Skip to main content
SpringerLink
Log in

Quantum Systems at the Brink

  • Conference paper
  • First Online:
Quantum Mathematics I (INdAM 2022)

Part of the book series: Springer INdAM Series ((SINDAMS,volume 57))

Included in the following conference series:

  • 187 Accesses

Abstract

We present a method to calculate the asymptotic behavior of eigenfunctions of Schrödinger operators that also works at the threshold of the essential spectrum. It can be viewed as a higher order correction to the well-known WKB method which does need a safety distance to the essential spectrum. We illustrate its usefulness on examples of quantum particles in a potential well with a long-range repulsive term outside the well.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Agmon, S.: Lower bounds for solutions of Schrödinger equations. J. Anal. Math. 23, 1–25 (1970). https://doi.org/10.1007/BF02795485

    Article  MATH  Google Scholar 

  2. Agmon, S.: Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators. In: Mathematical Notes, vol. 29. Princeton University Press, Princeton (1982)

    Google Scholar 

  3. Agmon, S.: Bounds on exponential decay of eigenfunctions of Schrödinger operators. In: Schrödinger Operators (Como, 1984). Lecture Notes in Math., vol. 1159, pp. 1–38. Springer, Berlin (1985). https://doi.org/10.1007/BFb0080331

  4. Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Commun. Pure Appl. Math. 35(2), 209–273 (1982). https://doi.org/10.1002/cpa.3160350206

    Article  MATH  Google Scholar 

  5. Baker, J.D., Freund, D.E., Hill, R.N., Morgan III, J.D.: Radius of convergence and analytic behavior of the 1 z expansion. Phys. Rev. A 41(3), 1247 (1990)

    Article  MathSciNet  Google Scholar 

  6. Barbaroux J.M., Hartig M.C., Hundertmark D., Vugalter, S.: Van der Waals-London interaction of atoms with pseudorelativistic kinetic energy. Analysis & PDE 15(6), 1375–1428 (2022). https://doi.org/10.2140/apde.2022.15.1375

    Article  MathSciNet  MATH  Google Scholar 

  7. Bellazzini, J., Frank, R.L., Lieb, E.H., Seiringer, R.: Existence of ground states for negative ions at the binding threshold. Rev. Math. Phys. 26(1), 1350021, 18 (2014). https://doi.org/10.1142/S0129055X13500219

  8. Bethe, H.: Berechnung der Elektronenaffinität des Wasserstoffs. Z. Phys. 57(11–12), 815–821 (1929)

    Article  MATH  Google Scholar 

  9. Bollé, D., Gesztesy, F., Schweiger, W.: Scattering theory for long-range systems at threshold. J. Math. Phys. 26(7), 1661–1674 (1985). https://doi.org/10.1063/1.526963

    Article  MathSciNet  MATH  Google Scholar 

  10. Burton, H.G.A.: Hartree–Fock critical nuclear charge in two-electron atoms. J. Chem. Phys. 154(11), 111103 (2021). https://doi.org/10.1063/5.0043105

    Article  Google Scholar 

  11. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, study edn. Texts and Monographs in Physics. Springer, Berlin (1987)

    Google Scholar 

  12. Dubau, J., Ivanov, I.A.: Numerical calculation of the complex energy of the resonance of a two-electron atom with nuclear charge below the threshold value. J. Phys. B: At. Mol. Opt. Phys. 31(15), 3335–3344 (1998). https://doi.org/10.1088/0953-4075/31/15/007

    Article  Google Scholar 

  13. Estienne, C.S., Busuttil, M., Moini, A., Drake, G.W.F.: Critical nuclear charge for two-electron atoms. Phys. Rev. Lett. 112, 173001 (2014). https://doi.org/10.1103/PhysRevLett.112.173001

    Article  Google Scholar 

  14. Grabowski, P.E., Burke, K.: Quantum critical benchmark for electronic structure theory. Phys. Rev. A 91(3), 032501 (2015)

    Article  Google Scholar 

  15. Gridnev, D.K., Garcia, M.E.: Rigorous conditions for the existence of bound states at the threshold in the two-particle case. J. Phys. A 40(30), 9003–9016 (2007). https://doi.org/10.1088/1751-8113/40/30/022

    Article  MathSciNet  MATH  Google Scholar 

  16. Griesemer, M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210(2), 321–340 (2004). https://doi.org/10.1016/j.jfa.2003.06.001

    Article  MathSciNet  MATH  Google Scholar 

  17. Hoffmann-Ostenhof, T.: A comparison theorem for differential inequalities with applications in quantum mechanics. J. Phys. A 13(2), 417–424 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Simon, B.: A multiparticle Coulomb system with bound state at threshold. J. Phys. A 16(6), 1125–1131 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hogreve, H.: On the (de) stabilization of quantum mechanical binding by potential barriers. Phys. Lett. A 201(2–3), 111–118 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hogreve, H.: On the maximal electronic charge bound by atomic nuclei. J. Phys. B: At. Mol. Opt. Phys. 31(10), L439–L446 (1998). https://doi.org/10.1088/0953-4075/31/10/001

