Journal of Spectral Theory


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Volume 3, Issue 4, 2013, pp. 465–484
DOI: 10.4171/JST/51

Published online: 2013-10-23

Spectral estimates for Dirichlet Laplacians and Schrödinger operators on geometrically nontrivial cusps

Pavel Exner[1] and Diana Barseghyan[2]

(1) Doppler Institute for Mathematical Physics and Applied Mathematics, Prague, Czech Republic
(2) University of Ostrava, Czech Republic

The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.

Keywords: Dirichlet Laplacian, cusp-shaped region, Lieb–Thirring inequalities, bending and twisting

Exner Pavel, Barseghyan Diana: Spectral estimates for Dirichlet Laplacians and Schrödinger operators on geometrically nontrivial cusps. J. Spectr. Theory 3 (2013), 465-484. doi: 10.4171/JST/51