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 higherorder 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.

A phenomenon of splitting resonant-tunneling one-point interactions

“…(iii): Finally, the third type of resonant-tunneling point interactions can be realized on the same resonance set defined by Eq. (16), however, for this type we assume that the limit sin(kr) → 0 proceeds faster (as a function of l 1 , l 2 ) than in the asymptotic representation (18). The limit diagonal elements λ 11 and λ 22 are given in this case by the same formulae (19).…”

The Propagators for δ and δ′ Potentials With Time-Dependent Strengths

“…The δ ′ perturbation of free Hamiltonian H 0 = − d 2 dx 2 is defined as a limit of short range potentials in the distributional sense [5][6][7]. Although there are some controversial issues about δ ′ interactions (see e.g., [8][9][10][11]), they are also getting considerable amount of interest. The ambiguities about δ ′ interactions have been summarized in a very recent article [11], where the integral form of the Schrödinger equation for δ ′ potential has been studied based on the work of Kurasov [12].…”

Schrödinger Operators with Singular Rank-Two Perturbations and Point Interactions

“…We want to especially note the paper [4, 9-12, 28, 29] and the references therein. This special case has attracted much attention recently [7,8,24,33]. Many authors have dealt with finite rank perturbations and their relationship with the point interactions.…”