On Spectrum of Schr\"odinger Operator on Manifold of a Special Type

A. V. Svetlov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 4, pp. 584-589.

Résumé

The main subject of the paper is spectrum of the Schrödinger operator on weighted quasimodel manifold with an end, which is warped product of a special type. We prove the criterion of discreteness for the spectrum of the operator in terms of metric coefficients and potential of the operator. As the conclusion we made some remarks on the corollaries of the proved theorem and on its extension to more complex quasimodel manifolds.

Bibliographie

[1] Pinsky M., “The spectrum of the Laplacian on a manifold of negative curvature, I”, J. Diff. Geom., 13 (1978), 87–91 | MR | Zbl

[2] Baider A., “Noncompact Riemannian manifolds with discrete spectra”, J. Diff. Geom., 14 (1979), 41–57 | MR | Zbl

[3] Brooks R., “A relation between growth and the spectrum of the Laplacian”, Math. Z., 178 (1981), 501–508 | DOI | MR | Zbl

[4] Svetlov A. V., “A discreteness criterion for the spectrum of the Laplace–Beltrami operator on quasimodel manifolds”, Siberian Math. J., 43:6 (2002), 1103–1111 | DOI | MR | Zbl

[5] Harmer M., “Discreteness of the spectrum of the Laplacian and stochastic incompleteness”, J. Geom. Anal., 19:2 (2009), 358–372 | DOI | MR | Zbl

[6] Kondratev V., Shubin M., “Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry”, Operator theory : Advances and Applications, 110 (1999), 185–226 | DOI | MR | Zbl

[7] Shen Z., “The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds”, Proc. Amer. Math. Soc., 131:11 (2003), 3447–3456 | DOI | MR | Zbl

[8] Svetlov A. V., “Discreteness criterion for the spectrum of the Schrödinger operator on weighted quasimodel manifolds”, Intern. J. Pure Appl. Math., 89:3 (2013), 393–400 | DOI | Zbl

[9] Losev A. G., “On some Liouville theorems on noncompact Riemannian manifolds”, Siberian Math. J., 39:1 (1998), 74–80 | DOI | MR | Zbl

[10] Losev A. G., Mazepa E. A., “Bounded solutions of the Schrödinger equation on Riemannian products”, St. Petersburg Math. J., 13:1 (2001), 57–73 | MR

[11] Korolkov S. A., Losev A. G., “Generalized harmonic functions of Riemannian manifolds with ends”, Mathematische Zeitschrift, 272:1–2 (2012), 459–472 | DOI | MR | Zbl

[12] Grigor'yan A. A., “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds”, Bull. Amer. Math. Soc., 36 (1999), 135–249 | DOI | MR | Zbl

[13] Svetlov A. V., “The spectrum of the Schrodinger operator on the warped products”, Vestnik VolGU. Ser. 1, Matematika. Fizika, 2002, no. 7, 12–19 (in Russian)

[14] Reed M. C., Simon B., Methods of Modern Mathematical Physics, v. 1, Functional analysis, Academic Press, London, 1980, 325 pp. | MR | MR | Zbl

[15] Schechter M., Spectra of partial differential operators, North-Holland, Amsterdam, 1971, 295 pp. | MR | Zbl

[16] Molchanov A. M., “On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order”, Tr. Mosk. Mat. Obs., 2, 1953, 169–199 (in Russian) | Zbl

[17] Svetlov A. V., “Discreteness criterion for the spectrum of the Schrödinger operator on manifolds of a special kind”, Modern problems of function theory and its applications, Texts of 17th Intern. Saratov Winter School, Saratov, 2014, 245–247 (in Russian)