A second eigenvalue bound for the Dirichlet Schroedinger operator

Details A second eigenvalue bound for the Dirichlet Schroedinger operator A second eigenvalue bound for the Dirichlet Schroedinger operator

2005-11-09

arxiv.org/abs/math-ph/0511032v1

Physics

Mathematical Physics

Scientific paper

10.1007/s00220-006-0041-1

Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\star$, we prove that $\lambda_2(\Omega,V) \le \lambda_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $\lambda_1(\Omega,V) = \lambda_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $\lambda_2(B_R, V) / \lambda_1(B_R, V)$ 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.

Affiliated with

Also associated with

No associations

Rating

A second eigenvalue bound for the Dirichlet Schroedinger operator does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A second eigenvalue bound for the Dirichlet Schroedinger operator, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A second eigenvalue bound for the Dirichlet Schroedinger operator will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-386579