Arkiv för Matematik

On the spectral gap for fixed membranes

Burgess Davis

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Abstract

The distance between the second and first eigenvalues for the Dirichlet Laplacian of a domain is called its (spectral) gap. We show that the gap of a convex planar domain D symmetric about both the x and y axes is no smaller than the gap of an oriented rectangle which contains D.

Article information

Source
Ark. Mat. Volume 39, Number 1 (2001), 65-74.

Dates
Received: 27 September 1999
Revised: 20 January 2000
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.afm/1485898709

Digital Object Identifier
doi:10.1007/BF02388791

Zentralblatt MATH identifier
1021.35040

Rights
2001 © Institut Mittag-Leffler

Citation

Davis, Burgess. On the spectral gap for fixed membranes. Ark. Mat. 39 (2001), no. 1, 65--74. doi:10.1007/BF02388791. http://projecteuclid.org/euclid.afm/1485898709.


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References

  • [AB] Ashbaugh, M. S. and Benguria, R., Optimal lower bounds for eigenvalue gaps for Schrödinger operators with symmetric single-well potentials and related results, in Maximum Principles and Eigenvalue Problems in Partial Differential Equations (Knoxville, Tenn., 1987) (Schaefer, P. W., ed.), pp. 134–145, Longman, Harlow, 1988.
  • [BM] Bañuelos, R. and Mendez, P., Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps, J. Funct. Anal. 176 (2000), 368–399.
  • [Be] van den Berg, M., On condensation in the free-boson gas and the spectrum of the Laplacian, J. Statist. Phys. 31 (1983), 623–637.
  • [Br] Breiman, L., Probability, Addison-Wesley, Reading, Mass., 1969.
  • [C] Cheng, S. Y., Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), 289–297.
  • [CFL] Courant, R., Friedrichs, K. and Lewy, H., On the partial difference equations of mathematical physics, Math. Ann. 100 (1928), 32–72.
  • [D] Davies, E. B.Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, 1989.
  • [DM] Davis, B. and McDonald, D., An elementary proof of the local central limit theorem, J. Theoret. Probab. 24 (1995), 693–701.
  • [DY] Dynkin, E. B. and Yuskevitch, A., Markov Processes, Theorems and Problems, Plenum Press, New York, 1969.
  • [KS] Kirsch, W. and Simon, B., Comparison theorems for the gap of Schrödinger operators, J. Funct. Anal. 73 (1987), 396–410.
  • [L] Ling, J., A lower bound for the gap between the first two eigenvalues of Schrödinger operators on convex domains in Sn or Rn, Michigan Math. J. 40 (1993), 259–270.
  • [Pa] Payne, L., On two conjectures in the fixed membrane eigenvalue problem, J. Appl. Math. Phys. 24 (1973), 720–729.
  • [Po] Pollard, D., Convergence of Stochastic Processes, Springer-Verlag, New York, 1984.
  • [Sa] Saloff-Coste, L., Precise estimates on the rate at which certain diffusions tend to equilibrium, Math. Z. 217 (1994), 641–677.
  • [SWYY] Singer, I. M., Wong, B., Yau, S. T. and 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. 12 (1985), 319–333.
  • [Sm] Smits, R., Spectral gaps and rates to equilibrium in convex domains, Michigan Math. J. 43 (1996), 141–157.
  • [YZ] Yu, Q. and Zhong, J. Q., Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator, Trans. Amer. Math. Soc. 294 (1986), 341–349.