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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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