Sharp lower bound of spectral gap for Schrödinger operator and related results

Abstract

We give an easy proof of Andrews and Clutterbuck’s main results [J. Amer. Math. Soc., 2011, 24(3): 899−916], which gives both a sharp lower bound for the spectral gap of a Schrödinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at the same estimates by the ‘double coordinate’ approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral gap. Besides, we also get a new lower bound of spectral gap for a class of Schödinger operator.

Key words： Schrödinger operator; Laplace operator; spectral gap; ground state; strictly convex domain; diameter of domain

References

1 Andrews B, Clutterbuck J. Proof of the fundamental gap conjecture. J Amer Math Soc, 2011, 24(3): 899−916

2 Ashbaugh M S, Benguria R. 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, 105(2): 419−424

3 Bañelos R, Kröger P. Gradient estimates for the ground state Schrödinger eigenfunction and applications. Comm Math Phys, 2001, 224(2): 545−550

4 Bañuelos R, Méndez-Hernández P J. Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps. J Funct Anal, 2000, 176(2): 368−399

5 Benguria R D, Linde H, Loewe B. Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator. Bull Math Sci, 2012, 2(1): 1−56

6 Berg M van den. On condensation in the free-boson gas and the spectrum of the Laplacian. J Stat Phys, 1983, 31(3): 623−637

7 Brascamp H, Liep E. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave function, and with an application to diffusion equation. J Funct Anal, 1976, 22: 366−389

8 Chen M F. Eigenvalues, Inequalities, and Ergodic Theory. Probability and Its Applications. New York: Springer-Verlag, 2005

9 Chen M F. General estimate of the first eigenvalue on manifolds. Front Math China, 2011, 6(6): 1025−1043

10 Chen M F, Wang F Y. Application of coupling method to the first eigenvalue on manifold. Sci China Ser A, 1994, 37(1): 1−14

11 Chen M F, Wang F Y. Estimation of spectral gap for elliptic operators. Trans Amer Math Soc, 1997, 349(3): 1239−1267

12 Chen Y Z. Second Order Parabolic Equations. Beijing: Peking University Press, 2003 (in Chinese)

13 Clutterbuck J. Parabolic Equations with Continuous Initial Data. Ph D Thesis, Australian National University. http://arxiv.org/abs/math/0504455

14 Courant R, Hilbert D. Methods of Mathematical Physics, I. New York: Interscience, 1953

15 Davis B. On the spectral gap for fixed membranes. Ark Mat, 2001, 39(1): 65−74

16 Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 2001

17 Gong F Z, Li H Q, Luo D J. The fundamental gap conjecture: a probabilistic approach via the coupling by reflection. http://arxiv.org/abs/1303.2459

18 Gong F Z, Liu Yong, Liu Yuan, Luo D J. Spectral gaps of Schrödinger operators and diffusion operators on abstract Wiener spaces. J Funct Anal, 2014, 266(9): 5639−5675

19 Henrot A. Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Basel: Birkhäuser Verlag, 2006

20 Henrot A, Pierre M. Variation et optimisation de formes: Une analyse géométrique. Mathématiques & Applications (Berlin), Vol 48. Berlin: Springer, 2005

21 Horváth M. On the first two eigenvalues of Sturm-Liouville operators. Proc Amer Math Soc, 2003, 131(4): 1215−1224

22 Hsu E P. Stochastic Analysis on Manifolds. Graduate Studies in Mathematics, 38. Providence: American Mathematical Society, 2002

23 Hu Bei. Blow-up Theories for Semilinear Parabolic Equations. Lecture Notes in Math, Vol 2018. Heidelberg: Springer, 2011

24 Lavine R. The eigenvalue gap for one-dimensional convex potentials. Proc Amer Math Soc, 1994, 121(3): 815−821

25 Lee Ki-ahm, Vázquez J L. Parabolic approach to nonlinear elliptic eigenvalue problems. Adv Math, 2008, 219(6): 2006−2028

26 Li P. A lower bound for the first eigenvalue of the Laplacian on a compact Riemannian manifold. Indiana Univ Math J, 1979, 28: 1013−1019

27 Li P, Yau S T. On the Schrödinger equation and the eigenvalue problem. Comm Math Phys, 1983, 88(3): 309−318

28 Lieberman G M. Second Order Parabolic Differential Equations. River Edge: World Scientific Publishing Co, Inc, 1996

29 Ling J. Estimates on the lower bound of the first gap. Comm Anal Geom, 2008, 16(3): 539−563

30 Lu Z Q. Eigenvalue Gaps (I). UCI PDE Learning Seminar, 2011-May-26

31 Ni L. Estimates on the modulus of expansion for vector fields solving nonlinear equations. J Math Pure Appl, 2013, 99(1): 1−16

32 Qian Z M, Zhang H C, Zhu X P. Sharp spectral gap and Li-Yau’s estimate on Alexandrov spaces. Math Z, 2013, 273(3−4): 1175−1195

33 Schoen R, Yau S T. Lectures on Differential Geometry. Conference Proceedings and ecture Notes in Geometry and Topology, Vol I. Boston: International Press, 1994

34 Shi Y M, Zhang H C. Lower bounds for the first eigenvalue on compact manifolds. Chinese Ann Math Ser A, 2007, 28(6): 863−866

35 Singer I M, Wong B, Yau S T, Yau S S T. An estimate of the gap of the first two eigenvalues in the Schrödinger operator. Ann Sc Norm Super Pisa Cl Sci (4), 1985, 12(2): 319−333

36 Wolfson J. Eigenvalue gap theorems for a class of non symmetric elliptic operators on convex domains. http://arxiv.org/abs/1212.1669

37 Yau S T. Nonlinear Analysis in Geometry. Monographies de L’Enseignement Math, Vol 33. L’Enseignement Mathématique, Geneva, 1986. Série des Conférences de l’Union Mathématique Internationale, 8

38 Yau S T. An estimate of the gap of the first two eigenvalues in the Schrödinger operator. Lectures on Partial Differential Equations. New Stud Adv Math, Vol 2. Somerville: Int Press, 2003, 223−235

39 Yau S T. Gap of the first two eigenvalues of the Schrödinger operator with nonconvex potential. Mat Contemp, 2008, 35: 267−285

40 Yu Q H, Zhong J Q. Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator. Trans Amer Math Soc, 1986, 294(1): 341−349