equation (11) provided that we set k = 0 and use φ ( x ) instead of ψ ( x ; k ) in (10). Thisgives Ψ ( x ; 0) = g ( x ; 0) (cid:20) φ ( x ) φ ′ ( x ) (cid:21) . (88)For potentials v belonging to L ( R ), the solutions of (9) tend to polynomials of the form a ± + ℓ − b ± x as x → ±∞ , where a ± and b ± are complex numbers, and ℓ is an arbitrary length scale [13].This implies, lim x →±∞ [ φ ( x ) − xφ ′ ( x )] = a ± , ℓ lim x →±∞ φ ′ ( x ) = b ± , (89)which motivate the following analog of the asymptotic boundary conditions (13).lim x →±∞ Ψ ( x ; 0) = (cid:20) a ± b ± (cid:21) . (90)The simplest choice for g ( x ; 0) that is consistent with this requirement is g ( x ; 0) := (cid:20) − x ℓ (cid:21) . (91)In view of (88), it implies Ψ ( x ; 0) = (cid:20) φ ( x ) − xφ ′ ( x ) ℓφ ′ ( x ) (cid:21) . (92)Furthermore, substituting (91) in (12) and setting k = 0, we find H ( x ; 0) = H ( x ) := − i v ( x ) (cid:20) x x /ℓ − ℓ − x (cid:21) = − i xv ( x ) X ( x ) K X ( x ) − , (93)where K is given by (17), and X ( x ) := (cid:20) x ℓ (cid:21) . (94)13imilarly to the matrix Hamiltonian (16), H ( x ) is non-Hermitian and non-diagonalizable, and itssquare equals the null matrix. If we introduce the transfer matrix M associated with the zero-energy Schr¨odinger equation (9)through the relation, (cid:20) a + b + (cid:21) = M (cid:20) a − b − (cid:21) , (95)we can use (90) to establish, M = lim x ± →±∞ U ( x + , x − ) , (96)where U ( x, x ) is the evolution operator for the matrix Hamiltonian (93), i.e., U ( x, x ) := T exp (cid:26) − i Z xx d ˜ x H (˜ x ) (cid:27) := I + ∞ X n =1 ( − i ) n Z xx dx n Z x n x dx n − · · · Z x x dx H ( x n ) H ( x n − ) · · · H ( x ) . (97)Next, we use (17) and (94) to show that x j K X ( x j ) − X ( x j − ) K = − ( x j − x j − ) K . (98)With the help of this identity and (93), we obtain the following expression for the integrand on theright-hand side of (97). H ( x n ) H ( x n − ) · · · H ( x ) = − i n x v n ( x , x , · · · , x n ) X ( x n ) K X ( x ) − = − i n v n ( x , x , · · · , x n ) (cid:20) x n x x n /ℓ − ℓ − x (cid:21) , (99)where v ( x ) := v ( x ), and for n ≥ v n ( x , x , · · · , x n ) := v ( x n )( x n − x n − ) v ( x n − ) · · · v ( x )( x − x ) v ( x )= v ( x n ) n − Y j =1 ( x j +1 − x j ) v ( x j ) . (100)Because v n ( x , x , · · · , x n ) is proportional to v ( x ) v ( x ) · · · v ( x n ), (97) gives a perturbative seriesexpansion for the evolution operator U ( x, x ). In particular, for the entries U ij ( x, x ) of U ( x, x )we have U ( x, x ) = 1 − ∞ X n =1 Z xx dx n Z x n x dx n − · · · Z x x dx x n v n ( x , x , · · · , x n ) , (101) U ( x, x ) = − ℓ ∞ X n =1 Z xx dx n Z x n x dx n − · · · Z x x dx x x n v n ( x , x , · · · , x n ) , (102) U ( x, x ) = ℓ ∞ X n =1 Z xx dx n Z x n x dx n − · · · Z x x dx v n ( x , x , · · · , x n ) , (103) U ( x, x ) = 1 + ∞ X n =1 Z xx dx n Z x n x dx n − · · · Z x x dx x v n ( x , x , · · · , x n ) . (104) If η ( x ) := σ X ( x ) − , the η ( x )-pseudo-adjoint of H ( x ), which is defined by H ( x ) ♯ := η ( x ) − H ( x ) † η ( x ),commutes with H ( x ). Therefore, H ( x ) is η ( x )-pseudo-normal. For real-valued potentials, H ( x ) ♯ = − H ( x ), i.e., i H ( x ) is η ( x )-pseudo-Hermitian [18]. M ij of the zero-energy transfer matrix M bysubstituting (101) – (104) in M ij = lim x ± →±∞ U ij ( x + , x − ) . (105)If, for j ∈ { , } , we use a j ± and b j ± to label the parameters a ± and b ± giving the asymptoticbehavior