Lower bounds for self-adjoint Sturm–Liouville operators

Behrndt, Jussi; Gesztesy, Fritz; Schmitz, Philipp; Trunk, Carsten

doi:10.1090/proc/16523

Lower bounds for self-adjoint Sturm–Liouville operators
HTML articles powered by AMS MathViewer

by Jussi Behrndt, Fritz Gesztesy, Philipp Schmitz and Carsten Trunk

Proc. Amer. Math. Soc. 151 (2023), 5313-5323

DOI: https://doi.org/10.1090/proc/16523

Published electronically: September 1, 2023

HTML | PDF | Request permission

Abstract:

In this note we provide estimates for the lower bound of the self-adjoint operator associated with the three-coefficient Sturm–Liouville differential expression \begin{equation*} \frac {1}{r} \left (-\frac {\mathrm d}{\mathrm dx} p \frac {\mathrm d}{\mathrm dx} + q\right ) \end{equation*} in the weighted $L^2$-Hilbert space $L^2(\mathbb R; rdx)$.

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
Jussi Behrndt, Philipp Schmitz, and Carsten Trunk, Spectral bounds for indefinite singular Sturm-Liouville operators with uniformly locally integrable potentials, J. Differential Equations 267 (2019), no. 1, 468–493. MR 3944278, DOI 10.1016/j.jde.2019.01.013
J. F. Barnes, H. J. Brascamp, and E. H. Lieb, Lower bound for the ground state energy of the Schrödinger equation using the sharp form of Young’s inequality, in Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann, E. H. Lieb, B. Simon, and A. S. Wightman (eds.), Princeton Series in Physics, Princeton Univ. Press, Princeton, N.J., 1976, pp. 83–90.
Jean-Claude Cuenin and Orif O. Ibrogimov, Sharp spectral bounds for complex perturbations of the indefinite Laplacian, J. Funct. Anal. 280 (2021), no. 1, Paper No. 108804, 26. MR 4157680, DOI 10.1016/j.jfa.2020.108804
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
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 188745
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, Eigenvalue bounds for Schrödinger operators with complex potentials. III, Trans. Amer. Math. Soc. 370 (2018), no. 1, 219–240. MR 3717979, DOI 10.1090/tran/6936
Rupert L. Frank, The Lieb-Thirring inequalities: recent results and open problems, Nine mathematical challenges—an elucidation, Proc. Sympos. Pure Math., vol. 104, Amer. Math. Soc., Providence, RI, [2021] ©2021, pp. 45–86. MR 4337417, DOI 10.1090/pspum/104/01877
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
F. Gesztesy, R. Nichols, and M. Zinchenko, Sturm–Liouville operators, their spectral theory, and some applications, monograph in preparation.
Philip Hartman, Differential equations with non-oscillatory eigenfunctions, Duke Math. J. 15 (1948), 697–709. MR 27927
Konrad Jörgens and Franz Rellich, Eigenwerttheorie gewöhnlicher Differentialgleichungen, Hochschultext. [University Textbooks], Springer-Verlag, Berlin-New York, 1976 (German). Überarbeitete und ergänzte Fassung der Vorlesungsausarbeitung “Eigenwerttheorie partieller Differentialgleichungen, Teil 1” von Franz Rellich (Wintersemester 1952/53); Bearbeitet von J. Weidmann. MR 499411, DOI 10.1007/978-3-642-66132-7
Ari Laptev, Spectral inequalities for partial differential equations and their applications, Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2, Amer. Math. Soc., Providence, RI, 2012, pp. 629–643. MR 2908096
D. B. Pearson, Quantum scattering and spectral theory, Techniques of Physics, vol. 9, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1988. MR 1099604
Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
Franz Rellich, Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung, Math. Ann. 122 (1951), 343–368 (German). MR 43316, DOI 10.1007/BF01342848
Martin Schechter, Operator methods in quantum mechanics, North-Holland Publishing Co., New York-Amsterdam, 1981. MR 597895
Gerald Teschl, Mathematical methods in quantum mechanics, 2nd ed., Graduate Studies in Mathematics, vol. 157, American Mathematical Society, Providence, RI, 2014. With applications to Schrödinger operators. MR 3243083, DOI 10.1090/gsm/157
Walter Thirring, A course in mathematical physics. Vol. 3, Lecture Notes in Physics, vol. 141, Springer-Verlag, New York-Vienna, 1981. Quantum mechanics of atoms and molecules; Translated from the German by Evans M. Harrell. MR 625662, DOI 10.1007/978-3-7091-7523-1
E. C. Titchmarsh, Eigenfunction expansions, part I, 2nd ed., Clarendon Press, Oxford, 1962.
Joachim Weidmann, Lineare Operatoren in Hilberträumen. Teil II, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 2003 (German). Anwendungen. [Applications]. MR 2382320, DOI 10.1007/978-3-322-80095-4
Anton Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, vol. 121, American Mathematical Society, Providence, RI, 2005. MR 2170950, DOI 10.1090/surv/121