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Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds
Authors:
Ahmad El Soufi, Evans M. Harrell II and Saïd Ilias
Journal:
Trans. Amer. Math. Soc. 361 (2009), 2337-2350
MSC (2000):
Primary 58J50, 58E11, 35P15
Posted:
December 16, 2008
MathSciNet review:
2471921
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Abstract: We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly's inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.
- 1.
Mark
S. Ashbaugh, The universal eigenvalue bounds of
Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang, Proc.
Indian Acad. Sci. Math. Sci. 112 (2002), no. 1,
3–30. Spectral and inverse spectral theory (Goa, 2000). MR 1894540
(2004c:35302), http://dx.doi.org/10.1007/BF02829638
- 2.
Mark
S. Ashbaugh and Rafael
D. Benguria, A sharp bound for the ratio of the
first two Dirichlet eigenvalues of a domain in a hemisphere of
𝑆ⁿ, Trans. Amer. Math. Soc.
353 (2001), no. 3,
1055–1087. MR 1707696
(2001f:35298), http://dx.doi.org/10.1090/S0002-9947-00-02605-2
- 3.
Mark
S. Ashbaugh and Lotfi
Hermi, A unified approach to universal inequalities for eigenvalues
of elliptic operators, Pacific J. Math. 217 (2004),
no. 2, 201–219. MR 2109931
(2005k:35305), http://dx.doi.org/10.2140/pjm.2004.217.201
- 4.
M. S. Ashbaugh and L. Hermi, On Yang-type bounds for eigenvalues with applications to physical and geometric problems, 2005 preprint.
- 5.
M. S. Ashbaugh and L. Hermi, On Harrell-Stubbe type inequalities for the discrete spectrum of a self-adjoint operator, 2007 preprint.
- 6.
Isaac
Chavel, Eigenvalues in Riemannian geometry, Pure and Applied
Mathematics, vol. 115, Academic Press Inc., Orlando, FL, 1984.
Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
(86g:58140)
- 7.
Bang-Yen
Chen, On the first eigenvalue of Laplacian of compact minimal
submanifolds of rank one symmetric spaces, Chinese J. Math.
11 (1983), no. 4, 1–15. MR 732862
(85h:58171)
- 8.
Qing-Ming
Cheng and Hongcang
Yang, Estimates on eigenvalues of Laplacian, Math. Ann.
331 (2005), no. 2, 445–460. MR 2115463
(2005i:58038), http://dx.doi.org/10.1007/s00208-004-0589-z
- 9.
Qing-Ming
Cheng and Hongcang
Yang, Inequalities for eigenvalues of Laplacian on domains and
compact complex hypersurfaces in complex projective spaces, J. Math.
Soc. Japan 58 (2006), no. 2, 545–561. MR 2228572
(2007k:58051)
- 10.
Yves
Colin de Verdière, Construction de laplaciens dont une
partie finie du spectre est donnée, Ann. Sci. École
Norm. Sup. (4) 20 (1987), no. 4, 599–615
(French). MR
932800 (90d:58156)
- 11.
E.
B. Davies, Spectral theory and differential operators,
Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge
University Press, Cambridge, 1995. MR 1349825
(96h:47056)
- 12.
A.
El Soufi and S.
Ilias, Immersions minimales, première valeur propre du
laplacien et volume conforme, Math. Ann. 275 (1986),
no. 2, 257–267 (French). MR 854009
(87j:53088), http://dx.doi.org/10.1007/BF01458460
- 13.
A.
El Soufi and S.
Ilias, Une inégalité du type “Reilly”
pour les sous-variétés de l’espace hyperbolique,
Comment. Math. Helv. 67 (1992), no. 2, 167–181
(French). MR
1161279 (93i:53059), http://dx.doi.org/10.1007/BF02566494
- 14.
Ahmad
El Soufi and Saïd
Ilias, Second eigenvalue of Schrödinger operators and mean
curvature, Comm. Math. Phys. 208 (2000), no. 3,
761–770. MR 1736334
(2001g:58050), http://dx.doi.org/10.1007/s002200050009
- 15.
Pavel
Exner and Petr
Šeba (eds.), Schrödinger operators, standard and
nonstandard, World Scientific Publishing Co. Inc., Teaneck, NJ, 1989.