of the solutions φ j of the zero-energy Schr¨odinger equation of Sec. 3, we find a − = b − = 1 , b − = a − = 0 , a j + = a j , b j + = b j , (106)where we have employed (68), (69), and (89). In light of (95) and (106), a = M , b = M , a = M , b = M . (107)These equations together with (75) – (77), (101) – (104), and (105) allow for a perturbative cal-culation of the low-energy expansions of the reflection and transmission amplitudes of any real orcomplex exponentially decaying potential up to and including quadratic terms in powers of kℓ andto arbitrary order of perturbation. We can also calculate the cubic term contributing to the trans-mission amplitude of the potential, if we can determine the coefficient g appearing in (77). In viewof (71) – (73), this requires the calculation of φ ( x ), φ ( x ), and φ ( x ) − xφ ′ ( x ) for all x ∈ R .We can use the identity Ψ ( x ; 0) = U ( x, x ) Ψ ( x ; 0) together with (92) to show that everysolution φ of the zero-energy Schr¨odinger equation (9) fulfills φ ( x ) = (cid:2) U ( x, x ) + ℓ − xU ( x, x ) (cid:3) [ φ ( x ) − x φ ′ ( x )] +[ ℓ U ( x, x ) + x U ( x, x )] φ ′ ( x ) , (108) φ ( x ) − xφ ′ ( x ) = U ( x, x )[ φ ( x ) − x φ ′ ( x )] + ℓ U ( x, x ) φ ′ ( x ) . (109)In particular, setting φ = φ j in these equations, taking the limit x → −∞ in the resultingexpressions, and making use of (68), we have φ ( x ) = U ( x, −∞ ) + ℓ − x U ( x, −∞ ) , (110) φ ( x ) = U ( x, −∞ ) + ℓ − x U ( x, −∞ ) , (111) φ ( x ) − xφ ′ ( x ) = U ( x, −∞ ) , (112)where U ij ( x, −∞ ) := lim x →−∞ U ij ( x, x ). Substituting (110) – (112) in (71) – (73) and employing(101) – (104), we can compute g and determine the term of order ( kℓ ) in the low-energy expansionof the transmission amplitude (77).Another consequence of (107) is the identification of the condition, b = 0, for the existence ofzero-energy resonances with M = 0. This proves the following theorem. Theorem 2:
Let v : R → C be an exponentially decaying potential and M ij be the entriesof its zero-energy transfer matrix. Then v has a zero-energy resonance if and only if M = 0.The delta-function potential, v ( x ) := z δ ( x − a ) , (113)with a possibly complex coupling constant z and center a ∈ R , provides a simple example which wecan use to check the validity of our approach to low-energy scattering. To do this, we first note that15he coupling constant z has the dimension of inverse of length. In the absence of any other relevantlength scale for this potential, we identify ℓ with | z | − , i.e., set ℓ := | z | − . This implies z = e iζ ℓ , (114)where ζ is a real number. The treatment of the scattering problem for the delta-function potential(113) with a real coupling constant is a standard textbook problem. Letting the coupling constanttake a complex value turns out not to cause any complications [30].The quickest way of solving the scattering problem for the delta-function potential (113) is todetermine its transfer matrix through the use of (21). For this potential, the matrix Hamiltonian(16) takes the form H ( x ; k ) = z k δ ( x − a ) e − ika σ K e ika σ . (115)This implies H ( x ; k ) H ( x ; k ) = , the Dyson series (21) for the transfer matrix terminates, and wefind M ( x ) = I − i z k e − ika σ K e ika σ = (cid:20) − i z / k − i z e − ika / ki z e ika / k i z / k (cid:21) . (116)Using this equation to read off the entries of M ( k ) and substituting them in (6), we obtain R l ( k ) = − z e ika z − ik = − e i ˆ a kℓ − ie − iζ kℓ = − − i (ˆ a + e − iζ ) kℓ + 2(ˆ a + 2ˆ ae − iζ + 2 e − iζ )( kℓ ) + O ( k ) , (117) R r ( k ) = − z e − ika z − ik = − e − i ˆ a kℓ − ie − iζ kℓ = − i (ˆ a − e − iζ ) kℓ + 2(ˆ a − ae − iζ + 2 e − iζ )( kℓ ) + O ( k ) , (118) T ( k ) = − ik z − ik = − ikℓ e − iζ − ie − iζ kℓ = − i e − iζ kℓ + 4 e − iζ ( kℓ ) + 8 ie − iζ ( kℓ ) + O ( k ) , (119)where ˆ a := a/ℓ .To compare (117) – (119) with the outcome of the application of (75) – (80) for the delta-functionpotential, we need to compute the coefficients a j , b j , and g .We can determine a j and b j using (107) provided that we calculate the zero-energy transfermatrix M . First, we observe that according to (93) and (114), H ( x ; 0) = − ia z δ ( x − a ) X ( a ) K X ( a ) − . Again because H ( x ; 0) H ( x ; 0) = , the Dyson series (97) terminates, and (96) and (107) give M = I − a z X ( a ) K X ( a ) − = I − ˆ ae iζ X ( a ) K X ( a ) − = (cid:20) − ˆ a e iζ − ˆ a e iζ e iζ a e iζ (cid:21) , (120) a = 1 − ˆ a e iζ , b = e iζ , a = − ˆ a e iζ , b = 1 + ˆ a e iζ . (121) The parameter a , which also has the dimension of length is not an admissible length scale, because its valuedepends on the choice of the origin of our coordinates.
16n particular, b = 0 and the low-energy series expansions for the reflection and transmissionamplitudes are given by (75) – (77).The calculation of g is more involved. First, we observe that, because for the delta-functionpotentian (113), v n ( x , x , · · · , x n ) = 0 for n ≥
1, Eqs. (101) – (104) give U ( x, −∞ ) = 1 − ˆ a e iζ θ ( x − a ) , U ( x, −∞ ) = − ˆ a e iζ θ ( x − a ) ,U ( x, −∞ ) = e iζ θ ( x − a ) , U ( x, −∞ ) = 1 + ˆ a e iζ θ ( x − a ) . Inserting these relations in (110) – (112), we find φ ( x ) = 1 + (cid:16) xℓ − ˆ a (cid:17) e iζ θ ( x − a ) ,φ ( x ) = xℓ + ˆ a (cid:16) xℓ − ˆ a (cid:17) e iζ θ ( x − a ) ,φ ( x ) − xφ ′ ( x ) = 1 − ˆ a e iζ θ ( x − a ) . These equations together with (71) – (73) imply g = ˆ a e iζ . Substituting this equation and (121)in (75) – (77) we recover (117) – (119). This provides a nontrivial check on the correctness of ourcalculations. Consider the Schr¨odinger equation in the half-line, − ∂ x ψ ( x ; k ) + V ( x ) ψ ( x ; k ) = k ψ ( x ; k ) , x ∈ R + , (122)where V : [0 , ∞ ) → C is a short-range potential, i.e., for x → + ∞ , |V ( x ) | tends to zero faster than1 /x . This implies that for every solution ψ ( x ; k ) of (122), there correspond coefficient functions A ( k ) and B ( k ) such that ψ ( x ; k ) → A ( k ) e ikx + B ( k ) e − ikx for x → + ∞ . (123)The Schr¨odinger equation (122) defines a scattering problem, if we impose a boundary condition ofthe form, α ( k ) ψ (0; k ) + k − β ( k ) ∂ x ψ (0; k ) = 0 , (124)where α ( k ) and β ( k ) are real- or complex-valued functions satisfying | α ( k ) | + | β ( k ) | 6 = 0. Here bythe scattering problem we mean the problem of determining the reflection amplitude, R ( k ) := A ( k ) B ( k ) . (125)Ref. [31] outlines a simple mapping of the scattering problem given by (122) and (124) on thehalf-line to a scattering problem defined by the Schr¨odinger equation (1) for the potential, v ( x ) := (cid:26) V ( x ) for x ≥ , x < , (126)17n the full line. This mapping allows for relating the reflection amplitude R ( k ) to the entries M ij ( k )of the transfer matrix (alternatively the reflection and transmission amplitudes, R l/r ( k ) and T ( k ))of the potential (126) according to R ( k ) = M ( k ) − γ ( k ) M ( k ) M ( k ) − γ ( k ) M ( k ) = R r ( k ) − T ( k ) R l ( k ) + γ ( k ) , (127)where γ ( k ) := α ( k ) + iβ ( k ) α ( k ) − iβ ( k ) . For the well-known Dirichlet and Neumann boundary conditions, which correspond to setting β ( k ) = α ( k ) − α ( k ) = β ( k ) − γ ( k ) = 1 and γ ( k ) = − V is an exponentially decaying potential, the same holds for v , and we can use (127) todetermine the low-energy expansion of the reflection amplitude R ( k ) in terms of the low-energyexpansion of the transfer matrix of v which we have derived in Sec. 3. In general, this requires theknowledge of the function γ ( k ).For situations where γ is a constant, we can choose α and β to be constant as well. In this case,we can read off the expressions for M ij ( k ) from (74) and substitute them in (127) to obtain thefollowing low-energy series expansion for R ( k ).For β = 0 : R ( k ) = O ( k ) for b = 0 , − − i a kℓ b + O ( k ) for b = 0 . (128)For β = 0 : R ( k ) = − − i a kℓ b + 2( a + iρ )( kℓ ) b + O ( k ) for b = 0 , − ρ b − iρ b + i − i b ( ρ a b + g ) kℓ ( ρ b + i ) + O ( k ) for b = 0 . (129)Here ρ := α/β , and a j and b j are the constants given by (69) provided that we identify φ j with thesolutions of the zero-energy Schr¨odinger equation (9) for the potential (126) that fulfill (68). Wecan use the results of Sec. 4 to calculate a j , b j , and g .According to (128), when we impose the Dirichlelt boundary condition at x = 0, i.e., set β = 0,the reflection amplitude R ( k ) displays different low-energy behavior for b = 0 and b = 0. Wecan trace the root of this phenomenon to the existence of a zero-energy resonance. By defini-tion, for the scattering problem defined by the Schr¨odinger equation (122) and Dirichlet boundarycondition ψ (0; k ) = 0, a zero-energy resonance arises when the Jost solution (7) of (1) satisfieslim k → ψ + (0; k ) = 0, [1]. Because, in view of (7) and (74),lim k → ψ + (0; k ) = lim k → [ M ( k ) − M ( k )] = b , this happens provided that b = 0. In view of (107), this proves the following theorem. Theorem 3:
Let V : [0 , ∞ ) → C be an exponentially decaying potential, v : R → C be itstrivial extension to R that is given by (126), φ be the solution of the zero-energy Schr¨odingerequation (9) satisfying φ (0) = 0 and φ ′ (0) = ℓ or some ℓ ∈ R + , and M ij be the entries ofthe zero-energy transfer matrix for v . Then the following assertions are equivalent.18. The scattering problem defined by the Schr¨odinger equation (122) on the half-line andthe Dirichlet boundary condition, ψ (0; k ) = 0, yields a zero-energy resonance.2. lim x → + ∞ φ ′ ( x ) = 0.3. M = 0.For the scattering problems in the half-line involving boundary conditions at x = 0 other thanDirichlet’s, we can identify the zero-energy resonances with situations where R ( k ) develops a dis-continuity at k = 0. In view of (129) this occurs when b = 0. This establishes the followingtheorem. Theorem 4:
Let V : [0 , ∞ ) → C be an exponentially decaying potential, v : R → C be itstrivial extension to R that is given by (126), φ be the solution of the zero-energy Schr¨odingerequation (9) satisfying φ (0) = 1 and φ ′ (0) = 0, and M ij be the entries of the zero-energytransfer matrix for v . Then the following assertions are equivalent.1. The scattering problem defined by the Schr¨odinger equation (122) on the half-line and aboundary condition of the type (124) with β = 0 gives rise to a zero-energy resonance.2. lim x → + ∞ φ ′ ( x ) = 0.3. M = 0.For the delta-function potential V ( x ) = z δ ( x − a ) with a >
