紧流形上的Schrödinger算子的谱间隙估计

Abstract

Abstract:

Let $M$ be an $n$-dimensional compact Riemannian manifold with strictly convex boundary. Suppose that the Ricci curvature of $M$ is bounded below by $(n-1)K$ for some constant $K\geq0$ and the first eigenfunction $f_1$ of Dirichlet (or Robin) eigenvalue problem of a Schrödinger operator on $M$ is log-concave. Then we obtain a lower bound estimate of the gap between the first two Dirichlet (or Robin) eigenvalues of such Schrödinger operator. This generalizes a recent result by Andrews et al. ([4]) for Laplace operator on a bounded convex domain in $\mathbb{R} ^n$.

Key words: Schrödinger operator, Dirichlet eigenvalue, Robin eigenvalue, Spectral gap, Manifold with convex boundary, Ricci curvature

CLC Number:

O186.1

Yue He,Hailong Her. An Estimate of Spectral Gap for Schrödinger Operators on Compact Manifolds[J].Acta mathematica scientia,Series A, 2020, 40(2): 257-270.

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References 30

1	Andrews B . Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations. Surv Differ Geom, 2014, 19, 1- 47 doi: 10.4310/SDG.2014.v19.n1.a1
2	Andrews B , Clutterbuck J . Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable. J Differential Equations, 2009, 246 (11): 4268- 4283 doi: 10.1016/j.jde.2009.01.024
3	Andrews B , Clutterbuck J . Proof of the fundamental gap conjecture. J Amer Math Soc, 2011, 24 (3): 899- 916 doi: 10.1090/S0894-0347-2011-00699-1
4	Andrews B, Clutterbuck J, Hauer D. Non-concavity of Robin eigenfunctions. 2017, arXiv: 1711.02779v1
5	Ashbaugh M S. The Fundamental Gap.[2006-12-12] http://www.aimath.org/WWN/loweigenvalues/
6	Ashbaugh M S , Benguria R D . 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
7	Van den Berg M . On condensation in the free-boson gas and the spectrum of the Laplacian. J Statist Phys, 1983, 31 (3): 623- 637
8	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 doi: 10.1016/0022-1236(76)90004-5
9	Cheng S Y , Oden K . Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain. J Geom Anal, 1997, 7 (2): 217- 239 doi: 10.1007/BF02921721
10	He C X, Wei G F. Fundamental Gap of Convex Domains in the Spheres (with Appendix B by Zhang Q S). 2017, https://arxiv.org/abs/1705.11152v1
11	Horváth M . On the first two eigenvalues of Sturm-Liouville operators. Proc Amer Math Soc, 2003, 131 (4): 1215- 1224 doi: 10.1090/S0002-9939-02-06637-6
12	Lavine R . The eigenvalue gap for one-dimensional convex potentials. Proc Amer Math Soc, 1994, 121 (3): 815- 821 doi: 10.1090/S0002-9939-1994-1185270-4
13	Lee Y I , Wang A N . Estimate of λ₂-λ₁ on spheres. Chinese J Math, 1987, 15 (2): 95- 97
14	Ling J . A lower bound for the gap between the first two eigenvalues of Schrödinger operators on the convex domains in Sⁿ or Rⁿ. Michigan Mathematical Journal, 1993, 40 (2): 259- 270 doi: 10.1307/mmj/1029004752
15	Ling J . A lower bound of the first Dirichlet eigenvalue of a compact manifold with positive Ricci curvature. International J Math, 2006, 17 (5): 605- 617 doi: 10.1142/S0129167X06003631
16	Ling J . Lower bounds of the eigenvalues of compact manifolds with positive Ricci curvature. Ann of Global Anal and Geo, 2007, 31 (4): 385- 408 doi: 10.1007/s10455-006-9047-3
17	Ling J . Estimates on the lower bound of the first gap. Comm Anal Geometry, 2008, 16 (3): 539- 563 doi: 10.4310/CAG.2008.v16.n3.a3
18	Ling J, Lu Z Q. Bounds of Eigenvalues on Riemannian Manifolds//Bian B J, et al. Trends in Partial Differential Equations. Beijing: Higher Education Press, 2009: 241-264
19	Ni L . Estimates on the modulus of expansion for vector fields solving nonlinear equations. J Math Pures Appl, 2013, 99 (1): 1- 16
20	Oden K , Sung C J , Wang J . Spectral gap estimates on compact manifolds. Trans Amer Math Soc, 1999, 351 (9): 3533- 3548 doi: 10.1090/S0002-9947-99-02039-5
21	Schoen R , Yau S T . Lectures on Differential Geometry. Cambridge: International Press, 1994
22	Seto S, Wang L L, Wei G F. Sharp fundamental gap estimate on convex domains of spheres. 2016, http://arxiv.org/abs/1606.01212v2
23	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 Scuola Norm Sup Pisa Cl Sci, 1985, 12 (2): 319- 333
24	Yang D G . Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature. Pacific J Math, 1999, 190 (2): 383- 398 doi: 10.2140/pjm.1999.190.383
25	Yau S T. Nonlinear Analysis in Geometry. Geneva: L'Enseignement Mathématique, 1986
26	Yau S T. An Estimate of the Gap of the First two Eigenvalues in the Schrödinger Operator//Chang Alice, et al. Lectures on Partial Differential Equations. Somerville, MA: Int Press, 2003: 223-235
27	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
28	Zhong J Q , Yang H C . On the estimate of the first eigenvalue of a compact Riemannian manifold. Sci Sinica Ser A, 1984, 27 (12): 1265- 1273
29	Wolfson J . Eigenvalue gap theorems for a class of nonsymmetric elliptic operators on convex domains. Comm Partial Differential Equations, 2015, 40 (4): 601- 628 doi: 10.1080/03605302.2014.978014
30	何跃. 正Ricci曲率的紧流形上第一特征值下界的新估计. 数学物理学报, 2016, 36A (2): 215- 230 doi: 10.3969/j.issn.1003-3998.2016.02.002
	He Yue . New estimates of lower bound for the first eigenvalue on compact manifolds with positive Ricci curvature. Acta Math Sci, 2016, 36A (2): 215- 230 doi: 10.3969/j.issn.1003-3998.2016.02.002

[1]	He Yue. New Estimates of Lower Bound for the First Eigenvalue on Compact Manifolds with Positive Ricci Curvature [J]. Acta mathematica scientia,Series A, 2016, 36(2): 215-230.
[2]	Wang Peihe;Shen Chunli. A Compensatory Phenomenon to the Classical Sphere Theorem [J]. Acta mathematica scientia,Series A, 2008, 28(3): 465-471.
[3]	LI Jin-Tang. F harmonic Maps for Minimal Submanifoldswith Positive Ricci Curvature [J]. Acta mathematica scientia,Series A, 2004, 24(2): 152-156.
[4]	Hu Zejun. A Liouville Theorem for a Class of Nonlinear Elliptic Equations [J]. Acta mathematica scientia,Series A, 2000, 20(4): 474-479.

An Estimate of Spectral Gap for Schrödinger Operators on Compact Manifolds

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