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.
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Communicated by B. Simon
R.B. was supported by FONDECYT project # 102-0844.
H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT.
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Benguria, R.D., Linde, H. A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator. Commun. Math. Phys. 267, 741–755 (2006). https://doi.org/10.1007/s00220-006-0041-1
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DOI: https://doi.org/10.1007/s00220-006-0041-1