0, the potential (126) coincideswith (113). Therefore, the coefficients a j and b j entering (128) and (129) are given by (121). Usingthese equations we obtain the following low-energy expression for the reflection amplitude of thedelta-function potential in the half-line.For β = 0 : R ( k ) = O ( k ) for z = − /a, − i ˆ a kℓ ˆ a + e − iζ + O ( k ) for z = − /a. For β = 0 : R ( k ) = − i (ˆ a − e − iζ ) kℓ + 2 (cid:2) (ˆ a − e − iζ ) + iρ e − iζ (cid:3) ( kℓ ) + O ( k ) . These formulas are in perfect agreement with the exact expression for R ( k ) that we obtain bysubstituting (117) – (119) in (127), namely R ( k ) = z (1 − γ e − iak ) + 2 ik z ( γ − e iak ) − iγk = 1 − γ e − i ˆ a kℓ + 2 ie − iζ kℓγ − e i ˆ a kℓ − iγe − iζ kℓ . Unlike the delta-function potential on the full line, the delta-function potential in the half-line with β = 0 (Dirichlet boundary condition) can support a zero-energy resonance. In view ofTheorem 3 and (121), this happens for ˆ ae iζ = −
1, which is equivalent to z = − /a .In contrast to the delta-function potential, the barrier potential defined on the half-line by V ( x ) := (cid:26) z for x ∈ [ a, a + L ] , x / ∈ [ a, a + L ] , a > , can develop zero-energy resonances for both β = 0 and β = 0. These are respectively characterizedby setting b and b equal to zero. In particular, for β = 0, the zero-energy resonances of the19arrier potential on the half line coincide with the ones we found for the same potential placed onthe full line. For β = 0, we can use (84) and (85) to identify the condition, b = 0, for the existenceof zero-energy resonances with the solutions of the transcendental equation tanh( √ z ) + √ z /a = 0.Because a is real and positive, this equation has nonzero solutions z that are real, negative, andinfinite in number. Writing the time-independent Schr¨odinger equation as the time-dependent Schr¨odinger equation fora two-level quantum system has a long history of applications in theoretical physics [32]. For a short-range potential there is a particular way of setting up this correspondence that allows for expressingthe transfer matrix of the potential in terms of the time-evolution operator for the two-level system.This leads to a dynamical formulation of stationary scattering theory in one dimensions [16, 17]with a number of interesting and useful applications [17, 20, 21] and generalizations [22, 23, 24].In this article, we have used this formulation to develop an approach to low-energy scattering forexponentially decaying real or complex potentials in one dimension. This subject was throughlyexamined for real potentials in the 1980’s by mathematicians [11, 12]. But the results employ certaintechnical tools of functional analysis which makes them extremely difficult to follow by physicistsinterested in performing low-energy scattering calculations.Our approach provides a simple iterative method of determining the coefficients of the low-energy series expansions for the transfer matrix and the reflection and transmission amplitudesof the potential. These coefficients depend on a pair of solutions of the zero-energy Schr¨odingerequation. We have introduced a transfer matrix M and a corresponding quantum dynamics thatyields series expansions for the coefficients of the low-energy series of the scattering data. Thistransfer matrix enjoys all