## 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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### References:

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