Duke Mathematical Journal

The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds

El Maati Ouhabaz

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Given a complete Riemannian manifold $M$ and a Schrödinger operator $-\Delta+m$ acting on $L^p(M)$, we study two related problems on the spectrum of $-\Delta+m$. The first one concerns the positivity of the $L^2$-spectral lower bound $s(-\Delta+m)$. We prove that if $M$ satisfies $L^2$-Poincaré inequalities and a local doubling property, then $s(-\Delta+m)>0$, provided that $m$ satisfies the mean condition

$\inf\substack {p\in M}\frac {1}{|B(p, r)|}\int \sb{B(p,r )}m(x)dx>0$

for some $r>0$. We also show that this condition is necessary under some additional geometrical assumptions on $M$.

The second problem concerns the existence of an $L^p$-principal eigenvalue, that is, a constant $\lambda\geq 0$ such that the eigenvalue problem $\Delta u=\lambda mu$ and equation above] has a positive solution $u\in L^p(M)$. We give conditions in terms of the growth of the potential $m$ and the geometry of the manifold $M$ which imply the existence of $L^p$-principal eigenvalues.

Finally, we show other results in the cases of recurrent and compact manifolds.

Article information

Duke Math. J. Volume 110, Number 1 (2001), 1-35.

First available in Project Euclid: 18 June 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47) 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]


Ouhabaz, El Maati. The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds. Duke Math. J. 110 (2001), no. 1, 1--35. doi:10.1215/S0012-7094-01-11011-9. http://projecteuclid.org/euclid.dmj/1087574811.

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  • W. Allegretto, Principal eigenvalues for indefinite-weight elliptic problems in $\mathbbR^n$, Proc. Amer. Math. Soc. 116 (1992), 701--706.
  • W. Arendt and C. J. K. Batty, Exponential stability of a diffusion equation with absorption, Differential Integral Equations 6 (1993), 1009--1024.
  • --. --. --. --., The spectral function and principal eigenvalues for Schrödinger operators, Potential Anal. 7 (1997), 415--436.
  • T. Aubin, Nonlinear Analysis on Manifolds: Monge-Ampère Equations, Grundlehren Math. Wiss. 252, Springer, New York, 1982.
  • R. Brooks, A relation between growth and the spectrum of the Laplacian, Math. Z. 178 (1981), 501--508.
  • K. J. Brown, C. Cosner, and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbbR^N$, Proc. Amer. Math. Soc. 109 (1990), 147--155.
  • K. J. Brown, D. Daners, and J. López-Gómez, Change of stability for Schrödinger semigroups, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 827--846.
  • P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), 213--230.
  • G. Carron, $L^2$-cohomologie et inégalités de Sobolev, Math. Ann. 314 (1999), 613--639.
  • J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15--53.
  • T. Coulhon and L. Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), 293--314.
  • D. Daners, Principal eigenvalues for some periodic-parabolic operators on $\mathbbR^N$ and related topics, J. Differential Equations 121 (1995), 293--313.
  • E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge Univ. Press, Cambridge, 1989.
  • P. G. Doyle, On the bass note of a Schottky group, Acta Math. 160 (1988), 249--284.
  • X. T. Duong and D. W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1996), 89--128.
  • M. Fukushima, Y. Ōshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math. 19, de Gruyter, Berlin, 1994.
  • S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, 2d ed., Universitext, Springer, Berlin, 1990.
  • A. A. Grigor'yan, Stochastically complete manifolds and summable harmonic functions, Math. USSR-Izv. 33 (1989), 425--432.
  • --. --. --. --., The heat equation on noncompact Riemannian manifolds, Math. USSR-Sb. 72 (1992), 47--77.
  • P. Hajłasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1211--1215.
  • R. Hempel and J. Voigt, The spectrum of a Schrödinger operator in $L_p(\mathbbR^\nu)$ is $p$-independent, Comm. Math. Phys. 104 (1986), 243--250.
  • P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), 999--1030.
  • D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503--523.
  • T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995.
  • P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153--201.
  • P. Maheux and L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique, Math. Ann. 303 (1995), 713--740.
  • I. McGillivray and E.-M. Ouhabaz, ``Existence of bounded invariant solutions for absorption semigroups'' in Differential Equations, Asymptotic Analysis, and Mathematical Physics (Potsdam, Germany, 1996), Math. Res. 100, Akademie, Berlin, 1997, 226--241.
  • G. Metafune and D. Pallara, On the location of the essential spectrum of Schrödinger operators, preprint, 2000.
  • E. M. Ouhabaz, On the spectral function of some higher order elliptic or degenerate-elliptic operators, Semigroup Forum 57 (1998), 305--314.
  • E. M. Ouhabaz and P. Stollmann, Stability of the essential spectrum of second-order complex elliptic operators, J. Reine Angew. Math. 500 (1998), 113--126.
  • E.-M. Ouhabaz, P. Stollmann, K.-T. Sturm, and J. Voigt, The Feller property for absorption semigroups, J. Funct. Anal. 138 (1996), 351--378.
  • Y. Pinchover, On criticality and ground states of second order elliptic equations, II, J. Differential Equations 87 (1990), 353--364.
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978.
  • G. Rozenblum and M. Solomyak, On principal eigenvalues for indefinite problems in Euclidean space, Math. Nachr. 192 (1998), 205--223.
  • L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 1992, 27--38.
  • --. --. --. --., Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), 417--450.
  • M. Schechter, Spectra of Partial Differential Operators, 2d ed., North-Holland Ser. Appl. Math. Mech. 14, North-Holland, Amsterdam, 1986.
  • M. A. Shubin, ``Spectral theory of elliptic operators on noncompact manifolds'' in Méthodes semi-classiques (Nantes, France, 1991), Vol. 1, Astérisque 207, Soc. Math. France, Montrouge, 1992, 5, 35--108.
  • B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 447--526., ; Erratum, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 426.
  • R. S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48--79.
  • K.-T. Sturm, On the $L^p$-spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal. 118 (1993), 442--453.
  • A. Tertikas, Critical phenomena in linear elliptic problems, J. Funct. Anal. 154 (1998), 42--66.
  • N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, Cambridge, 1992.
  • J. Voigt, Absorption semigroups, their generators, and Schrödinger semigroups, J. Funct. Anal. 67 (1986), 167--205.
  • F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal. 170 (2000), 219--245.