Accurate estimates for magnetic bottles in connection with superconductivity

TL;DR

This paper provides a comprehensive asymptotic expansion for low-lying eigenvalues of the Schrödinger operator with magnetic field in generic domains, advancing understanding in superconductivity theory, especially in two dimensions.

Contribution

It establishes an all-orders asymptotic expansion for eigenvalues in generic domains with boundary curvature conditions, filling a key gap in semi-classical analysis related to superconductivity.

Findings

Asymptotic expansion to all orders for eigenvalues

Results apply to domains with boundary curvature having a unique maximum

Advances the mathematical understanding of magnetic bottles in superconductivity

Abstract

Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, many papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schr\"odinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg and Helffer-Morame and also Bauman-Phillips-Tang for the case of a disc. In the present paper we settle one important part of this question completely by proving an asymptotic expansion to all orders for low-lying eigenvalues for generic domains. The word `generic' means in this context that the curvature of the boundary of the domain has a unique non-degenerate maximum.