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Nodal sets for ground states of Schrödinger operators with zero magnetic field in non simply connected domains. (English) Zbl 1042.81012
Let \(\Omega \subset {\mathbb R}^2\) be a region with smooth boundary which is homeomorphic to a disk with \(k\) holes. Let \(H = (i\nabla +A)^2+V\) be magnetic Schrödinger operator on \(L^2(\Omega)\) with Neumann boundary conditions, smooth potential \(V\) and smooth magnetic vector potential \(A\) such that \(curl\;A = 0\). Let \(\Phi_n\) be circulation of \(A\) around \(n\)th hole. For the case \(\Phi_n = 1/2, n=1,...k\) the authors obtain characterisation of the nodal set and bounds on the multiplicity of the ground state. If \(k=1\) it is shown that first eigenvalue takes it highest value for \(\Phi_1 = 1/2\).

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35P99 Spectral theory and eigenvalue problems for partial differential equations
35J10 Schrödinger operator, Schrödinger equation
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