Absence of eigenvalues of non-selfadjoint Schrödinger operators on the boundary of their numerical range

Hansmann, Marcel

doi:10.1090/S0002-9939-2014-11885-5

Absence of eigenvalues of non-selfadjoint Schrödinger operators on the boundary of their numerical range
HTML articles powered by AMS MathViewer

by Marcel Hansmann

Proc. Amer. Math. Soc. 142 (2014), 1321-1335

DOI: https://doi.org/10.1090/S0002-9939-2014-11885-5

Published electronically: January 29, 2014

PDF | Request permission

Abstract:

We use a classical result of Hildebrandt to establish simple conditions for the absence of eigenvalues of non-selfadjoint discrete and continuous Schrödinger operators on the boundary of their numerical range.

References

A. A. Abramov, A. Aslanyan, and E. B. Davies, Bounds on complex eigenvalues and resonances, J. Phys. A 34 (2001), no. 1, 57–72. MR 1819914, DOI 10.1088/0305-4470/34/1/304
W. O. Amrein, A.-M. Berthier, and V. Georgescu, $L^{p}$-inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 3, vii, 153–168 (English, with French summary). MR 638622
A. Borichev, L. Golinskii, and S. Kupin, A Blaschke-type condition and its application to complex Jacobi matrices, Bull. Lond. Math. Soc. 41 (2009), no. 1, 117–123. MR 2481997, DOI 10.1112/blms/bdn109
David Damanik, Rowan Killip, and Barry Simon, Schrödinger operators with few bound states, Comm. Math. Phys. 258 (2005), no. 3, 741–750. MR 2172016, DOI 10.1007/s00220-005-1366-x
E. Brian Davies, Linear operators and their spectra, Cambridge Studies in Advanced Mathematics, vol. 106, Cambridge University Press, Cambridge, 2007. MR 2359869, DOI 10.1017/CBO9780511618864
Michael Demuth, Marcel Hansmann, and Guy Katriel, On the discrete spectrum of non-selfadjoint operators, J. Funct. Anal. 257 (2009), no. 9, 2742–2759. MR 2559715, DOI 10.1016/j.jfa.2009.07.018
I. Egorova and L. Golinskii, On the location of the discrete spectrum for complex Jacobi matrices, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3635–3641. MR 2163601, DOI 10.1090/S0002-9939-05-08181-5
Paul Erdős and John C. Oxtoby, Partitions of the plane into sets having positive measure in every non-null measurable product set, Trans. Amer. Math. Soc. 79 (1955), 91–102. MR 72928, DOI 10.1090/S0002-9947-1955-0072928-3
Rupert L. Frank, Eigenvalue bounds for Schrödinger operators with complex potentials, Bull. Lond. Math. Soc. 43 (2011), no. 4, 745–750. MR 2820160, DOI 10.1112/blms/bdr008
Rupert L. Frank, Ari Laptev, Elliott H. Lieb, and Robert Seiringer, Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials, Lett. Math. Phys. 77 (2006), no. 3, 309–316. MR 2260376, DOI 10.1007/s11005-006-0095-1
Rupert L. Frank, Ari Laptev, and Robert Seiringer, A sharp bound on eigenvalues of Schrödinger operators on the half-line with complex-valued potentials, Spectral theory and analysis, Oper. Theory Adv. Appl., vol. 214, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 39–44. MR 2808462, DOI 10.1007/978-3-7643-9994-8_{3}
Leonid Golinskii and Stanislav Kupin, Lieb-Thirring bounds for complex Jacobi matrices, Lett. Math. Phys. 82 (2007), no. 1, 79–90. MR 2367876, DOI 10.1007/s11005-007-0189-4
Marcel Hansmann, An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators, Lett. Math. Phys. 98 (2011), no. 1, 79–95. MR 2836430, DOI 10.1007/s11005-011-0494-9
Marcel Hansmann and Guy Katriel, Inequalities for the eigenvalues of non-selfadjoint Jacobi operators, Complex Anal. Oper. Theory 5 (2011), no. 1, 197–218. MR 2773062, DOI 10.1007/s11785-009-0040-2
Stefan Hildebrandt, Über den numerischen Wertebereich eines Operators, Math. Ann. 163 (1966), 230–247 (German). MR 200725, DOI 10.1007/BF02052287
Lars Hörmander, The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR 1996773, DOI 10.1007/978-3-642-61497-2
David Jerison and Carlos E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2) 121 (1985), no. 3, 463–494. With an appendix by E. M. Stein. MR 794370, DOI 10.2307/1971205
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
Ari Laptev and Oleg Safronov, Eigenvalue estimates for Schrödinger operators with complex potentials, Comm. Math. Phys. 292 (2009), no. 1, 29–54. MR 2540070, DOI 10.1007/s00220-009-0883-4
V. G. Maz’ya and I. E. Verbitsky, Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator, Invent. Math. 162 (2005), no. 1, 81–136. MR 2198326, DOI 10.1007/s00222-005-0439-y
Oleg Safronov, Estimates for eigenvalues of the Schrödinger operator with a complex potential, Bull. Lond. Math. Soc. 42 (2010), no. 3, 452–456. MR 2651940, DOI 10.1112/blms/bdq007
Oleg Safronov, On a sum rule for Schrödinger operators with complex potentials, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2107–2112. MR 2596049, DOI 10.1090/S0002-9939-10-10248-2
Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8
Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536, DOI 10.1090/surv/072
Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1443208