Abstract
We consider the Schrödinger operator
where V is bounded from below and prove a lower bound on the first eigenvalue λ 1 in terms of sublevel estimates: if w V (y) = |{x ∈ [a, b] : V (x) ≤ y}|, then
The result is sharp up to a universal constant if {x ∈ [a, b] : V(x) ≤ y} is an interval for the value of y solving the minimization problem. An immediate application is as follows: let \({\Omega } \subset \mathbb {R}^{2}\) be a convex domain and let \(u:{\Omega } \rightarrow \mathbb {R}\) be the first eigenfunction of the Laplacian − Δ on Ω with Dirichlet boundary conditions on ∂Ω. We prove
which answers a question of van den Berg in the special case of two dimensions.
Similar content being viewed by others
References
Chiti, G.: A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators. J. Appl. Math. Phys. 33, 143–148 (1982)
Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. (2) 106, 93–100 (1977)
Grieser, D., Jerison, D.: The size of the first eigenfunction of a convex planar domain. J. Amer. Math. Soc. 11(1), 41–72 (1998)
Jerison, D.: The diameter of the first nodal line of a convex domain. Ann. Math. (2) 141(1), 1–33 (1995)
Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics, pp. 269–303. Princeton University Press (1976)
Lieb, E.: Bounds on the eigenvalues of the Laplace and Schrodinger operators. Bull. Amer. Math. Soc. 82(5), 751– 753 (1976)
Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)
Rozenblum, G.: Distribution of the discrete spectrum of singular differential operators. Dokl. Acad. Nauk SSSR 202, 1012–1015 (1972). Translation in Soviet Math. Dokl., 13 (1972), 245–249
van den Berg, M.: On the L ∞−Norm of the First Eigenfunction of the Dirichlet Laplacian. Poten. Anal. 13, 361–366 (2000)
Acknowledgements
The first and second authors gratefully acknowledge the Max Planck Institute for Mathematics, Bonn for providing ideal working conditions. The third author was partially supported by an AMS Simons Travel grant and INET Grant #INO15-00038.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Georgiev, B., Mukherjee, M. & Steinerberger, S. A Spectral Gap Estimate and Applications. Potential Anal 49, 635–645 (2018). https://doi.org/10.1007/s11118-017-9670-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-017-9670-6