Sharp boundary trace theory and Schrödinger operators on bounded Lipschitz domains / Jussi Behrndt, Fritz Gesztesy, Marius Mitrea.

Format:

Book

Author/Creator:

Behrndt, Jussi, author.

Gesztesy, Fritz, 1953- author.

Mitrea, Marius, author.

Series:

Memoirs of the American Mathematical Society ; v. 1550.

Memoirs of the American Mathematical Society ; v. 1550

Language:

English

Subjects (All):

Functional analysis.

Physical Description:

v, 208 pages ; 26 cm

Place of Publication:

Providence, RI : American Mathematical Society, 2025.

Summary:

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 Schrödinger differential expression −Δ+V. In turn, this boundary trace theory serves as a platform for developing a spectral theory for Schrödinger 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 Schrödinger 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 Schrödinger operators on bounded Lipschitz domains.

Notes:

Includes bibliographical references (pages 201-208).

ISBN:

1470472678

9781470472672

OCLC:

1511789063