The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Author:
Zhongwei Shen

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3447-3456

MSC (2000):
Primary 35P20, 35J10

DOI:
https://doi.org/10.1090/S0002-9939-03-06968-5

Published electronically:
February 20, 2003

MathSciNet review:
1990634

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a noncompact complete Riemannian manifold. We consider the Schrödinger operator acting on , where is a nonnegative, locally integrable function on . We obtain some simple conditions which imply that , the bottom of the spectrum of , is strictly positive. We also establish upper and lower bounds for the counting function .

**[A-B]**W. Arendt and C.J.K. Batty,*Exponential stability of a diffusion equation with absorption*, Diff. Integral Eqs.**6**(1993), 1009-1024. MR**94k:35038****[Bu]**P. Buser,*A note on the isoperimetric constant*, Ann. Sci. École Norm. Sup.**15**(1982), 213-230. MR**84e:58076****[C-G-T]**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. Diff. Geom.**17**(1982), 15-53. MR**84b:58109****[Da]**E. B. Davies,*Heat Kernels and Spectral Theory*, Cambridge Univ. Press, 1990. MR**92a:35035****[Fe]**C. Fefferman,*The uncertainty principle*, Bull. Amer. Math. Soc.**9**(1983), 129-206. MR**85f:35001****[K-S]**V. Kondratev and M. Shubin,*Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry*, Operator Theory: Advances and Applications**110**(1999), 185-226. MR**2001c:58030****[Le-So]**D. Levin and M. Solomyak,*Rozenblum-Lieb-Cwikel inequality for Markov generators*, J. d'Analyse Math.**71**(1997), 173-193. MR**98j:47090****[Li-Ya]**P. Li and S-T. Yau,*On the Schrödinger equation and the eigenvalue problem*, Comm. Math. Phys.**88**(1983), 309-318. MR**84k:58225****[Ma]**V. G. Maz'ya,*Sobolev Spaces*, Springer Verlag, Berlin, 1985. MR**87g:47056****[Ou]**E. M. Ouhabaz,*The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds*, Duke Math. J.**110(1)**(2001). MR**2002i:58038****[Re-Si]**M. Reed and B. Simon,*Methods of Modern Mathematical Physics, vol. II*, Academic Press, 1975. MR**58:12429b****[Sh1]**Z. Shen,*estimates for Schrödinger operators with certain potentials*, Ann. Inst. Fourier**45(2)**(1995), 513-546. MR**96h:35037****[Sh2]**-,*On the eigenvalue asymptotics of Schrödinger operators*, unpublished (1995).**[Sh3]**-,*Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields*, Trans. Amer. Math. Soc.**348(11)**(1996), 4465-4488. MR**97d:35162****[Sh4]**-,*On bounds of**for a magnetic Schrödinger operator*, Duke Math. J.**94(3)**(1998), 479-507. MR**99e:35171**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35P20,
35J10

Retrieve articles in all journals with MSC (2000): 35P20, 35J10

Additional Information

**Zhongwei Shen**

Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Email:
shenz@ms.uky.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-06968-5

Received by editor(s):
May 27, 2002

Published electronically:
February 20, 2003

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2003
American Mathematical Society