Isoperimetric Inequalities for the Eigenvalues of Natural Schrödinger Operators on Surfaces
نویسنده
چکیده
This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first and the second eigenvalues are maximized by (round) spheres.
منابع مشابه
Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane
To determine the sharp constants for the one dimensional Lieb– Thirring inequalities with exponent γ ∈ (1/2, 3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be attained by potentials having only one bound state. Here we exhibit a connection between the Lieb–Thirring conjecture for γ = 1 and an isporimetric inequality fo...
متن کاملImproved energy bounds for Schrödinger operators
Given a potential V and the associated Schrödinger operator −∆+V , we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example V or V −1 enjoys suitable summability properties, the problem has a positive answer. In this paper we show that the corresponding isoperimetric-like inequalities can be improved by means of quantitative...
متن کاملEigenvalue Estimates for Schrödinger Operators on Metric Trees
We consider Schrödinger operators on regular metric trees and prove LiebThirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit.
متن کاملLieb-Thirring type inequalities for non-selfadjoint perturbations of magnetic Schrödinger operators
Let H := H0 + V and H⊥ := H0,⊥ + V be respectively perturbations of the free Schrödinger operators H0 on L2 ( R2d+1 ) and H0,⊥ on L2 ( R2d ) , d ≥ 1 with constant magnetic field of strength b > 0, and V is a complex relatively compact perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum ofH andH⊥. In particular, these estimates give a priori information on the distri...
متن کاملSchrödinger Operators with Many Bound States
Consider the Schrödinger operators H± = −d/dx ± V (x). We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of H±.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009