Scattering and delay time for 1D asymmetric potentials: the step-linear and the step-exponential cases

We analyze the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential constituted by a step plus (i) a linear barrier or (ii) an exponential barrier. We solve the energy eigenvalue equation by means of the integral representation method, classifying the independent solutions as equivalence classes of homotopic paths in the complex plane. We discuss the structure of the bound states as function of the height U_0 of the step and we study the propagation of a sharp-peaked wave packet reflected by the barrier. For both the linear and the exponential barrier we provide an explicit formula for the delay time \tau(E) as a function of the peak energy E. We display the resonant behavior of \tau(E) at energies close to U_0. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, \tau(E) approaches the classical value for E ->\infty, thus diverging for the step-linear case and vanishing for the step-exponential one.


I. INTRODUCTION
In a previous paper 1 we have analyzed the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential formed by a step plus a harmonic barrier (the "step-harmonic" potential), by using the integral representation method. 2 We investigated the behavior of the discrete energy levels (as a function of the height of the step) and of the delay time τ of a wave packet coming from infinity and bouncing back on the harmonic barrier, as a function of the packet's peak energy and of the height U 0 of the step.
Among the convex or concave locally bounded symmetric and confining potentials the harmonic oscillator is a threshold, in that it gives rise to classical isochronous oscillations and thus to evenly spaced energy levels when quantized. 3,4 In our quantum mechanical step variant of the problem we recover both these features in the limit in which the step height U 0 → ∞, and the potential reduces to the half-space harmonic oscillator. Then, it is conceivable that the harmonic one is the only confining barrier which displays a constant nonvanishing delay τ in the limit of high energies. For steeper barriers we expect τ to vanish at high energies, while for milder ones we expect the delay to become infinite in this limit, in accordance with the corresponding classical situations. Similarly, we expect that, as U 0 → ∞, the spacing between two neighboring discrete levels tends to infinity in the former case and to zero in the latter.
In the present paper we corroborate this conjecture by analyzing two examples, the "steplinear" potential and the "step-exponential" one. Both these problems can be exactly solved by the use of the integral representation method.

II. THE STEP-LINEAR POTENTIAL
Consider the "step-linear" potential U 0 U(x) x Fig. 1: The "step-linear" potential.
where M and U 0 are positive parameters. If E denotes the energy of the particle and m its mass, the time-independent Schrödinger equation is This, for x < 0, writes It is convenient to define α := 2mM which allows us to recast Eq. (3) as follows: which is the Airy equation. 5,6 The general solution of Eq. (5) is where C and D are arbitrary integration constants, and the two linearly independent solutions Ai and Bi are expressed in appendix A in terms of the integral representation method.
Since Bi(x) → +∞ for x → +∞, in order for Eq. (6) to be an eigenfunction, we must set D = 0. Hence, we obtain For x > 0, the solution of Eq. (2) has the following from: where A, B and F are arbitrary integration constants and A. The case E < U 0 : bound states and level spacing The requirement of continuity of u(x) and of its first derivative in x = 0 is expressed by System (11) has a non trivial solution iff where β 0 := αU 0 /M. The energy levels are determined graphically by the intersections of the curves at the two sides of Eq. (12). An example is depicted in Fig. 2 for β 0 = 6. In the limit U 0 → ∞ the step of Eq. (1) becomes an infinite barrier. In this case, the energy levels correspond to the zeros β n of Ai(−β), the denominator of Eq. (12). As expected, these energy levels are the ones of the symmetric confining potential U(x) = M|x| corresponding to the odd eigenfunctions of the latter (appendix B).
It is instructive to study how their separation behaves for large energies. To this purpose, consider the asymptotic expansion of Ai(−β) for large values of β: 7 Ai(−β) where ζ := 2β 3/2 /3 and the coefficients of the series expansions are given by Note that expansion (13) can be proved by easy but rather tedious calculations starting from Eq. (A5).
The inversion of the asymptotic expansion (13) allows us to find, for large values of β, the following approximate solution of the equation Ai(−β) = 0 : 7 and Thus, at the leading order of Eq. (16) the approximate zeros have the form This is an excellent approximation to the zeros of Ai(−x), already from the first one (see Table I). For n → ∞ we get from Eq. (17), to leading order in 1/n, n β n (exact) β n (approximate) Relative Error  β n+1 − β n ∼ 8 3n Therefore, the level spacing decreases with n and vanishes in the limit. The spacing behavior n −1/3 is the threshold between concave and convex potentials.
B. The case E > U 0 : scattering and delay In this case, the (improper) eigenfunctions have the form The junction conditions in x = 0 are: Solving for the constants, the normalized (with respect to k) improper eigenfunctions are given by: where As expected, the continuous part of the spectrum (E > U 0 ) is simple. Note that From Eq. (21) a generic wave packet has the form Then, writing c(k) = |c(k)|e iγ(k) , ψ in and ψ refl take the following form: where If c(k) is sufficiently regular and non-vanishing only in a small neighborhood of somek, then ψ in and ψ refl represent wave packets which move according to the following equations of motion: 8,9 x for the "incoming" wave packet, and for the reflected "outgoing" one.
The solution thus built represents a particle of well defined momentump = k which approaches the origin from the right, interacts with the linear potential (at t = t 0 ) 12 , and is totally reflected. The phase shift results in a delay τ in the rebound, 13 caused by the interaction with the confining linear barrier. From Eqs. (4) and (9) it follows that whereβ := β(k). We compute τ from Eq. (23). Taking into account the Airy equation where we have suppressed the tilde on the packet peak energyβ.
We are interested in the behavior of the interaction time for large values of β, i.e. for incoming packets with high energy. To this end, we need the asymptotic expansion of where ζ := 2β 3/2 /3 and the coefficients d k are with c k given in Eq. (14).
Then, dividing the two asymptotic expansions of Ai ′ (−β) and Ai(−β) we obtain to leading Thus, Eq. (32) becomes Hence, reintroducing the physical variables, the high-energy behavior of the interaction time is which is exactly the time a classical particle arriving from infinity with energy E would spend in the x < 0 region. In Conversely, as U 0 increases, the lifetime of the resonance closest to the height of the step becomes progressively longer and then infinite when the resonance turns into the next bound state. This behavior is evident in Fig. 3, in which the first three plots correspond to values of β 0 for which there is only one bound state. In the successive three plots the resonance at β ≃ η 1 has disappeared, having turned into the second bound state.

