On traces and modified Fredholm determinants for half-line Schrödinger operators with purely discrete spectra
Authors:
Fritz Gesztesy and Klaus Kirsten
Journal:
Quart. Appl. Math. 77 (2019), 615-630
MSC (2010):
Primary 47A10, 47B10, 47G10; Secondary 34B27, 34L40
DOI:
https://doi.org/10.1090/qam/1520
Published electronically:
September 6, 2018
MathSciNet review:
3962585
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Abstract: After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schrödinger operators $(- d^2/dx^2) + q$ on $(0,\infty )$ with purely discrete spectra. Roughly speaking, the class considered is generated by potentials $q$ that, for some fixed $C_0 > 0$, $\varepsilon > 0$, $x_0 \in (0, \infty )$, diverge at infinity in the manner that $q(x) \geq C_0 x^{(2/3) + \varepsilon _0}$ for all $x \geq x_0$. We treat all self-adjoint boundary conditions at the left endpoint $0$.
References
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- 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
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
- Barry Simon, Notes on infinite determinants of Hilbert space operators, Advances in Math. 24 (1977), no. 3, 244–273. MR 482328, DOI https://doi.org/10.1016/0001-8708%2877%2990057-3
- Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153
- Barry Simon, Operator theory, A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI, 2015. MR 3364494
- Joachim Weidmann, Lineare Operatoren in Hilberträumen. Teil II, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 2003 (German). Anwendungen. [Applications]. MR 2382320
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References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- D. Burghelea, L. Friedlander, and T. Kappeler, On the determinant of elliptic boundary value problems on a line segment, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3027–3038. MR 1301012, DOI https://doi.org/10.2307/2160656
- Stephen Clark, Fritz Gesztesy, Roger Nichols, and Maxim Zinchenko, Boundary data maps and Krein’s resolvent formula for Sturm-Liouville operators on a finite interval, Oper. Matrices 8 (2014), no. 1, 1–71. MR 3202926, DOI https://doi.org/10.7153/oam-08-01
- E. Brian Davies, Linear operators and their spectra, Cambridge Studies in Advanced Mathematics, vol. 106, Cambridge University Press, Cambridge, 2007. MR 2359869
- Tommy Dreyfus and Harry Dym, Product formulas for the eigenvalues of a class of boundary value problems, Duke Math. J. 45 (1978), no. 1, 15–37. MR 0481232
- M. S. P. Eastham, The asymptotic solution of linear differential systems, London Mathematical Society Monographs. New Series, vol. 4, The Clarendon Press, Oxford University Press, New York, 1989. Applications of the Levinson theorem; Oxford Science Publications. MR 1006434
- F. Gesztesy and K. Kirsten, Effective computation of traces, determinants, and $\zeta$-functions for Sturm–Liouville operators, arxiv:1712.00928, J. Funct. Anal., DOI 10.1016/j.jfa.2018.02.009 (to appear).
- Fritz Gesztesy and Konstantin A. Makarov, Erratum: “(Modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels revisited” [Integral Equations Operator Theory 47 (2003), no. 4, 457–497; MR2021969], Integral Equations Operator Theory 48 (2004), no. 3, 425–426. MR 2038511, DOI https://doi.org/10.1007/s00020-003-1278-0
- F. Gesztesy and R. Weikard, Floquet theory revisited, Differential equations and mathematical physics (Birmingham, AL, 1994) Int. Press, Boston, MA, 1995, pp. 67–84. MR 1703573
- Fritz Gesztesy and Maxim Zinchenko, Symmetrized perturbation determinants and applications to boundary data maps and Krein-type resolvent formulas, Proc. Lond. Math. Soc. (3) 104 (2012), no. 3, 577–612. MR 2900237, DOI https://doi.org/10.1112/plms/pdr024
- Israel Gohberg, Seymour Goldberg, and Naum Krupnik, Traces and determinants of linear operators, Integral Equations Operator Theory 26 (1996), no. 2, 136–187. MR 1410588, DOI https://doi.org/10.1007/BF01191855
- Israel Gohberg, Seymour Goldberg, and Naum Krupnik, Hilbert-Carleman and regularized determinants for linear operators, Integral Equations Operator Theory 27 (1997), no. 1, 10–47. MR 1426017, DOI https://doi.org/10.1007/BF01195742
- Israel Gohberg, Seymour Goldberg, and Nahum Krupnik, Traces and determinants of linear operators, Operator Theory: Advances and Applications, vol. 116, Birkhäuser Verlag, Basel, 2000. MR 1744872
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
- H. R. Grümm, Two theorems about ${\mathcal {C}}_{p}$, Rep. Mathematical Phys. 4 (1973), 211–215. MR 0327208
- K. Kirsten, Spectral Functions in Mathematics and Physics, CRC Press, Boca Raton, 2001.
