Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 6 (2014), 1397-1420.

On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces

Gerasim Kokarev

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Abstract

A classical result by Cheng in 1976, improved later by Besson and Nadirashvili, says that the multiplicities of the eigenvalues of the Schrödinger operator (Δg+ν), where ν is C-smooth, on a compact Riemannian surface M are bounded in terms of the eigenvalue index and the genus of M. We prove that these multiplicity bounds hold for an Lp-potential ν, where p>1. We also discuss similar multiplicity bounds for Laplace eigenvalues on singular Riemannian surfaces.

Article information

Source
Anal. PDE, Volume 7, Number 6 (2014), 1397-1420.

Dates
Received: 19 November 2013
Revised: 5 May 2014
Accepted: 12 July 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731587

Digital Object Identifier
doi:10.2140/apde.2014.7.1397

Mathematical Reviews number (MathSciNet)
MR3270168

Zentralblatt MATH identifier
1301.58016

Subjects
Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 35P99: None of the above, but in this section 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

Keywords
Schrödinger equation eigenvalue multiplicity nodal set Riemannian surface

Citation

Kokarev, Gerasim. On multiplicity bounds for Schrödinger eigenvalues on Riemannian surfaces. Anal. PDE 7 (2014), no. 6, 1397--1420. doi:10.2140/apde.2014.7.1397. https://projecteuclid.org/euclid.apde/1513731587


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References

  • M. Aizenman and B. Simon, “Brownian motion and Harnack inequality for Schrödinger operators”, Comm. Pure Appl. Math. 35:2 (1982), 209–273.
  • A. D. Aleksandrov and V. A. Zalgaller, Intrinsic geometry of surfaces, Translations of Mathematical Monographs 15, American Mathematical Society, Providence, RI, 1967.
  • P. Bérard and D. Meyer, “Inégalités isopérimétriques et applications”, Ann. Sci. École Norm. Sup. $(4)$ 15:3 (1982), 513–541.
  • L. Bers, “Local behavior of solutions of general linear elliptic equations”, Comm. Pure Appl. Math. 8 (1955), 473–496.
  • G. Besson, “Sur la multiplicité de la première valeur propre des surfaces riemanniennes”, Ann. Inst. Fourier $($Grenoble$)$ 30:1 (1980), 109–128.
  • C. Carathéodory, “Über die Begrenzung einfach zusammenhängender Gebiete”, Math. Ann. 73:3 (1913), 323–370.
  • S. Chanillo and E. Sawyer, “Unique continuation for $\Delta+v$ and the C. Fefferman–Phong class”, Trans. Amer. Math. Soc. 318:1 (1990), 275–300.
  • J. Cheeger, “Spectral geometry of singular Riemannian spaces”, J. Differential Geom. 18:4 (1983), 575–657.
  • S. Y. Cheng, “Eigenfunctions and nodal sets”, Comment. Math. Helv. 51:1 (1976), 43–55.
  • R. Courant and D. Hilbert, Methods of mathematical physics, I, Interscience, New York, 1953.
  • D. B. A. Epstein, “Prime ends”, Proc. London Math. Soc. $(3)$ 42:3 (1981), 385–414.
  • P. Giblin, Graphs, surfaces and homology, 3rd ed., Cambridge University Press, 2010.
  • W. K. Hayman and P. B. Kennedy, Subharmonic functions, I, London Mathematical Society Monographs 9, Academic Press, New York, 1976.
  • M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof, “Local properties of solutions of Schrödinger equations”, Comm. Part. Diff. Eq. 17:3-4 (1992), 491–522.
  • M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, “Interior Hölder estimates for solutions of Schrödinger equations and the regularity of nodal sets”, Comm. Part. Diff. Eq. 20:7-8 (1995), 1241–1273.
  • M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, “On the multiplicity of eigenvalues of the Laplacian on surfaces”, Ann. Global Anal. Geom. 17:1 (1999), 43–48.
  • T. Hoffmann-Ostenhof, P. W. Michor, and N. Nadirashvili, “Bounds on the multiplicity of eigenvalues for fixed membranes”, Geom. Funct. Anal. 9:6 (1999), 1169–1188.
  • A. Huber, “Zum potentialtheoretischen Aspekt der Alexandrowschen Flächentheorie”, Comment. Math. Helv. 34 (1960), 99–126.
  • M. Karpukhin, G. Kokarev, and I. Polterovich, “Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces”, preprint, 2013. To appear in Ann. Inst. Fourier.
  • T. Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften 132, Springer, New York, 1976.
  • G. Kokarev, “Variational aspects of Laplace eigenvalues on Riemannian surfaces”, Adv. Math. 258 (2014), 191–239.
  • G. Kokarev, “Eigenvalue problems on Alexandrov surfaces of bounded integral curvature”. In preparation.
  • V. G. Maz'ja, Sobolev spaces, Springer, Berlin, 1985.
  • J. Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies 160, Princeton University Press, 2006.
  • N. S. Nadirashvili, “Multiple eigenvalues of the Laplace operator”, Mat. Sb. $($N.S.$)$ 133(175):2 (1987), 223–237, 272. In Russian; translated in Math. USSR-Sb. 61:1 (1988), 225–238.
  • Y. G. Reshetnyak, “Isothermal coordinates on manifolds of bounded curvature, I and II”, Sibirsk. Mat. Zh. 1 (1960), 88–116 and 248–276. In Russian.
  • Y. G. Reshetnyak, “Two-dimensional manifolds of bounded curvature”, pp. 3–163 in Geometry, IV, edited by Y. G. Reshetnyak, Encyclopaedia Math. Sci. 70, Springer, Berlin, 1993.
  • E. T. Sawyer, “Unique continuation for Schrödinger operators in dimension three or less”, Ann. Inst. Fourier $($Grenoble$)$ 34:3 (1984), 189–200.
  • B. Sévennec, “Multiplicity of the second Schrödinger eigenvalue on closed surfaces”, Math. Ann. 324:1 (2002), 195–211.
  • B. Simon, “Schrödinger semigroups”, Bull. Amer. Math. Soc. $($N.S.$)$ 7:3 (1982), 447–526.
  • M. Troyanov, “Les surfaces a courbure intégrale bornée au sens d'Alexandrov”, preprint, 2009.
  • Y. Colin de Verdière, “Sur la multiplicité de la première valeur propre non nulle du laplacien”, Comment. Math. Helv. 61:2 (1986), 254–270.
  • Y. Colin de Verdière, “Construction de laplaciens dont une partie finie du spectre est donnée”, Ann. Sci. École Norm. Sup. $(4)$ 20:4 (1987), 599–615.