Proof of the fundamental gap conjecture

Andrews, Ben¹ ; Clutterbuck, Julie²

¹ Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia and Mathematical Sciences Center, Tsinghua University, Beijing, 100084, Peoples Republic of China and Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, 100190, Peoples Republic of China
² Centre for Mathematics and Its Applications, Australian National University, ACT 0200, Australia

Journal of the American Mathematical Society, Tome 24 (2011) no. 3, pp. 899-916

Résumé

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential. More generally, for an arbitrary smooth potential in higher dimensions, our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction, in terms of the diameter of the domain and a modulus of convexity for the potential.

Détail

DOI : 10.1090/S0894-0347-2011-00699-1

Bibliographie

[1] Andrews, Ben, Clutterbuck, Julie Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable J. Differential Equations 2009 4268 4283

[2] Andrews, Ben, Clutterbuck, Julie Time-interior gradient estimates for quasilinear parabolic equations Indiana Univ. Math. J. 2009 351 380

[3] Ashbaugh, Mark S. The Fundamental Gap 2006

[4] Ashbaugh, Mark S., Benguria, Rafael Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials Proc. Amer. Math. Soc. 1989 419 424

[5] Baã±Uelos, Rodrigo, Krã¶Ger, Pawel Gradient estimates for the ground state Schrödinger eigenfunction and applications Comm. Math. Phys. 2001 545 550

[6] Baã±Uelos, Rodrigo, Mã©Ndez-Hernã¡Ndez, Pedro J. Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps J. Funct. Anal. 2000 368 399

[7] Van Den Berg, M. On condensation in the free-boson gas and the spectrum of the Laplacian J. Statist. Phys. 1983 623 637

[8] Brascamp, Herm Jan, Lieb, Elliott H. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation J. Functional Analysis 1976 366 389

[9] Davis, Burgess On the spectral gap for fixed membranes Ark. Mat. 2001 65 74

[10] Gilbarg, David, Trudinger, Neil S. Elliptic partial differential equations of second order 2001

[11] Harrell, Evans M. Double wells Comm. Math. Phys. 1980 239 261

[12] Henrot, Antoine Extremum problems for eigenvalues of elliptic operators 2006

[13] Henrot, Antoine, Pierre, Michel Variation et optimisation de formes 2005

[14] Horvã¡Th, Miklã³S On the first two eigenvalues of Sturm-Liouville operators Proc. Amer. Math. Soc. 2003 1215 1224

[15] Korevaar, Nicholas J. Convex solutions to nonlinear elliptic and parabolic boundary value problems Indiana Univ. Math. J. 1983 603 614

[16] Krã¶Ger, Pawel On the spectral gap for compact manifolds J. Differential Geom. 1992 315 330

[17] Lavine, Richard The eigenvalue gap for one-dimensional convex potentials Proc. Amer. Math. Soc. 1994 815 821

[18] Li, Peter A lower bound for the first eigenvalue of the Laplacian on a compact manifold Indiana Univ. Math. J. 1979 1013 1019

[19] Li, Peter, Yau, Shing Tung Estimates of eigenvalues of a compact Riemannian manifold 1980 205 239

[20] Ling, Jun Estimates on the lower bound of the first gap Comm. Anal. Geom. 2008 539 563

[21] Payne, L. E., Weinberger, H. F. An optimal Poincaré inequality for convex domains Arch. Rational Mech. Anal. 1960

[22] Singer, I. M., Wong, Bun, Yau, Shing-Tung, Yau, Stephen S.-T. An estimate of the gap of the first two eigenvalues in the Schrödinger operator Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1985 319 333

[23] Smits, Robert G. Spectral gaps and rates to equilibrium for diffusions in convex domains Michigan Math. J. 1996 141 157

[24] Yau, S.-T. Gap of the first two eigenvalues of the Schrödinger operator with nonconvex potential Mat. Contemp. 2008 267 285

[25] Yau, Shing-Tung Nonlinear analysis in geometry 1986 54

[26] Yau, Shing-Tung An estimate of the gap of the first two eigenvalues in the Schrödinger operator 2003 223 235

[27] Yu, Qi Huang, Zhong, Jia Qing Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator Trans. Amer. Math. Soc. 1986 341 349

[28] Zhong, Jia Qing, Yang, Hong Cang On the estimate of the first eigenvalue of a compact Riemannian manifold Sci. Sinica Ser. A 1984 1265 1273