Papers from the conference held in Dubna, September 6–10, 1988. MR 1091986
(91j:81005)
- 16.
Pavel
Exner, Evans
M. Harrell, and Michael
Loss, Optimal eigenvalues for some Laplacians and Schrödinger
operators depending on curvature, Mathematical results in quantum
mechanics (Prague, 1998) Oper. Theory Adv. Appl., vol. 108,
Birkhäuser, Basel, 1999, pp. 47–58. MR 1708787
(2000e:58045)
- 17.
Evans
M. Harrell II, Commutators, eigenvalue gaps, and mean curvature in
the theory of Schrödinger operators, Comm. Partial Differential
Equations 32 (2007), no. 1-3, 401–413. MR 2304154
(2008i:35041), http://dx.doi.org/10.1080/03605300500532889
- 18.
Evans
M. Harrell II and Patricia
L. Michel, Commutator bounds for eigenvalues, with applications to
spectral geometry, Comm. Partial Differential Equations
19 (1994), no. 11-12, 2037–2055. MR 1301181
(95i:58182), http://dx.doi.org/10.1080/03605309408821081
- 19.
Evans
M. Harrell II and Joachim
Stubbe, On trace identities and universal
eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997), no. 5, 1797–1809. MR 1401772
(97i:35129), http://dx.doi.org/10.1090/S0002-9947-97-01846-1
- 20.
David
S. Jerison, The Dirichlet problem for the Kohn Laplacian on the
Heisenberg group. I, J. Funct. Anal. 43 (1981),
no. 1, 97–142. MR 639800
(83c:58081a), http://dx.doi.org/10.1016/0022-1236(81)90040-9
- 21.
Tosio
Kato, Perturbation theory for linear operators, Classics in
Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
(96a:47025)
- 22.
J.
J. Kohn, Boundaries of complex manifolds, Proc. Conf. Complex
Analysis (Minneapolis, 1964) Springer, Berlin, 1965,
pp. 81–94. MR 0175149
(30 #5334)
- 23.
Pui
Fai Leung, On the consecutive eigenvalues of the Laplacian of a
compact minimal submanifold in a sphere, J. Austral. Math. Soc. Ser. A
50 (1991), no. 3, 409–416. MR 1096895
(92d:58212)
- 24.
Peter
Li, Eigenvalue estimates on homogeneous manifolds, Comment.
Math. Helv. 55 (1980), no. 3, 347–363. MR 593051
(81k:58067), http://dx.doi.org/10.1007/BF02566692
- 25.
Michael
Levitin and Leonid
Parnovski, Commutators, spectral trace identities, and universal
estimates for eigenvalues, J. Funct. Anal. 192
(2002), no. 2, 425–445. MR 1923409
(2003g:47040), http://dx.doi.org/10.1006/jfan.2001.3913
- 26.
Elliott
H. Lieb, Inequalities, Springer-Verlag, Berlin, 2002. Selecta
of Elliott H. Lieb; Edited, with a preface and commentaries, by M. Loss and
M. B. Ruskai. MR
1922236 (2003f:01063)
- 27.
Claus
Müller, Spherical harmonics, Lecture Notes in
Mathematics, vol. 17, Springer-Verlag, Berlin, 1966. MR 0199449
(33 #7593)
- 28.
Pengcheng
Niu and Huiqing
Zhang, Payne-Polya-Weinberger type inequalities for eigenvalues of
nonelliptic operators, Pacific J. Math. 208 (2003),
no. 2, 325–345. MR 1971668
(2004c:35306), http://dx.doi.org/10.2140/pjm.2003.208.325
- 29.
L.
E. Payne, G.
Pólya, and H.
F. Weinberger, On the ratio of consecutive eigenvalues, J.
Math. and Phys. 35 (1956), 289–298. MR 0084696
(18,905c)
- 30.
Robert
C. Reilly, On the first eigenvalue of the Laplacian for compact
submanifolds of Euclidean space, Comment. Math. Helv.
52 (1977), no. 4, 525–533. MR 0482597
(58 #2657)
- 31.
Kunio
Sakamoto, Planar geodesic immersions, Tôhoku Math. J.
