Sharp Boundary Trace Theory and Schr\"{o}dinger Operators on Bounded Lipschitz Domains

TL;DR

This paper develops a comprehensive boundary trace and spectral theory for Schrödinger operators on bounded Lipschitz domains, allowing for broader function regularity and explicit boundary conditions for all self-adjoint extensions.

Contribution

It introduces a sharp boundary trace theory accommodating limited regularity and extends the spectral analysis to all self-adjoint Schrödinger operator extensions on Lipschitz domains.

Findings

Extended Dirichlet and Neumann trace operators for Schrödinger operators.

Explicit boundary conditions for all self-adjoint extensions.

Applicable to variable coefficient Schrödinger operators on Lipschitz domains.

Abstract

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian on the functions in question which, nonetheless, works perfectly with the Dirichlet and Neumann realizations of the Schr\"odinger differential expression $-\Delta+V$. In turn, this boundary trace theory serves as a platform for developing a spectral theory for Schr\"odinger operators on bounded Lipschitz domains, along with their associated Weyl-Titchmarsh operators. Overall, this pushes the present state of knowledge a significant step further. For example, we succeed in extending the Dirichlet and Neumann trace operators in such a way that all self-adjoint extensions of a Schr\"odinger operator on a bounded Lipschitz domain may be described with explicit boundary conditions, thus providing a final answer to a problem that has been investigated for more than 60 years in the mathematical literature. Along the way, a number of other open problems are solved. The most general geometric and analytic setting in which the theory developed here yields satisfactory results is that of Lipschitz subdomains of Riemannian manifolds and for the corresponding Laplace-Beltrami operator (in place of the standard flat-space Laplacian). In particular, such an extension yields results for variable coefficient Schr\"odinger operators on bounded Lipschitz domains.