Abstract
New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrödinger operators are obtained in terms of L p -norms of the potentials. The results cover and improve those known previously, in particular, due to Frank (Bull Lond Math Soc 43(4):745–750, 2011), Safronov (Proc Am Math Soc 138(6):2107–2112, 2010), Laptev and Safronov (Commun Math Phys 292(1):29–54, 2009). We mention the estimations of the eigenvalues situated in the strip around the real axis (in particular, the essential spectrum). The method applied for this case involves the unitary group generated by the Laplacian. The results are extended to the more general case of polyharmonic operators. Schrödinger operators with slowly decaying potentials and belonging to weak Lebesgue’s classes are also considered.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramov A.A., Aslanyan A., Davies E.B.: Bounds on complex eigenvalues and resonances. J. Phys. A 34(1), 57–72 (2001)
Babenko K.I.: An inequality in the theory of Fourier integrals. Izv. Akad. Nauk SSSR Ser. Mat. 25, 531–542 (1961)
Beckner W.: Inequalities in Fourier analysis. Ann. Math. 2 102(1), 159–182 (1975)
Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976).(Grundlehren der Mathematischen Wissenschaften, No. 223)
Berezin, F.A., Shubin, M.A.: The Schrödinger equation. In: Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991)
Davies E.B.: Non-self-adjoint differential operators. Bull. London Math. Soc. 34(5), 513–532 (2002)
Davies E.B., Nath J.: Schrödinger operators with slowly decaying potentials. J. Comput. Appl. Math. 148(1), 1–28 (2002)
Frank R.L., Laptev A., Lieb E.H., Seiringer R.: Lieb–Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77(3), 309–316 (2006)
Frank, R.L., Laptev, A., Seiringer, R.: A sharp bound on eigenvalues of Schrödinger operators on the half-line with complex-valued potentials. In: Spectral Theory and Analysis, vol. 214, pp. 39–44. Birkhäuser/Springer Basel AG, Basel (2011); Oper. Theory Adv. Appl.
Folland, G.B.: Real analysis. In: Pure and Applied Mathematics (New York), 2nd edn. Wiley, New York (1999)
Frank R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical Society, Providence (1974)
Jörgens, K., Weidmann, J.: Spectral properties of Hamiltonian operators. In: Lecture Notes in Mathematics, vol. 313. Springer, Berlin (1973)
Kato, T.: Perturbation theory for linear operators. In: Classics in Mathematics. Springer, Berlin (1995). (Reprint of the 1980 edition)
Keller J.B.: Lower bounds and isoperimetric inequalities for eigenvalues of the Schrödinger equation. J. Math. Phys. 2, 262–266 (1961)
Konno R., Konno R.: On the finiteness of perturbed eigenvalues. J. Fac. Sci. Univ. Tokyo Sect. I 13, 55–63 (1966)
Kenig C.E., Ruiz A., Sogge C.D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55(2), 329–347 (1987)
Laptev A., Safronov O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys. 292(1), 29–54 (2009)
Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics, pp. 269–303. Princeton University Press, Princeton (1976)
O’Neil, R.: Convolution operators and L(p, q) spaces. Duke Math. J. 30, 129–142 (1963)
Prosser R.T.: Convergent perturbation expansions for certain wave operators. J. Math. Phys. 5, 708–713 (1964)
Rejto P.A.: On partly gentle perturbations. III. J. Math. Anal. Appl. 27, 21–67 (1969)
Safronov O.: Estimates for eigenvalues of the Schrödinger operator with a complex potential. Bull. Lond. Math. Soc. 42(3), 452–456 (2010)
Safronov O.: On a sum rule for Schrödinger operators with complex potentials. Proc. Am. Math. Soc. 138(6), 2107–2112 (2010)
Schechter M.: Essential spectra of elliptic partial differential equations. Bull. Am. Math. Soc. 73, 567–572 (1967)
Schechter, M.: Spectra of partial differential operators. In: North-Holland Series in Applied Mathematics and Mechanics, vol. 14, 2nd edn. North-Holland Publishing Co., Amsterdam (1986)
Stummel F.: Singuläre elliptische differential-operatoren in Hilbertschen R äumen. Math. Ann. 132, 150–176 (1956)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Enblom, A. Estimates for Eigenvalues of Schrödinger Operators with Complex-Valued Potentials. Lett Math Phys 106, 197–220 (2016). https://doi.org/10.1007/s11005-015-0810-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-015-0810-x