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- W2043669271 abstract "Let λ i (Ω,V) be the i th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $$Omega subset mathbb{R}^n$$ and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V *, we prove that λ2(Ω,V) ≤ λ2(S 1,V *). Here S 1 denotes the ball, centered at the origin, that satisfies the condition λ1(Ω,V)=λ1(S 1,V *). Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ2(B R , V) / λ1(B R , V) decreases when the radius R of the ball B R increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density." @default.
- W2043669271 created "2016-06-24" @default.
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- W2043669271 date "2006-05-23" @default.
- W2043669271 modified "2023-10-11" @default.
- W2043669271 title "A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator" @default.
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- W2043669271 doi "https://doi.org/10.1007/s00220-006-0041-1" @default.
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