## Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2.(English)Zbl 1210.81034

Summary: The aim of this paper is to establish estimates of the lowest eigenvalue of the Neumann realization of $$(i\nabla + B{\mathbf A})^2$$ on an open bounded subset $$\Omega \subset \mathbb R^2$$ with smooth boundary as $$B$$ tends to infinity. We introduce a “magnetic” curvature mixing the curvature of $$\partial \Omega$$ and the normal derivative of the magnetic field and obtain an estimate analogous with the one of constant case. Actually, we give a precise estimate of the lowest eigenvalue in the case where the restriction of magnetic field to the boundary admits a unique minimum which is non degenerate. We also give an estimate of the third critical field in Ginzburg-Landau theory in the variable magnetic field case.

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35J10 Schrödinger operator, Schrödinger equation
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