Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 2 (2014), 435-460.

Spectral estimates on the sphere

Jean Dolbeault, Maria Esteban, and Ari Laptev

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In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the d-dimensional unit sphere. These estimates depend on Lp norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semiclassical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space.

Article information

Anal. PDE, Volume 7, Number 2 (2014), 435-460.

Received: 7 January 2013
Accepted: 13 June 2013
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 81Q35: Quantum mechanics on special spaces: manifolds, fractals, graphs, etc.
Secondary: 47A75: Eigenvalue problems [See also 47J10, 49R05] 26D10: Inequalities involving derivatives and differential and integral operators 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 58E35: Variational inequalities (global problems) 81Q20: Semiclassical techniques, including WKB and Maslov methods

spectral problems partial differential operators on manifolds quantum theory estimation of eigenvalues Sobolev inequality interpolation Gagliardo–Nirenberg–Sobolev inequalities logarithmic Sobolev inequality Schrödinger operator ground state one bound state Keller–Lieb–Thirring inequality


Dolbeault, Jean; Esteban, Maria; Laptev, Ari. Spectral estimates on the sphere. Anal. PDE 7 (2014), no. 2, 435--460. doi:10.2140/apde.2014.7.435.

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  • M. Abramowitz and I. A. Stegun (editors), Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, US Government Printing Office, Washington, DC, 1964. Reprinted by Dover, New York, 1974.
  • T. Aubin, “Problèmes isopérimétriques et espaces de Sobolev”, J. Differential Geometry 11:4 (1976), 573–598.
  • D. Bakry, “Functional inequalities for Markov semigroups”, pp. 91–147 in Probability measures on groups: recent directions and trends (Mumbai, 2002), edited by S. G. Dani and P. Graczyk, Tata Inst. Fund. Res. 18, Narosa, New Delhi, 2006.
  • D. Bakry and M. Ledoux, “Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator”, Duke Math. J. 85:1 (1996), 253–270.
  • J. F. Barnes, “Appendix A: Numerical studies”, pp. 295–301 in [LiebThirring76?].
  • W. Beckner, “Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality”, Ann. of Math. $(2)$ 138:1 (1993), 213–242.
  • R. D. Benguria and M. Loss, “Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane”, pp. 53–61 in Partial differential equations and inverse problems (Santiago, 2003), edited by C. Conca et al., Contemp. Math. 362, Amer. Math. Soc., Providence, RI, 2004.
  • A. Bentaleb and S. Fahlaoui, “Integral inequalities related to the Tchebychev semigroup”, Semigroup Forum 79:3 (2009), 473–479.
  • A. Bentaleb and S. Fahlaoui, “A family of integral inequalities on the circle ${\mathbb S}^1$”, Proc. Japan Acad. Ser. A Math. Sci. 86:3 (2010), 55–59.
  • M.-F. Bidaut-Véron and L. Véron, “Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations”, Invent. Math. 106:3 (1991), 489–539.
  • G. A. Bliss, “An integral inequality”, J. London Math. Soc. 5:1 (1930), 40–46.
  • C. Brouttelande, “The best-constant problem for a family of Gagliardo–Nirenberg inequalities on a compact Riemannian manifold”, Proc. Edinb. Math. Soc. $(2)$ 46:1 (2003), 117–146.
  • C. Brouttelande, “On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds”, Bull. Sci. Math. 127:4 (2003), 292–312.
  • J. Dolbeault and M. J. Esteban, “Extremal functions for Caffarelli–Kohn–Nirenberg and logarithmic Hardy inequalities”, Proc. Roy. Soc. Edinburgh Sect. A 142:4 (2012), 745–767.