the useful properties of the standard transfer matrix M for the potential.In particular, it satisfies the same composition property. Furthermore, the zero-energy resonancesof the potential turn out to be given by the zeros of M entry of M .Our method also applies to potential scattering defined on the half-line [0 , ∞ ). In this case thescattering problem is specified by the choice of a potential V in the half-line and a homogeneousboundary condition at x = 0. We have mapped this problem to a scattering problem defined bythe trivial extension of V to the full line and used this mapping to determine the low-energy seriesexpansion for the reflection amplitude of the corresponding scattering problem. This has led us toa useful characterization of the corresponding zero-energy resonances in terms of the zeros of theentries of the zero-energy transfer matrix M for the trivial extension of V .A class of low-energy scattering problems that we can easily apply our method to arises in thestudy of the scattering properties of traversable wormholes [33]. For simple models allowing foran analytic investigation, this turns out to be equivalent to the scattering problem for a scatteringpotential of the form v ( x ) = r ′′ ( x ) /r ( x ), where r ( x ) is a given real-valued function specifying thegeometry of the wormhole. For a large family of wormholes this function tends to first-degree polyno-mials, a ± + b ± x , as x → ±∞ . Because r ( x ) solves the zero-energy Schr¨odinger equation (9) for v ( x ),we can easily obtain the general solution of this equation as, φ ( x ) = c r ( x ) + c r ( x ) R x −∞ d ˜ x r (˜ x ) − ,where c and c are arbitrary constants [25]. This observation allows for the determination of the20olutions φ and φ that enter the expression for the coefficients of the low-energy series expansionof the scattering data. Because only low-energy waves can pass through the wormhole’s throat [33],the ability to compute low-energy scattering properties of the wormhole is of basic importance. Ourmethod can be directly applied to the wormhole scattering problems in which v ( x ) is a finite-rangeor exponentially decaying potential.For situations where v ( x ) belongs to L σ ( R ) for some σ ≥
2, which encompasses the scatteringof scalar waves by some well-known wormhole spacetimes, we can use our method to determinethe coefficients of the leading-order and next-to-leading-order terms in the low-energy asymptoticexpansion of the transmission amplitude [33]. This follows from the fact that whenever v belongsto L n + ǫ ) ( R ) for some n ∈ Z + and ǫ ∈ [0 , P n − m = − M ( m ) ( kℓ ) m + o ( k n − ǫ ), where o ( k µ ) is a matrix-valued function of k such that lim k → o ( k µ ) /k µ = , [11]. For these potentials, our method is capable of calculating thecoefficient matrices M ( m ) .In view of the recent progress in developing dynamical formulations of higher dimensional [22],long-range [24], and electromagnetic [23] scattering, a natural direction of further research is to tryto generalize the analysis of the present article to low-energy scattering problems for scalar andelectromagnetic waves in higher dimensions, and for long-range potentials. Acknowledgements
This work has been supported by the Scientific and Technological Research Council of Turkey(T ¨UB˙ITAK) in the framework of the project 120F061 and by Turkish Academy of Sciences (T ¨UBA).
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