Some Aspects of Theory of Schrödinger Operators on Riemannian Manifold

Farah Diyab; B. Surender Reddy

doi:10.26713/cma.v13i5.2258

Authors

Farah Diyab Department of Mathematics, Osmania University, Hyderabad, India https://orcid.org/0000-0001-5564-2519 (unauthenticated)
B. Surender Reddy Department of Mathematics, Osmania University, Hyderabad, India https://orcid.org/0000-0002-6078-9934 (unauthenticated)

DOI:

https://doi.org/10.26713/cma.v13i5.2258

Keywords:

Manifolds, Spectral theory, Laplacian, Spectral geometry

Abstract

This paper deals with a given Riemannian manifold \(\mathcal{M}\). One of the main tasks is description of spectrum of several classes of Schrödinger operator \(P=\frac{-h^{2}}{2}\Delta _{g}+V\) where \(\Delta _{g}\) is Laplace Beltrami operator and \(V\) is potential on manifold. We illustrate the inverse and direct problems of \(\Delta _{g}\) and the way to discover the geometry of Riemannian manifold from spectral data.

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References

F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Springer Science+Business Media, B.V., 555 pages (1991).

M. Braverman, O. Milatovic and M. Shubin, Essential self-adjointness of Schrödinger-type operators on manifolds, Russian Mathematical Surveys 57(4) (2002), 641, DOI: 10.1070/RM2002v057n04ABEH000532.

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 1st edition, Springer, New York, xiv + 600 (2011), DOI: 10.1007/978-0-387-70914-7.

P. Cartier, Problèmes mathématiques de la théorie quantique des champs, Séminaire N. Bourbaki 23(388) (1970-71), 107 – 122, URL: http://www.numdam.org/article/SB_1970-1971__13__107_0.pdf.

I. Chavel and E. A. Feldman, Spectra of domains in compact manifolds, Journal of Functional Analysis 30(2) (1978), 198 – 222, DOI: 10.1016/0022-1236(78)90070-8.

I. Chavel and E. A. Feldman, Spectra of manifolds less a small domain, Duke Mathematical Journal 56(2) (1988), 399 – 414, DOI: 10.1215/S0012-7094-88-05617-7.

S.-Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Mathematische Zeitschrift 143(3) (1975), 289 – 297, DOI: 10.1007/BF01214381.

P. R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, Journal of Functional Analysis 12(4) (1973), 401 – 414, DOI: 10.1016/0022-1236(73)90003-7.

H. O. Cordes, Spectral Theory of Linear Differential Operators and Comparison Algebras, London Mathematical Society Lecture Note Series, Vol. 76, Cambridge University Press (1987), DOI: 10.1017/CBO9780511662836.

G. Courtois, Spectrum of manifolds with holes, Journal of Functional Analysis 134(1) (1995), 194 – 221, URL: 10.1006/jfan.1995.1142.

M. G. Dabkowski and M. T. Lock, The lowest eigenvalue of Schrödinger operators on compact manifolds, Potential Analysis 50(4) (2019), 621 – 630, DOI: 10.1007/s11118-018-9698-2.

K. Friedrichs, Spektraltheorie halbbeschränkter operatoren und anwendung auf die spektralzerlegung von differentialoperatoren, Mathematische Annalen 109(1) (1934), 465 – 487, DOI: 10.1007/BF01449150.

F. Gesztesy and Z. Zhao, Domain perturbations, Brownian motion, capacities, and ground states of Dirichlet Schrödinger operators, Mathematische Zeitschrift 215(1) (1994), 143 – 150, DOI: 10.1007/BF02571703.

C. S. Gordon, Survey of isospectral manifolds, Chapter 6 of Handbook of Differential Geometry, Vol. 1, F. J. E. Dillen and L. C. A. Verstraelen (editors), 747 – 778, North-Holland, Amsterdam (2000), DOI: 10.1016/S1874-5741(00)80009-6.

M. Kac, Can one hear the shape of a drum?, The American Mathematical Monthly 73(4-2) (1966), 1 – 23, URL: https://doi.org/10.2307/2313748.

M. Kac, Probabilistic methods in some problems of scattering theory, The Rocky Mountain Journal of Mathematics 4(3) (1974), 511 – 537, URL: https://www.jstor.org/stable/44236400.

T. Kato, Perturbation Theory for Linear Operators, Vol. 132, Springer Science & Business Media, xvi+591 (1966), URL: https://books.google.co.in/books?hl=en&lr=&id=k-7nCAAAQBAJ&oi=fnd&pg=PR17&ots=w7NIsbgcAF&sig=gXdJnJt3n8FBgucXGDUGutO126w#v=onepage&q&f=false.

T. Kato, Schrödinger operators with singular potentials, Israel Journal of Mathematics 13(1) (1972), 135 – 148, DOI: 10.1007/BF02760233.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, 2nd edition, (Course of Theoretical Physics, Vol. 3), Peragamon Press (1965), URL: https://power1.pc.uec.ac.jp/~toru/notes/LandauLifshitz-QuantumMechanics.pdf.

J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences of the United States of America 51(4) (1964), 542, DOI: 10.1073/pnas.51.4.542.

I. M. Oleinik, On the connection of the classical and quantum mechanical completeness of a potential at infinity on complete Riemannian manifolds, Mathematical Notes 55(4) (1994), 380 – 386, DOI: 10.1007/BF02112477.

J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, Journal of Functional Analysis 18(1) (1975), 27 – 59, DOI: 10.1016/0022-1236(75)90028-2.

M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press, New York — San Francisco — London (1975), URL: https://books.google.co.in/books?hl=en&lr=&id=zHzNCgAAQBAJ&oi=fnd&pg=PP1&ots=RbLz1r5glE&sig=gvmVSKUPY8LqR5j9VwenkGxdlM8#v=onepage&q&f=false.

E. Schrödinger, Space-time Structure, Cambridge University Press, 119 pages (first published 1950, reprinted 1997), URL: http://strangebeautiful.com/other-texts/schrodinger-st-struc.pdf.

T. Sunada, Riemannian coverings and isospectral manifolds, Annals of Mathematics 121(1) (1985), 169 – 186, URL: https://www.jstor.org/stable/1971195.