  • J. Dolbeault, P. Felmer, M. Loss, and E. Paturel, “Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems”, J. Funct. Anal. 238:1 (2006), 193–220.
  • J. Dolbeault, M. J. Esteban, M. Kowalczyk, and M. Loss, “Sharp interpolation inequalities on the sphere: new methods and consequences”, Chin. Ann. Math. Ser. B 34:1 (2013), 99–112.
  • P. Federbush, “Partially alternate derivation of a result of Nelson”, J. Math. Phys. 10 (1969), 50–52.
  • P. Funk, “Beiträge zur Theorie der Kugelfunktionen”, Mathematische Annalen 77:1 (1915), 136–152.
  • G. H. Hardy and J. E. Littlewood, “Notes on the theory of series, XII: On certain inequalities connected with the calculus of variations”, J. London Math. Soc. 5:1 (1930), 34–39.
  • E. Hecke, “Über orthogonal-invariante Integralgleichungen”, Math. Ann. 78:1 (1917), 398–404.
  • A. A. Ilyin, “Lieb–Thirring inequalities on the $N$-sphere and in the plane, and some applications”, Proc. London Math. Soc. $(3)$ 67:1 (1993), 159–182.
  • A. A. Ilyin, “Lieb–Thirring inequalities on some manifolds”, J. Spectr. Theory 2:1 (2012), 57–78.
  • J. B. Keller, “Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation”, J. Math. Phys. 2 (1961), 262–266.
  • M. Ledoux, “The geometry of Markov diffusion generators”, Ann. Fac. Sci. Toulouse Math. $(6)$ 9:2 (2000), 305–366.
  • D. Levin, “On some new spectral estimates for Schrödinger-like operators”, Cent. Eur. J. Math. 4:1 (2006), 123–137.
  • D. Levin and M. Solomyak, “The Rozenblum–Lieb–Cwikel inequality for Markov generators”, J. Anal. Math. 71 (1997), 173–193.
  • E. H. Lieb, “Bounds on the eigenvalues of the Laplace and Schroedinger operators”, Bull. Amer. Math. Soc. 82:5 (1976), 751–753.
  • E. H. Lieb, “Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities”, Ann. of Math. $(2)$ 118:2 (1983), 349–374.
  • E. H. Lieb, “On characteristic exponents in turbulence”, Comm. Math. Phys. 92:4 (1984), 473–480.
  • E. H. Lieb and W. E. Thirring, “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities”, pp. 269–303 in Studies in mathematical physics: essays in honor of Valentine Bargmann, edited by E. H. Lieb et al., Princeton University Press, 1976. Reprinted as pp. 205–239 in The stability of matter: from atoms to stars (Selecta of Elliott H. Lieb), edited by W. Thirring, Springer, Berlin, 2005.
  • C. E. Mueller and F. B. Weissler, “Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the $n$-sphere”, J. Funct. Anal. 48:2 (1982), 252–283.
  • E. M. Ouhabaz and C. Poupaud, “Remarks on the Cwikel–Lieb–Rozenblum and Lieb–Thirring estimates for Schrödinger operators on Riemannian manifolds”, Acta Appl. Math. 110:3 (2010), 1449–1459.
  • G. Rosen, “Minimum value for $c$ in the Sobolev inequality $\phi\sp{3}\Vert \leq c\nabla \phi\Vert \sp{3}$”, SIAM J. Appl. Math. 21 (1971), 30–32.
  • O. S. Rothaus, “Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators”, J. Funct. Anal. 42:1 (1981), 110–120.
  • G. Talenti, “Best constant in Sobolev inequality”, Ann. Mat. Pura Appl. $(4)$ 110 (1976), 353–372.
  • E. J. M. Veling, “Lower bounds for the infimum of the spectrum of the Schrödinger operator in $\R\sp N$ and the Sobolev inequalities”, J. Inequal. Pure Appl. Math. 3:4 (2002), Article ID #63.
  • E. J. M. Veling, “Corrigendum on the paper: “Lower bounds for the infimum of the spectrum of the Schrödinger operator in $\R\sp N$ and the Sobolev inequalities” [J. Inequal. Pure Appl. Math. 3:4 (2002), Article ID #63]”, J. Inequal. Pure Appl. Math. 4:5 (2003), Article ID #109.