Dirichlet problems for stationary von Neumann-Landau wave equations

TL;DR

This paper studies the Dirichlet problem for a stationary von Neumann-Landau wave equation, extending quantum mechanics formalism, and proves existence and uniqueness of solutions using functional analysis in anti-inner product spaces.

Contribution

It introduces anti-inner product spaces to establish the existence and uniqueness of solutions for the Dirichlet problem of the stationary von Neumann-Landau wave equation.

Findings

Proved existence of solutions using functional-analytic methods.

Established uniqueness of solutions in the introduced framework.

Extended mathematical formalism of quantum mechanics to boundary value problems.

Abstract

It is known that von Neumann-Landau wave equation can present a mathematical formalism of motion of quantum mechanics, that is an extension of Schr\"{o}dinger's wave equation. In this paper, we concern with the Dirichlet problem of the stationary von Neumann-Landau wave equation: {(- \triangle_x + \triangle_y) \Phi (x, y) = 0, x, y \in \Omega, \Phi|_{\partial \Omega \times \partial \Omega} = f, where $\Omega$ is a bounded domain in $\mathbf{R}^n.$ By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods.