Abstract
The central problem we consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are explained and elaborated. One approach uses complex analysis to study a holomorphic function whose zeros can be identified with the eigenvalues of the linear operator. The second method is an operator theoretic approach involvingthe numerical range. General results obtained by the two methods are derived and compared. Applications to non-selfadjoint Jacobi and Schrödinger operators are considered. Some possible directions for future research are discussed.
Mathematics Subject Classification (2010). 47A10, 47A75, 47B36, 81Q12.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
References
A.A. Abramov, A. Aslanyan, and E.B. Davies. Bounds on complex eigenvalues and resonances. J. Phys. A, 34(1):57–72, 2001.
R. Bhatia and Ch. Davis. Perturbation of extended enumerations of eigenvalues. Acta Sci. Math. (Szeged), 65(1-2):277–286, 1999.
R. Bhatia and L. Elsner. The Hoffman–Wielandt inequality in infinite dimensions. Proc. Indian Acad. Sci. Math. Sci., 104(3):483–494, 1994.
R. Bhatia and K.B. Sinha. A unitary analogue of Kato’s theorem on variation of discrete spectra. Lett. Math. Phys., 15(3):201–204, 1988.
A. Borichev, L. Golinskii, and S. Kupin. A Blaschke-type condition and its application to complex Jacobi matrices. Bull. London Math. Soc., 41:117–123, 2009.
R. Bouldin. Best approximation of a normal operator in the Schatten p-norm. Proc. Amer. Math. Soc., 80(2):277–282, 1980.
V. Bruneau and E.M. Ouhabaz. Lieb–Thirringe stimates for non-self-adjoint Schrödinger operators. J. Math. Phys., 49(9):093504, 10, 2008.
E.B. Davies. Linear operators and their spectra. Cambridge University Press, Cambridge, 2007.
M. Demuth, M. Hansmann, and G. Katriel. On the discrete spectrum of nonselfadjoint operators. J. Funct. Anal., 257:2742–2759, 2009.
M. Demuth, M. Hansmann, and G. Katriel. Lieb–ThirringTy pe Inequalities for Schrödinger Operators with a Complex-Valued Potential. Integral Equations Operator Theory, 75(1):1–5, 2013.
N. Dunford and J.T. Schwartz. Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space. Interscience Publishers John Wiley & Sons New York- London, 1963.
D.E. Edmunds and W.D. Evans. Spectral theory and differential operators. The Clarendon Press Oxford University Press, New York, 1987.
K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations. Springer-Verlag, New York, 2000.
S. Favorov and L. Golinskii. A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators. In Linear and complex analysis, volume 226 of American Mathematical Society Translations: Series 2, pages 37–47. Amer. Math. Soc., Providence, RI, 2009.
S. Favorov and L. Golinskii. Blaschke-type conditions in unbounded domains, generalized convexity and applications in perturbation theory. Preprint, arXiv:1204.4283, 2012.
R.L. Frank. Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc., 43(4):745–750, 2011.
R.L. Frank, A. Laptev, E.H. Lieb, and R. Seiringer. Lieb–Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys., 77(3):309– 316, 2006.
M.I. Gil’. Upper and lower bounds for regularized determinants. J. Inequal. Pure Appl. Math., 9(1), 2008.
I.C. Gohberg, S. Goldberg, and M.A. Kaashoek. Classes of linear operators. Vol. I. Birkh¨auser Verlag, Basel, 1990.
I.C. Gohberg, S. Goldberg, and N. Krupnik. Traces and determinants of linear operators. Birkh¨auser Verlag, Basel, 2000.
I.C. Gohbergan d M.G. Krein. Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, R.I., 1969.
L. Golinskii and S. Kupin. Lieb–Thirringb ounds for complex Jacobi matrices. Lett. Math. Phys., 82(1):79–90, 2007.
L. Golinskii and S. Kupin. A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator. J. Math. Anal. Appl., 2011. DOI 10.1016/j.jmaa.2011.12.011.
K.E. Gustafson and D.K.M. Rao. Numerical range. Universitext. Springer-Verlag, New York, 1997. The field of values of linear operators and matrices.
M. Hansmann. On the discrete spectrum of linear operators in Hilbert spaces. Dissertation, TU Clausthal, 2010. See “http://nbn-resolving.de/urn:nbn:de:gbv:104- 1097281” for an electronic version.
M. Hansmann. An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators. Lett. Math. Phys., 98(1):79–95, 2011.
M. Hansmann. Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators. Preprint, arXiv:1202.1118., 2012.
M. Hansmann and G. Katriel. Inequalities for the eigenvalues of non-selfadjoint Jacobi operators. Complex Anal. Oper. Theory, 5(1):197–218, 2011.
D. Hundertmark. Some bound state problems in quantum mechanics. In Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, volume 76 of Proc. Sympos. Pure Math., pages 463–496. Amer. Math. Soc., Providence, RI, 2007.
D. Hundertmark and B. Simon. Lieb–Thirringi nequalities for Jacobi matrices. J. Approx. Theory, 118(1):106–130, 2002.
T. Kato. Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Amer. Math. Soc., 70:195–211, 1951.
T. Kato. Variation of discrete spectra. Comm. Math. Phys., 111(3):501–504, 1987.
T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin, 1995.
A. Laptev and O. Safronov. Eigenvalue estimates for Schrödinger operators with complex-valued potentials. Commun. Math. Phys., 292(1):29–54, 2009.
A. Laptev and T. Weidl. Recent results on Lieb-Thirringin equalities. In Journées “ Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), pages Exp. No. XX, 14. Univ. Nantes, Nantes, 2000.
E.H. Lieb and W. Thirring. Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett., 35:687–689, 1975.
J. Mawhin. Spectra in mathematics and in physics: From the dispersion of light to nonlinear eigenvalues. In CIM Bulletin, No. 29, pages 03–13. Centro Internacional de Matemática, 2011.
A. Pietsch. Eigenvalues and s-numbers, volume 43 of Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology]. Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987.
Ch. Pommerenke. Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992.
W. Rudin. Real and complex analysis. McGraw-Hill Book Co., New York, third edition, 1987.
O. Safronov. Estimates for eigenvalues of the Schrödinger operator with a complex potential. Bull. Lond. Math. Soc., 42(3):452–456, 2010.
O. Safronov. On a sum rule for Schrödinger operators with complex potentials. Proc. Amer. Math. Soc., 138(6):2107–2112, 2010.
B. Simon. Notes on infinite determinants of Hilbert space operators. Advances in Math., 24(3):244–273, 1977.
B. Simon. Trace ideals and their applications. American Mathematical Society, Providence, RI, second edition, 2005.
D.R. Yafaev. Mathematical scattering theory. American Mathematical Society, Providence, RI, 1992.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Demuth, M., Hansmann, M., Katriel, G. (2013). Eigenvalues of Non-selfadjoint Operators: A Comparison of Two Approaches. In: Demuth, M., Kirsch, W. (eds) Mathematical Physics, Spectral Theory and Stochastic Analysis. Operator Theory: Advances and Applications(), vol 232. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0591-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0591-9_2
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0590-2
Online ISBN: 978-3-0348-0591-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)