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On the number of negative eigenvalues for a Schrödinger operator with magnetic field. (English) Zbl 0863.35071

Summary: We consider the Schrödinger operator with magnetic field \[ H=\sum^n_{j=1} \Biggl({1\over i} {\partial\over\partial x_j}-a_j\Biggr)^2+V\quad\text{in }\mathbb{R}^n. \] Under certain conditions on the magnetic field \(B=\text{curl }a\), we generalize the Fefferman-Phong estimates [C. L. Fefferman, Bull. Am. Math. Soc. 9, 129-206 (1983; Zbl 0526.35080)] on the number of negative eigenvalues for \(-\Delta+V\) to the operator \(H\). Upper and lower bounds are established. Our estimates incorporate the contribution from the magnetic field. The conditions on \(B\) in particular are satisfied if the magnetic potentials \(a_j(x)\) are polynomials.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35Q40 PDEs in connection with quantum mechanics

Citations:

Zbl 0526.35080
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References:

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