Distribution theory for Schrödinger's integral equation

Abstract

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger's equation. This paper, in contrast, investigates the integral form of Schrödinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger's integral equation is viable tool for studying singular interactions in quantum mechanics.

23 pages

References in corpus (3)

• Potential theory, path integrals and the Laplacian of the indicator
• Distributional approach to point interactions in one-dimensional quantum mechanics
• On S-matrix of Schrodinger Operators with Non-Symmetric Zero-Range Potentials

Cited by in corpus (10)

• Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials
• Point interactions with bias potentials
• Schrödinger operators with singular rank-two perturbations and point interactions
• Two-parametric ${\delta }^{\prime }$-interactions: approximation by Schrödinger operators with localized rank-two perturbations
• Scattering data and bound states of a squeezed double-layer structure
• Origin of resonant tunneling through single-point barriers
• Wigner current in multidimensional quantum billiards
• Conditions for realizing one-point interactions from a multi-layer structure model
• Regularization of energy-dependent pointlike interactions in 1D quantum mechanics
• Bound States and Resonance Analysis of One-Dimensional Relativistic Parity-Symmetric Two Point Interactions