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Diamagnetic behavior of sums of Dirichlet eigenvalues. (English) Zbl 0957.35104

From the introduction: We prove a modest extension of the Li-Yau result to the magnetic Dirichlet Laplacian with a constant magnetic field. More specifically, for any domain \(U\subset\mathbb{R}^n\) of finite volume we consider the operator \[ H=\bigl(-i \nabla+A (x)\bigr)^2 \] on \(L^2(U)\) given by the closure of the form \[ (\psi,H \psi):= \int_U\biggl |\bigl(-i\nabla+ A(x)\bigr) \psi(x) \biggr|^2 d x\tag{1} \] on \(C_p^\infty(U)\). The one-form \(A(x)\) satisfies \(dA= B\), where \(B\) is a constant two-form. Our main result is the following:
Theorem. Let \(H\) be given by (1) where \(A\) generates a constant magnetic field. Then for any \(N\) orthonormal functions \(\{\varphi_j\}^N_{j=1}\) in the form domain of \(H\) we have the inequality \[ \sum^N_{j=1} (\varphi_j, H\varphi_j) \geq {n\over n+2}C_n N^{n+2\over n}|U|^{-{2\over n}}, \] with \(C_n\). The constant \(C_n\) is the best possible, \(C_n:=(2\pi)^2|B_n|^{-2/n}\), \(B_n\) is the unit ball in \(\mathbb{R}^n\) and \(|B_n|\) is its volume.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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