- Klaus Kirsten and Alan J. McKane, Functional determinants by contour integration methods, Ann. Physics 308 (2003), no. 2, 502–527. MR 2018683, DOI https://doi.org/10.1016/S0003-4916%2803%2900149-0
- Matthias Lesch, Determinants of regular singular Sturm-Liouville operators, Math. Nachr. 194 (1998), 139–170. MR 1653090, DOI https://doi.org/10.1002/mana.19981940110
- Matthias Lesch and Jürgen Tolksdorf, On the determinant of one-dimensional elliptic boundary value problems, Comm. Math. Phys. 193 (1998), no. 3, 643–660. MR 1624851, DOI https://doi.org/10.1007/s002200050342
- Matthias Lesch and Boris Vertman, Regular singular Sturm-Liouville operators and their zeta-determinants, J. Funct. Anal. 261 (2011), no. 2, 408–450. MR 2793118, DOI https://doi.org/10.1016/j.jfa.2011.03.011
- S. Levit and U. Smilansky, A theorem on infinite products of eigenvalues of Sturm-Liouville type operators, Proc. Amer. Math. Soc. 65 (1977), no. 2, 299–302. MR 0457836, DOI https://doi.org/10.2307/2041911
- Govind Menon, The Airy function is a Fredholm determinant, J. Dynam. Differential Equations 28 (2016), no. 3-4, 1031–1038. MR 3537363, DOI https://doi.org/10.1007/s10884-015-9458-6
- J. Östensson and D. R. Yafaev, A trace formula for differential operators of arbitrary order, in A Panorama of Modern Operator Theory and Related Topics. The Israel Gohberg Memorial Volume, H. Dym, M. A. Kaashoek, P. Lancaster, H. Langer, and L. Lerer (eds.), Operator Theory: Advances and Appls., Vol. 218, Birkhäuser, Springer, 2012, pp. 541–570.
- 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
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
- Barry Simon, Notes on infinite determinants of Hilbert space operators, Advances in Math. 24 (1977), no. 3, 244–273. MR 0482328, DOI https://doi.org/10.1016/0001-8708%2877%2990057-3
- Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153
- Barry Simon, Operator theory, A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI, 2015. MR 3364494
- Joachim Weidmann, Lineare Operatoren in Hilberträumen. Teil II, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 2003 (German). Anwendungen. [Applications]. MR 2382320
- D. R. Yafaev, Mathematical scattering theory, Translations of Mathematical Monographs, vol. 105, American Mathematical Society, Providence, RI, 1992. General theory; Translated from the Russian by J. R. Schulenberger. MR 1180965
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Additional Information
Fritz Gesztesy
Affiliation:
Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
MR Author ID:
72880
ORCID:
[object Object]
Email:
Fritz_Gesztesy@baylor.edu
Klaus Kirsten
Affiliation:
GCAP-CASPER, Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
MR Author ID:
102045
Email:
Klaus_Kirsten@baylor.edu
Keywords:
Traces,
(modified) Fredholm determinants,
semi-separable integral kernels,
Sturm–Liouville operators,
discrete spectrum.
Received by editor(s):
April 18, 2018
Received by editor(s) in revised form:
July 21, 2018
Published electronically:
September 6, 2018
Additional Notes:
The second author was supported by the Baylor University Summer Sabbatical Program.
Article copyright:
© Copyright 2018
Brown University