The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Shen, Zhongwei

doi:10.1090/S0002-9939-03-06968-5

The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds
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by Zhongwei Shen

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

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

Published electronically: February 20, 2003

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Abstract:

Let $M$ be a noncompact complete Riemannian manifold. We consider the Schrödinger operator $-\Delta +V$ acting on $L^{2}(M)$, where $V$ is a nonnegative, locally integrable function on $M$. We obtain some simple conditions which imply that $\inf \text {Spec} (-\Delta +V)$, the bottom of the spectrum of $-\Delta +V$, is strictly positive. We also establish upper and lower bounds for the counting function $N(\lambda )$.

References

Wolfgang Arendt and Charles J. K. Batty, Exponential stability of a diffusion equation with absorption, Differential Integral Equations 6 (1993), no. 5, 1009–1024. MR 1230476
Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635, DOI 10.24033/asens.1426
Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR 1103113
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
Vladimir Kondrat′ev and Mikhail Shubin, Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry, The Maz′ya anniversary collection, Vol. 2 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 110, Birkhäuser, Basel, 1999, pp. 185–226. MR 1747895
Daniel Levin and Michael Solomyak, The Rozenblum-Lieb-Cwikel inequality for Markov generators, J. Anal. Math. 71 (1997), 173–193. MR 1454250, DOI 10.1007/BF02788029
Peter Li and Shing Tung Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309–318. MR 701919, DOI 10.1007/BF01213210
G. O. Okikiolu, Semigroups of functions with generators defined in terms of multiplier transforms, Bull. Math. 15 (1984), 1–44. MR 775201
El Maati Ouhabaz, The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds, Duke Math. J. 110 (2001), no. 1, 1–35. MR 1861087, DOI 10.1215/S0012-7094-01-11011-9
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Zhong Wei Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546 (English, with English and French summaries). MR 1343560, DOI 10.5802/aif.1463
—, On the eigenvalue asymptotics of Schrödinger operators, unpublished (1995).
Zhongwei Shen, Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4465–4488. MR 1370650, DOI 10.1090/S0002-9947-96-01709-6
Zhongwei Shen, On bounds of $N(\lambda )$ for a magnetic Schrödinger operator, Duke Math. J. 94 (1998), no. 3, 479–507. MR 1639527, DOI 10.1215/S0012-7094-98-09420-0