(2) 29 (1977), no. 1, 25–56. MR 0470913
(57 #10657)
- 32.
Shin-sheng
Tai, Minimum imbeddings of compact symmetric spaces of rank
one, J. Differential Geometry 2 (1968), 55–66.
MR
0231395 (37 #6950)
- 33.
Walter
Thirring, A course in mathematical physics. Vol. 3,
Springer-Verlag, New York, 1981. Quantum mechanics of atoms and molecules;
Translated from the German by Evans M. Harrell; Lecture Notes in Physics,
141. MR
625662 (84m:81006)
- 34.
H. C. Yang, An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of the Intl. Centre for Theoretical Physics, 1991. Revised version, preprint 1995.
- 35.
Paul
C. Yang and Shing
Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces
and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
7 (1980), no. 1, 55–63. MR 577325
(81m:58084)
- 1.
- M. S. Ashbaugh, Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H. C. Yang, Proc. Indian Acad. Sci. (Math. Sci.) 112 (2002) 3-30. MR 1894540 (2004c:35302)
- 2.
- M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of
 , Trans. Amer. Math. Soc. 353 (2001) 1055-1087. MR 1707696 (2001f:35298)
- 3.
- M. S. Ashbaugh and L. Hermi, A unified approach to universal inequalities for eigenvalues of elliptic operators, Pacific J. Math. 217 (2004) 201-219. MR 2109931 (2005k:35305)
- 4.
- M. S. Ashbaugh and L. Hermi, On Yang-type bounds for eigenvalues with applications to physical and geometric problems, 2005 preprint.
- 5.
- M. S. Ashbaugh and L. Hermi, On Harrell-Stubbe type inequalities for the discrete spectrum of a self-adjoint operator, 2007 preprint.
- 6.
- I. Chavel, Eigenvalues in Riemannian Geometry. Orlando: Academic Press, 1984. MR 768584 (86g:58140)
- 7.
- B. Y. Chen, On the first eigenvalue of Laplacian of compact minimal submanifolds of rank one symmetric spaces, Chinese J. Math. 11 (1983) 259-273. MR 732862 (85h:58171)
- 8.
- Q. M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005) 445-460. MR 2115463 (2005i:58038)
- 9.
- Q. M. Cheng and H. C. Yang, Inequalities for eigenvalues of Laplacian on domains and complex hypersurfaces in complex projective spaces, J. Math. Soc. Japan 58 (2006) 545-561. MR 2228572 (2007k:58051)
- 10.
- Y. Colin de Verdière, Construction de laplaciens dont une partie finie du spectre est donnée. Ann. Sci. École Norm. Sup. 20(4) (1987) 599-615. MR 932800 (90d:58156)
- 11.
- E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics 42. Cambridge: Cambridge University Press, 1995. MR 1349825 (96h:47056)
- 12.
- A. El Soufi and S. Ilias, Immersions minimales, première valeur propre du Laplacien et volume conforme, Math. Ann. 276 (1986) 257-267. MR 854009 (87j:53088)
- 13.
- A. El Soufi, and S. Ilias, Une inégalité du type ``Reilly'' pour les sous-variétés de l'espace hyperbolique, Comment. Math. Helv. 67 (1992) 167-181. MR 1161279 (93i:53059)
- 14.
- A. El Soufi, and S. Ilias, Second eigenvalue of Schrödinger operators and mean curvature, Commun. Math. Phys. 208 (2000) 761-770. MR 1736334 (2001g:58050)
- 15.
- P. Exner and P. Šeba, Electrons in semiconductor microstructures: a challenge to operator theorists, pp. 85-106 in Schrödinger Operators, Standard and Non-Standard, World Scientific, Singapore, 1989. MR 1091986 (91j:81005)
- 16.
- P. Exner, E. M. Harrell II, and M. Loss, Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature, pp. 47-58 in: Mathematical Results in Quantum Mechanics, J. Dittrich, P. Exner, M. Tater, eds. Basel: Birkhäuser, 1999. MR 1708787 (2000e:58045)
- 17.
- E. M. Harrell II, Commutators, eigenvalue gaps and mean curvature in the theory of Schrödinger operators, Commun. Part. Diff. Eq. 32 (2007) 401-413. MR 2304154
- 18.
