Eigenvalue bounds for Schrödinger operators with complex potentials. III

Frank, Rupert

doi:10.1090/tran/6936

Eigenvalue bounds for Schrödinger operators with complex potentials. III
HTML articles powered by AMS MathViewer

by Rupert L. Frank PDF

Trans. Amer. Math. Soc. 370 (2018), 219-240

Abstract:

We discuss the eigenvalues $E_j$ of Schrödinger operators $-\Delta +V$ in $L^2(\mathbb {R}^d)$ with complex potentials $V\in L^p$, $p<\infty$. We show that (A) $\operatorname {Re} E_j\to \infty$ implies $\operatorname {Im} E_j\to 0$, and (B) $\operatorname {Re} E_j\to E\in [0,\infty )$ implies $(\operatorname {Im} E_j)\in \ell ^q$ for some $q$ depending on $p$. We prove quantitative versions of (A) and (B) in terms of the $L^p$-norm of $V$.

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
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
E. B. Davies and Jiban Nath, Schrödinger operators with slowly decaying potentials, J. Comput. Appl. Math. 148 (2002), no. 1, 1–28. On the occasion of the 65th birthday of Professor Michael Eastham. MR 1946184, DOI 10.1016/S0377-0427(02)00570-8
Michael Demuth and Guy Katriel, Eigenvalue inequalities in terms of Schatten norm bounds on differences of semigroups, and application to Schrödinger operators, Ann. Henri Poincaré 9 (2008), no. 4, 817–834. MR 2413204, DOI 10.1007/s00023-008-0373-9
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
Michael Demuth, Marcel Hansmann, and Guy Katriel, Eigenvalues of non-selfadjoint operators: a comparison of two approaches, Mathematical physics, spectral theory and stochastic analysis, Oper. Theory Adv. Appl., vol. 232, Birkhäuser/Springer Basel AG, Basel, 2013, pp. 107–163. MR 3077277, DOI 10.1007/978-3-0348-0591-9_{2}
D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR 929030
Alexandra Enblom, Estimates for eigenvalues of Schrödinger operators with complex-valued potentials, Lett. Math. Phys. 106 (2016), no. 2, 197–220. MR 3451537, DOI 10.1007/s11005-015-0810-x
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
R. L. Frank, Eigenvalues of Schrödinger operators with complex surface potentials, in Functional Analysis and Operator Theory for Quantum Physics, J. Dittrich, et al. (eds.), EMS, 2017, pp. 245–260.
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 and Alexander Pushnitski, Trace class conditions for functions of Schrödinger operators, Comm. Math. Phys. 335 (2015), no. 1, 477–496. MR 3314510, DOI 10.1007/s00220-014-2205-8
R. L. Frank and J. Sabin, Restriction theorems for orthonormal functions, Strichartz inequalities and uniform Sobolev estimates. Amer. J. Math., to appear, http://arxiv.org/pdf/ 1404.2817.pdf
R. L. Frank and B. Simon, Eigenvalue bounds for Schrödinger operators with complex potentials. II, J. Spectr. Theory, to appear.
Fritz Gesztesy, Helge Holden, and Roger Nichols, On factorizations of analytic operator-valued functions and eigenvalue multiplicity questions, Integral Equations Operator Theory 82 (2015), no. 1, 61–94. MR 3335509, DOI 10.1007/s00020-014-2200-7
F. Gesztesy, Y. Latushkin, M. Mitrea, and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys. 12 (2005), no. 4, 443–471. MR 2201310
I. C. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of linear operators. Vol. 1, Birkhäuser Verlag, Basel, 2000.
Franz Hanauska, On the closure of the discrete spectrum of nuclearly perturbed operators, Oper. Matrices 9 (2015), no. 2, 359–364. MR 3338569, DOI 10.7153/oam-09-21
M. Hansmann, On the discrete spectrum of linear operators in Hilbert spaces, PhD thesis, TU Clausthal, 2010, http://d-nb.info/1001898664/34/
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
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329–347. MR 894584, DOI 10.1215/S0012-7094-87-05518-9
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
Y. Latushkin and A. Sukhtayev, The algebraic multiplicity of eigenvalues and the Evans function revisited, Math. Model. Nat. Phenom. 5 (2010), no. 4, 269–292. MR 2662459, DOI 10.1051/mmnp/20105412
Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
E. H. Lieb aaand W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics. Princeton University Press, Princeton (1976), pp. 269–303.
R. M. Martirosjan, On the spectrum of the non-selfadjoint operator $-\Delta u+cu$ in three dimensional space, Izv. Akad. Nauk Armyan. SSR. Ser. Fiz.-Mat. Nauk 10 (1957), no. 1, 85–111 (Russian, with Armenian summary). MR 0103332
R. M. Martirosjan, On the spectrum of various perturbations of the Laplace operator in spaces of three or more dimensions, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 897–920 (Russian). MR 0133592
M. A. Naĭmark, Investigation of the spectrum and the expansion in eigenfunctions of a nonselfadjoint operator of the second order on a semi-axis, Trudy Moskov. Mat. Obšč. 3 (1954), 181–270 (Russian). MR 0062311
B. S. Pavlov, On a non-selfadjoint Schrödinger operator, Probl. Math. Phys., No. I, Spectral Theory and Wave Processes (Russian), Izdat. Leningrad. Univ., Leningrad, 1966, pp. 102–132 (Russian). MR 0203530
B. S. Pavlov, On a non-selfadjoint Schrödinger operator. II, Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems (Russian), Izdat. Leningrad. Univ., Leningrad, 1967, pp. 133–157 (Russian). MR 0234319
B. S. Pavlov, On a nonselfadjoint Schrödinger operator. III, Problems of mathematical physics, No. 3: Spectral theory (Russian), Izdat. Leningrad. Univ., Leningrad, 1968, pp. 59–80 (Russian). MR 0348554
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
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
Barry Simon, Notes on infinite determinants of Hilbert space operators, Advances in Math. 24 (1977), no. 3, 244–273. MR 482328, DOI 10.1016/0001-8708(77)90057-3
Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
D. R. Yafaev, Mathematical scattering theory, Mathematical Surveys and Monographs, vol. 158, American Mathematical Society, Providence, RI, 2010. Analytic theory. MR 2598115, DOI 10.1090/surv/158