Abstract
Let \({\Omega }\subset \mathbb{R} ^{n}\) be a strictly convex domain with smooth boundary and diameter D. The fundamental gap conjecture claims that if \(V:\bar {\Omega }\to \mathbb{R} \) is convex, then the spectral gap of the Schrödinger operator −Δ+V with Dirichlet boundary condition is greater than \(\frac {3\pi ^{2}}{D^{2}}\). Using analytic methods, Andrews and Clutterbuck recently proved in (J. Amer. Math. Soc. 24(3), 889–916 2011) a more general spectral gap comparison theorem which implies this conjecture. In the first part of the current work, we shall give a probabilistic proof of their result via the coupling by reflection of the diffusion processes. Moreover, we also present in the second part a simpler probabilistic proof of the original conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, B., Clutterbuck, J.: Proof of the fundamental gap conjecture. J. Amer. Math. Soc. 24(3), 899–916 (2011)
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. 105(2), 419–424 (1989)
Bakry, D., Emery, M.: Diffusions hypercontractives, Séminaires de Probabilités XIX. Springer Lect. Notes Math. 1123, 177–206 (1985)
Brascamp, H.J., Lieb, E.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. Funct. Anal. 22(4), 366–389 (1976)
Carlen, E.A.: Conservative diffusions. Commun. Math. Phys. 94, 293–315 (1984)
Chen, M.-F.: Eigenvalues, Inequalties, and Ergodic Theory. Probability and its Applications (New York). Springer–Verlag London, Ltd., London (2005)
Chen, M.-F., Li, S.-F.: Coupling methods for multidimensional diffusion processes. Ann. Probab. 17(1), 151–177 (1989)
Chen, M.-F., Wang, F.-Y.: Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A 37(1), 1–14 (1994)
Chen, M.-F., Wang, F.-Y.: Estimation of spectral gap for elliptic operators. Trans. Amer. Math. Soc. 349(3), 1239–1267 (1997)
Gong, F.-Z., Liu, Y., Liu, Y., Luo, D.-J: Spectral gaps of Schrödinger operators and diffusion operators on abstract Wiener space. J. Funct. Anal. 266, 5639–5675 (2014)
Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1061–1083 (1975)
He, Y.: Sharp lower bound of spectral gap for Schr¨odinger operator and related results. Front. Math. China. (2015). doi:10.1007/s11464-015-0455-1
Hsu, E.P.: Stochastic analysis on manifolds. Graduate Studies in Mathematics, 38. American Mathematical Society, Providence, RI (2002)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, 2nd. North-Holland Publishing Co. (1989)
Lee, Ki-ahm., Vázquez, J.L.: Parabolic approach to nonlinear elliptic eigenvalue problems. Adv. Math. 219, 2006–2028 (2008)
Lindvall, T., Rogers, L.C.G.: Coupling of multidimensional diffusions by reflection. Ann. Probab. 14(3), 860–872 (1986)
Lu, Z.-Q.: Eigenvalue gaps (I), UCI PDE Learning Seminar, May 26, 2011. http://www.math.uci.edu/~zlu/talks2011-UCI-PDE/uci-pde-seminar-1.pdf
Meyer, P.A., Zheng, W.A.: Construction de processus de Nelson réversibles. Séminaire de probabilités, XIX, 1983/84, 12–26, vol. 1123. Springer, Berlin (1985)
Van den Berg, M.: On condensation in the free-boson gas and the spectrum of the Laplacian. J. Statist. Phys. 31(3), 623–637 (1983)
Wang, F.-Y.: Logarithmic Sobolev inequalities: conditions and counterexamples. J. Oper. Theory 46(1), 183–197 (2001)
Wang, F.-Y.: Gradient estimates and the first Neumann eigenvalue on manifolds with boundary. Stoch. Process Appl. 115, 1475–1486 (2005)
Wang, F.-Y.: Second fundamental form and gradient of Neumann semigroups. J. Funct. Anal. 256, 3461–3469 (2009)
Yau, S.-T.: Nonlinear analysis in geometry. Monographies de L’Enseignement Mathématique, vol. 33, L’Enseignement Mathématique, Geneva, 1986. Série des Conférences de l’Union Mathématique Internationale, 8
Yosida, K.: Functional analysis. Sixth edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123. Springer-Verlag, Berlin-New York (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partly supported by the Key Laboratory of RCSDS, CAS (2008DP173182), NSFC (11021161) and 973 Project (2011CB808000).
Partly supported by the NSFC (11401403) and the Australian Research Council (ARC) grant (DP130101302).
Partly supported by the Key Laboratory of RCSDS, CAS (2008DP173182), NSFC (11101407) and AMSS (Y129161ZZ1).
Rights and permissions
About this article
Cite this article
Gong, F., Li, H. & Luo, D. A Probabilistic Proof of the Fundamental Gap Conjecture Via the Coupling by Reflection. Potential Anal 44, 423–442 (2016). https://doi.org/10.1007/s11118-015-9476-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-015-9476-3