Scattering in highly singular potentials (original) (raw)

Scattering from singular potentials in quantum mechanics

Journal of Physics A: Mathematical and General, 1998

In non-relativistic quantum mechanics, singular potentials in problems with spherical symmetry lead to a Schrödinger equation for stationary states with non-Fuchsian singularities both as r → 0 and as r → ∞. In the sixties, an analytic approach was developed for the investigation of scattering from such potentials, with emphasis on the polydromy of the wave function in the r-variable. The present paper extends those early results to an arbitrary number of spatial dimensions. The Hill-type equation which leads, in principle, to the evaluation of the polydromy parameter, is obtained from the Hill equation for a two-dimensional problem by means of a simple change of variables. The asymptotic forms of the wave function as r → 0 and as r → ∞ are also derived. The Darboux technique of intertwining operators is then applied to obtain an algorithm that makes it possible to solve the Schrödinger equation with a singular potential containing many negative powers of r, if the exact solution with even just one term is already known.

Scattering at Zero Energy for Attractive Homogeneous Potentials

Annales Henri Poincaré, 2009

We compute up to a compact term the zero-energy scattering matrix for a class of potentials asymptotically behaving as −γ|x| −μ with 0 < μ < 2 and γ > 0. It turns out to be the propagator for the wave equation on the sphere at time μπ 2−μ. 2 J. Dereziński and E. Skibsted Ann. Henri Poincaré see [4, Example 4.3]. Whence the deflection angle of such trajectories equals − μπ 2−μ. In particular, for attractive Coulomb potentials it equals −π, which corresponds to the well-known fact that in this case zero-energy orbits are parabolas (see [15, p. 126] for example). One can ask whether a similar behavior can be seen at the quantum level. Our analysis shows that indeed this is the case. Our main result is stated in terms of the unitary group e iθΛ generated by a certain self-adjoint operator Λ on L 2 (S d−1). The operator Λ is defined by setting ΛY = (l + d/2 − 1)Y if Y is a spherical harmonic of order l. e iθΛ can be called the propagator for the wave equation on the sphere. Note that for any θ, the distributional kernel of e iθΛ can be computed explicitly and its singularities appear at ω • ω = cos θ. This is expressed in the following fact [16]: Proposition 1.1. e iθΛ equals 1. c θ I, where I is the identity, if θ ∈ π2Z; 2. c θ P , where P is the parity operator (given by τ (ω) → τ (−ω)), if θ ∈ π(2Z+1); 3. the operator whose Schwartz kernel is of the form c θ (ω • ω − cos θ + i0) − d 2 if θ ∈]π2k, π(2k + 1)[ for some k ∈ Z; 4. the operator whose Schwartz kernel is of the form c θ (ω • ω − cos θ − i0) − d 2 if θ ∈]π(2k − 1), π2k[ for some k ∈ Z. We also remark that for all θ, the operator e iθΛ belongs to the class of Fourier integral operators of order 0 in the sense of Hörmander [11, 12]. Let us now briefly recall some points of the time-dependent scattering theory for Schrödinger operators. Set H 0 := − 1 2 Δ and H = H 0 + V (x). If the potential V (x) is short-range, following the standard formalism, we can define the usual scattering operator. In the long-range case the usual formalism does not apply. Nevertheless, one can use one of the modified formalisms, which leads to a modified scattering operator S.

On scattering from the one-dimensional multiple Dirac delta potentials

European Journal of Physics

In this paper, we propose a pedagogical presentation of the Lippmann-Schwinger equation as a powerful tool so as to obtain important scattering information. In particular, we consider a one dimensional system with a Schrödinger type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann-Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for arbitrary number of centers and study threshold anomaly for N = 2 and N = 4 cases. We also study further features like quantum metastable states like resonances, including their corresponding Gamow functions, and virtual or antibound states. The use of the Lippmann-Schwinger equation simplifies enormously our analysis and gives exact results for an arbitrary number of Dirac delta potential.

Scattering theory for the Schrödinger equation with repulsive potential

Journal de Mathématiques Pures et Appliquées, 2005

We consider the scattering theory for the Schrödinger equation with −∆ − |x| α as a reference Hamiltonian, for 0 < α ≤ 2, in any space dimension. We prove that when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of α, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity and describe its spectrum. Some results are generalized to the case where −|x| α is replaced by a general second order polynomial.

Scattering theory for Schrödinger operators with Bessel-type potentials

Journal für die reine und angewandte Mathematik (Crelles Journal), 2012

We show that for the Schrödinger operators on the half-axis with Bessel-type potentials κ(κ+1)/x 2 , κ ∈ [− 1 2 , 1 2), there exists a meaningful direct and inverse scattering theory. Several new phenomena not observed in the "classical case" of Faddeev-Marchenko potentials arise here; in particular, for κ = 0 the scattering function S takes two different values on the positive and negative semi-axes and is thus discontinuous both at the origin and at infinity.

Double-Delta Potentials: One Dimensional Scattering

International Journal of Theoretical Physics, 2011

The path is explored between one-dimensional scattering through Dirac-δ walls and one-dimensional quantum field theories defined on a finite length interval with Dirichlet boundary conditions. It is found that two δ's are related to the Casimir effect whereas two δ's plus the first transparent Pösch-Teller well arise in the context of the sine-Gordon kink fluctuations, both phenomena subjected to Dirichlet boundary conditions. One or two delta wells will be also explored in order to describe absorbent plates, even though the wells lead to non unitary Quantum Field Theories.

Inverse scattering at a fixed energy for long-range potentials

Inverse Problems and Imaging, 2007

In this paper we consider the inverse scattering problem at a fixed energy for the Schrödinger equation with a long-range potential in R d , d ≥ 3. We prove that the long-range part can be uniquely reconstructed from the leading forward singularity of the scattering amplitude at some positive energy.

On the solution of the integral equation of scattering for finite range potentials

2010

In the problem of the scattering of a particle in the presence of a finite range central potential, the integral equation for the lth partial wave is studied. By using a matrix method, the exact external solutions are expressed in terms of the Fredholm determinant ⌬, and the phase of ⌬ turns out to be equal to the phase shift. As an example, an array of delta-shell potentials is considered.

Asymptotics of the scattering coefficients for a generalized Schrödinger equation

Journal of Mathematical Physics, 1999

The generalized Schr odinger equation d 2 =dx 2 + F (k) = ikP (x) + Q(x)] is considered, where P and Q are integrable potentials with nite rst moments and F satis es certain conditions. The behavior of the scattering coe cients near zeros of F is analyzed. It is shown that in the so-called exceptional case, the values of the scattering coe cients at a zero of F may be a ected by P (x): The location of the k-values in the complex plane where the exceptional case can occur is studied. Some examples are provided to illustrate the theory.