III. THE STEP-EXPONENTIAL POTENTIAL
Consider the "step-exponential" potential x Fig. 4: The "step-exponential" potential.
where κ, σ and U 0 are positive parameters. For x < 0, it is convenient to introduce the following dimensionless quantities: in terms of which the time-independent Schrödinger equation writes as Setting ν 2 := −β, Eq. (40) can be cast in the form of a modified Bessel equation: 5,10 whose general solution is 5,10 where C and D are arbitrary integration constants and K ν and I ν are the modified Bessel The function I ν (z) diverges exponentially for z → +∞. 5 For this reason, in order for u(x) to be a proper (or improper) eigenfunction, we must set D = 0. Therefore Eq. (42) reduces to Like the Airy functions, the solutions of Eq. (41) can be conveniently studied by means of the integral representation method (see appendix C for details). A. The case E < U 0 : bound states The junction conditions in x = 0 give which, setting β 0 := 8m(U 0 + κ)σ 2 / 2 can be recast in the form: To study how the energy levels behave for large energies, we employ the following formula for the asymptotic behavior of the function K i √ β (α) for large β: 5 for β → ∞ . Note that expansion (47) can be proved starting from Eq. (C8). Therefore, the zeros of K i √ β (α), as a function of β, are asymptotically the solutions of the following equation: Solving for β n we obtain where W (x) is the Lambert function. 5 . Then, since W (x) ∼ log x − log log x for x → ∞, from Eq. (48) we have for large n that We see from Eq. (50) that the potential U(x) = κ e |x|/σ − 1 behaves for large x as an infinite square well whose width, up to inessential factors, grows as log n, an intuitive fact. Moreover, for n → ∞, proving thus that the level spacing diverges.

B. The case E > U 0 : scattering and delay
The unbound eigenstates have the form and the junction conditions are Therefore, the normalized (with respect to k) improper eigenfunctions are given by: where k = 2m(E − U 0 ) and . (55) Hence, Then, following the same argument adopted for the step-linear case, we obtain the following formula for the delay τ of the rebound of an incoming wavepacket with peak energy β: Using Eq. (47), we obtain for large values of β from which A comment is here in order. In general, taking the derivative of an asymptotic expansion with respect to the variable or a parameter may lead to wrong results. However, in our case this procedure can be justified using the integral representation of Eq. (C9) (we leave this as an exercise for the interested reader).
Thus, inserting Eq. (59) into Eq. (57) we obtain the asymptotic behavior of the delay time for large β's, namely or, in terms of the energy of the particle As expected, Eq. (61) coincides with the large energy value of the half period of the classical particle subjected to the confining potential U(x) = κ e |x|/σ − 1 . An entirely similar discussion can be applied to the step variant of any symmetric potential V (x) (V (x) = V (−x)) such that lim x→±∞ V (x) = +∞. Indeed the energy eigenvalue equation for V (x) has two linearly independent solutions u L (x) and v L (x) the first of which approaches zero very rapidly as x → −∞ whereas the second one diverges steadily without oscillating, and two linearly independent solutions u R (x) and v R (x) having a corresponding behavior for x → +∞. 8,11 Since the energy eigenvalues are the roots E n of the equation b(E) = 0. Since the potential is symmetric, these roots correspond to even and odd eigenfunctions alternatively, the ground state being even. However, in the general case the eigenvectors cannot be found explicitly.
Therefore, for example, no explicit formula is available in general for the delay time of the reflected packet in the corresponding step variant potential (62).
Appendix A: Derivation of the Airy functions by the integral representation method We look for a solution of Eq. (5) of the form where γ is a generic path in the complex plane and f is a function holomorphic in a suitable domain. Inserting Eq. (A1) into Eq. (5) we find Integrating by parts, we obtain: Therefore, E(y) is a solution of Eq. (5) if e ty f (t) ∂γ = 0 and Hence, a class of solutions of the Airy equation is of the form where γ is a suitable path for which the contour term vanishes.
The integrand of Eq. (A5) is an entire function; therefore, by virtue of Cauchy theorem, every closed path represents the trivial solution E(y) = 0.
Consider an unbounded path. In order for [e ty f (t)] ∂γ to vanish, we require the leading term in the exponent of f (t) (i.e. t 3 ) to have a negative real part. Therefore, the acceptable unbounded paths are those whose phase φ is confined to the regions π 6 < φ + 2 3 nπ < π 2 (n = 0, 1, 2). These possible paths are showed in Fig. 7, where the allowed sectors π 6 < φ + 2 3 nπ < π 2 (n = 0, 1, 2) are shaded. Paths with both endpoints in the same sector (e.g. Γ 4 in Fig. 7) can be closed at infinity using Jordan's Lemma; therefore, they correspond to the trivial solution. The only nontrivial paths are those which link different sectors. There are only 3 non-equivalent classes of such paths which we dub Γ 1 , Γ 2 and Γ 3 respectively (see Fig. 7). Taking into account Cauchy theorem, these paths satisfy the relation Γ 1 + Γ 2 = Γ 3 in the sense that the corresponding solutions are not independent. The conventional Airy functions Ai(z) and Bi(z) are the independent solutions of w ′′ (z) − zw(z) = 0 such that 5 Denoting by E (i) β (y) the solutions in Eq. (A5) corresponding to the paths Γ i (i = 1, 2), it is not difficult to show that We leave the details 6 to the interested reader (hint: Check the above expressions and their first derivatives in Eq. (A5) for y = β = 0. In this case the integrals E The eigenfunctions can be written as where C 1 and C 2 are integration constants to be determined by means of the junction conditions in x = 0. The latter can be resumed as follows: • If Ai(−β) = 0, then the requirement of continuity of u(x) in x = 0 implies that Moreover, the continuity of the derivative implies Ai ′ (−β) = 0. This condition determines the even eigenfunctions and the corresponding eigenvalues.
• If Ai(−β) = 0, then the continuity of the derivative implies C 1 = −C 2 . This condition determines the odd eigenfunctions and the corresponding eigenvalues. Up to a normalization, the solution is f (t) = sinh(t) 2ν . If ν is integer then f (t) is either entire (ν ≥ 0) or meromorphic (ν < 0), otherwise it has infinite branch points located at t n = inπ (n = 0, ±1, . . . ), see Fig. 9. In the latter instance, the usual procedure is to define a domain in which the function is holomorphic by cutting the t-plane and thus forbidding loops around the branch points. In Fig. 9 a convenient choice for the cuts is also shown.
Recalling that Re(z) > 0, the contour condition z ν+1 e −z cosh(t) f (t) sinh(t) ∂γ = 0 writes e −z cosh t sinh(t) 2ν+1 | ∂γ = 0 , There are 4 different classes of paths for which Eq. (C7) is satisfied and is well defined. The "paths zoology" is more complicated and rich than in the linear and in the harmonic case. 1 the two kinds of finite paths are proportional to the integral along the "fundamental" path [0, iπ]. Moreover, the integrals along any one of the infinite paths are linear combinations of the ones performed along the finite and semi-infinite paths.
In conclusion, two linear independent solutions of Eq. (C1) are K ν (z) = π 1/2 (z/2) ν Γ(ν + 1/2) ∞ 0 dx e −z cosh x sinh(x) 2ν , and I ν (z) = (z/2) ν π 1/2 Γ(ν + 1/2) π 0 dx e −z cos x sin(x) 2ν , where K ν and I ν correspond, respectively, to the integrals performed along the solid blue and the dashed red thick lines in Fig. 9 (a semi-infinite path and a finite one). It can be shown that both solutions (derived here for z > 0) can be analytically continued throughout the whole z-plane cut along the negative real axis. 2 Appendix D: The confining symmetric exponential potential Consider the confining symmetric potential (see Fig. 10) whose eigenfunctions can be written as U(x) x Fig. 10: The "exponential" well.
where C 1 and C 2 are integration constants to be determined by means of the junction conditions in x = 0. The latter can be resumed as follows (compare the analogous situation for the linear well in appendix B): • If K i √ β (α) = 0, then the requirement of continuity of u(x) in x = 0 implies that C 1 = C 2 . Moreover, the continuity of the derivative implies K ′ i √ β (α) = 0. This condition determines the even eigenfunctions and the corresponding eigenvalues.
• If K i