Absence of eigenvalues for the periodic Schrödinger operator with singular potential in a rectangular cylinder. (English. Russian original) Zbl 1273.35200
Funct. Anal. Appl. 47, No. 2, 104-112 (2013); translation from Funkts. Anal. Prilozh. 47, No. 2, 27-37 (2013).
Summary: We consider the periodic Schrödinger operator on a \(d\)-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form \(\sigma(x,y)\delta_\Sigma(x,y)\), where \(\Sigma\) is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that \(\Sigma\) is sufficiently smooth and \(\sigma\in L_{p,{\mathrm{loc}}}(\Sigma)\), \(p>d-1\).
MSC:
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 47A60 | Functional calculus for linear operators |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
References:
| [1] | M. Sh. Birman and T. A. Suslina, ”Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity,” Algebra i Analiz, 11:2 (1999), 1–40; English transl.: St. Petersburg Math. J., 11:2 (2000), 203–232. · Zbl 0941.35015 |
| [2] | M. Sh. Birman, T. A. Suslina, and R. G. Shterenberg, ”Absolute continuity of the spectrum of a two-dimensional Schrödinger operator with potential supported on a periodic system of curves,” Algebra i Analiz, 12:6 (2000), 140–177; English transl.: St. Petersburg Math. J., 12:6 (2001), 983–1012. · Zbl 0998.35006 |
| [3] | N. Burq, P. Gérard, and N. Tzvetkov, ”Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds,” Duke Math. J., 138:3 (2007), 445–486. · Zbl 1131.35053 · doi:10.1215/S0012-7094-07-13834-1 |
| [4] | L. Danilov, ”On absolute continuity of the spectrum of a periodic magnetic Schrödinger operator,” J. Phys. A: Math. Theor., 42:27 (2009), 275204. · Zbl 1173.35615 · doi:10.1088/1751-8113/42/27/275204 |
| [5] | I. Kachkovskiĭ, ”Stein-Tomas theorem for a torus and the periodic Schrödinger operator with singular potential,” Algebra i Analiz, 24:6 (2012), 124–138. |
| [6] | I. Kachkovskiĭ and N. Filonov, ”Absolute continuity of the Schrdinger operator spectrum in a multidimensional cylinder,” Algebra i Analiz, 21:1 (2009), 133–152; English transl.: St. Petersburg Math. J., 21:1 (2010), 95–109. |
| [7] | I. Kachkovskiĭ and N. Filonov, ”Absolute continuity of the spectrum of the periodic Scrödinger operator in a layer and in a smooth cylinder, Boundary-value problems of mathematical physics and related problems of function theory. Part 41,” Zap. Nauchn. Sem. POMI, 385 (2010), 69–82; English transl.: J. Math. Sci. (N.Y.), 178:3 (2011), 274–281. |
| [8] | T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1966. · Zbl 0148.12601 |
| [9] | P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhauser, Basel, 1993. · Zbl 0789.35002 |
| [10] | P. Kuchment and S. Levendorskii, ”On the structure of spectra of periodic elliptic operators,” Trans. Amer. Math. Soc., 354:2 (2001), 537–569. · Zbl 1058.35174 · doi:10.1090/S0002-9947-01-02878-1 |
| [11] | A. Prokhorov and N. Filonov, On the spectrum of the periodic Maxwell operator in a cylinder (to appear). · Zbl 1360.35265 |
| [12] | M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4: Analysis of Operators, Academic Press, New York-London, 1978. · Zbl 0401.47001 |
| [13] | E. Shargorodsky and A. V. Sobolev, ”Quasiconformal mappings and periodic spectral problems in dimension two,” J. Anal. Math., 91 (2003), 67–103. · Zbl 1048.30010 · doi:10.1007/BF02788782 |
| [14] | Z. Shen, ”On absolute continuity of the periodic Schrödinger operators,” Intern. Math. Res. Notes, 2001:1 (2001), 1–31. · Zbl 0998.35007 · doi:10.1155/S1073792801000010 |
| [15] | R. G. Shterenberg, ”Absolute continuity of the spectrum of two-dimensional periodic Schrödinger operators with strongly subordinate magnetic potential,” Investigations on Linear Operators and Function Theory. Part 31, Zap. Nauchn. Sem. POMI, 303 (2003), 279–320; English transl.: J. Math. Sci. (N.Y.), 129:4 (2005), 4087–4109. |
| [16] | A. Sobolev and J. Walthoe, ”Absolute continuity in periodic waveguides,” Proc. London Math. Soc., 85:3 (2002), 717–741. · Zbl 1247.35077 · doi:10.1112/S0024611502013631 |
| [17] | T. A. Suslina, ”On the absence of eigenvalues of a periodic matrix Schrödinger operator in a layer,” Russ. J. Math. Phys., 8:4 (2001), 463–486. · Zbl 1186.81059 |
| [18] | T. A. Suslina, ”Absolute continuity of the spectrum of the periodic Maxwell operator in a layer,” Boundary-value problems of mathematical physics and related problems of function theory. Part 32, Zap. Nauchn. Sem. POMI, 288 (2002), 232–255; English transl.: J. Math. Sci. (N.Y.), 123:6 (2004), 4654–4667. |
| [19] | T. A. Suslina and R. G. Shterenberg, ”Absolute continuity of the spectrum for the Schrödinger operator with potential supported by a periodic system of hypersurfaces,” Algebra i Analiz, 13:5 (2001), 197–240; English transl.: St. Petersburg Math. J., 13:5 (2002), 859–891. · Zbl 0998.35006 |
| [20] | T. A. Suslina and R. G. Shterenberg, ”Absolute continuity of the spectrum of the magnetic Schrödinger operator with a metric in a two-dimensional periodic waveguide,” Algebra i Analiz, 14:2 (2002), 159–206; English transl.: St. Petersburg Math. J., 14:2 (2003), 305–343. · Zbl 1057.35024 |
| [21] | L. Thomas, ”Time dependent approach to scattering from impurities in a crystal,” Comm. Math. Phys., 33 (1973), 335–343. · doi:10.1007/BF01646745 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
