Approximations of Strongly Singular Evolution Equations

TL;DR

This paper develops methods to approximate solutions of evolution equations involving strongly singular self-adjoint operators, which are important in quantum physics and PDEs with distributional coefficients, by higher-order systems.

Contribution

It introduces explicit approximation schemes for evolution equations with strongly singular operators and proves their strong convergence in the context of quantum and PDE applications.

Findings

Approximation of singular operators by higher-order evolution systems.

Strong convergence of the approximated evolution operators.

Applicability to quantum field theory and PDEs with distributional coefficients.

Abstract

The problem of specification of self-adjoint operators corresponding to singular bilinear forms is very important for applications, such as quantum field theory and theory of partial differential equations with coefficient functions being distributions. In particular, the formal expression $-\Delta + g\delta({\bold x})$ corresponds to a non-trivial self-adjoint operator $\hat{H}$ in the space $L^2({\Bbb R}^d)$ only if $d\le 3$. For spaces of larger dimensions (this corresponds to the strongly singular case), the construction of $\hat{H}$ is much more complicated: first one should consider the space $L^2({\Bbb R}^d)$ as a subspace of a wider Pontriagin space, then one implicitly specifies $\hat{H}$. It is shown in this paper that Schrodinger, parabolic and hyperbolic equations containing the operator $\hat{H}$ can be approximated by explicitly defined systems of evolution equations of a larger order. The strong convergence of evolution operators taking the initial condition of the Cauchy problem to the solution of the Cauchy problem is proved.