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On negative eigenvalues of the Schrodinger operator. (English) Zbl 1465.35322

Summary: In the paper, we consider the Schrödinger operator in an electric field, depending on the intensity of the magnetic field. Under certain constraints imposed on the electric potential, we prove its selfadjointness. It is proved that if the exact lower bound of the electric field is negative, then with an increase in the intensity of the magnetic field below the threshold of the essential spectrum, negative eigenvalues appear. We establish that as the magnetic field intensity increases, the number of negative eigenvalues of the Schrödinger operator increases.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
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