- E. M. Harrell II and P. L. Michel, Commutator bounds for eigenvalues with applications to spectral geometry, Commun. in Part. Diff. Eqs. 19 (1994) 2037-2055. MR 1301181 (95i:58182)
- 19.
- E. M. Harrell II, and J. Stubbe, On trace identities and universal eigenvalue estimates for some partial differential operators, Trans. Amer. Math. Soc. 349 (1997) 1797-1809. MR 1401772 (97i:35129)
- 20.
- D. S. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I, J. Funct. Anal. 43 (1981) 97-142. MR 639800 (83c:58081a)
- 21.
- T. Kato, Perturbation theory for linear operators, 2nd edition, Berlin, Heidelberg, New York: Springer Verlag, 1995. MR 1335452 (96a:47025)
- 22.
- J. J. Kohn, Boundaries of complex manifolds. 1965, pp. 81-94 in: Proc. Conf. Complex Analysis (Minneapolis, 1964). Berlin: Springer Verlag, 1965. MR 0175149 (30:5334)
- 23.
- P.-F. Leung, On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere, J. Austral. Math. Soc. (ser. A) 50 (1991) 409-416. MR 1096895 (92d:58212)
- 24.
- P. Li, Eigenvalue estimates on homogeneous manifolds, Comment. Math. Helvetici 55 (1980) 347-363. MR 593051 (81k:58067)
- 25.
- M. Levitin and L. Parnovski, Commutators, spectral trace identities and universal estimates for eigenvalues, J. Funct. Anal. 192 (2002) 425-445. MR 1923409 (2003g:47040)
- 26.
- M. Loss and M. B. Ruskai, eds., Inequalities, Selecta of Elliott H. Lieb. Berlin, Heidelberg, and New York: Springer, 2002. MR 1922236 (2003f:01063)
- 27.
- K. Müller, Spherical Harmonics, Springer Lecture Notes In Mathematics 17, Berlin: Springer Verlag, 1966. MR 0199449 (33:7593)
- 28.
- P. C. Niu and H. Q. Zhang, Payne-Polya-Weinberger type inequalities for eigenvalues of eigenvalues of nonelliptic operators, Pac. J. Math. 208, 325-345 (2003). MR 1971668 (2004c:35306)
- 29.
- L. E. Payne, G. Pólya, and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956) 289-298. MR 0084696 (18:905c)
- 30.
- R. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977) 525-533. MR 0482597 (58:2657)
- 31.
- K. Sakamoto, Planar geodesic immersions, Tohoku Math. J. 29 (1977) 25-56. MR 0470913 (57:10657)
- 32.
- S. S. Tai, Minimal imbedding of compact symmetric spaces of rank one. J. Diff. Geom. 2 (1968) 55-66. MR 0231395 (37:6950)
- 33.
- W. Thirring, Quantum Mechanics of Atoms and Molecules, A Course in Mathematical Physics 3, New York and Vienna: Springer, 1979. MR 625662 (84m:81006)
- 34.
- H. C. Yang, An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of the Intl. Centre for Theoretical Physics, 1991. Revised version, preprint 1995.
- 35.
- P. C. Yang and S.-T. Yau, Eigenvalues of the Laplacian of a compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa, cl. sci. 4 (1980) 55-63. MR 577325 (81m:58084)
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Additional Information
Ahmad El Soufi
Affiliation:
Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France
Email:
elsoufi@univ-tours.fr
Evans M. Harrell II
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email:
harrell@math.gatech.edu
Saïd Ilias
Affiliation:
Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais de Tours, UMR-CNRS 6083, Parc de Grandmont, 37200 Tours, France
Email:
ilias@univ-tours.fr
DOI:
http://dx.doi.org/10.1090/S0002-9947-08-04780-6
PII:
S 0002-9947(08)04780-6
Keywords:
Spectrum,
eigenvalue,
Laplacian,
Schr\"{o}dinger operator,
Reilly inequality,
Kohn Laplacian.
Received by editor(s):
January 16, 2007
Posted:
December 16, 2008
Article copyright:
© Copyright 2008 by the authors
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