    Article  Google Scholar 

  21. Hundertmark, D., Lee, Y.R.: On non-local variational problems with lack of compactness related to non-linear optics. J. Nonlinear Sci. 22(1), 1–38 (2012). https://doi.org/10.1007/s00332-011-9106-1

    Article  MathSciNet  MATH  Google Scholar 

  22. Hundertmark, D., Jex, M., Lange, M.: Quantum systems at The Brink. existence and decay rates of bound states at thresholds; atoms, p. 14 (2019). Preprint arXiv:1908.05016

    Google Scholar 

  23. Hundertmark, D., Jex, M., Lange, M.: Quantum systems at the brink: Helium–type systems, p. 62 (2021). Preprint arXiv:1908.04883v2

    Google Scholar 

  24. Hundertmark, D., Jex, M., Lange, M.: Quantum systems at The Brink: existence of bound states, critical potentials and dimensionality, Forum of Mathematics, Sigma 11, e61 (2023). https://doi.org/10.1017/fms.2023.39

    MATH  Google Scholar 

  25. Ismagilov, R.S.: Conditions for the semiboundedness and discreteness of the spectrum in the case of one-dimensional differential operators. Dokl. Akad. Nauk SSSR 140, 33–36 (1961)

    MathSciNet  Google Scholar 

  26. Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3), 583–611 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kaleta, K., Lőrinczi, J.: Zero-energy bound state decay for non-local Schrödinger operators. Commun. Math. Phys. 374(3), 2151–2191 (2020). https://doi.org/10.1007/s00220-019-03515-3

  28. Kenig, C.E.: Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation. In: Harmonic Analysis and Partial Differential Equations (El Escorial, 1987). Lecture Notes in Mathematics, vol. 1384, pp. 69–90. Springer, Berlin (1989). https://doi.org/10.1007/BFb0086794

  29. Klaus, M., Simon, B.: Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case. Ann. Phys. 130(2), 251–281 (1980). https://doi.org/10.1016/0003-4916(80)90338-3

    MATH  Google Scholar 

  30. Knowles, I.: On the location of eigenvalues of second-order linear differential operators. Proc. R. Soc. Edinb. A 80(1–2), 15–22 (1978). https://doi.org/10.1017/S030821050001009X

    Article  MathSciNet  MATH  Google Scholar 

  31. Knowles, I.: On the number of \(L^2\)-solutions of second order linear differential equations. Proc. R. Soc. Edinb. A 80(1–2), 1–13 (1978). https://doi.org/10.1017/S0308210500010088

  32. Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53(4), 603–641 (1981). https://doi.org/10.1103/RevModPhys.53.603

    Article  MathSciNet  MATH  Google Scholar 

  33. Lieb, E.H.: Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A 29, 3018–3028 (1984). https://doi.org/10.1103/PhysRevA.29.3018

    Article  Google Scholar 

  34. Mirtschink, A., Umrigar, C.J., Morgan, J.D., Gori-Giorgi, P.: Energy density functionals from the strong-coupling limit applied to the anions of the he isoelectronic series. J. Chem. Phys. 140(18), 18A532 (2014). https://doi.org/10.1063/1.4871018

  35. Morgan III, J.D.: Schrödinger operators whose potentials have separated singularities. J. Oper. Theory 1(1), 109–115 (1979)

    MATH  Google Scholar 

  36. Nakamura, S.: Low energy asymptotics for Schrödinger operators with slowly decreasing potentials. Commun. Math. Phys. 161(1), 63–76 (1994)

    Article  MATH  Google Scholar 

  37. Newton, R.G.: Nonlocal interactions; the generalized Levinson theorem and the structure of the spectrum. J. Math. Phys. 18(8), 1582–1588 (1977). https://doi.org/10.1063/1.523466

    Article  MathSciNet  Google Scholar 

  38. Persson, A.: Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8, 143–153 (1960). https://doi.org/10.7146/math.scand.a-10602

    Article  MathSciNet  MATH  Google Scholar 

  39. Ramm, A.G.: Sufficient conditions for zero not to be an eigenvalue of the Schrödinger operator. J. Math. Phys. 28(6), 1341–1343 (1987). https://doi.org/10.1063/1.527817

    Article  MathSciNet  MATH  Google Scholar 

  40. Ramm, A.G.: Conditions for zero not to be an eigenvalue of the Schrödinger operator. II. J. Math. Phys. 29(6), 1431–1432 (1988). https://doi.org/10.1063/1.527935

    Article  MATH  Google Scholar 

  41. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic, New York-London (1978)

    Google Scholar 

  42. Sergeev, A.V., Kais, S.: Resonance states of atomic anions. Int. J. Quantum Chem. 82(5), 255–261 (2001). https://doi.org/10.1002/qua.1047

    Article  Google Scholar 

  43. Sigal, I.M.: Geometric methods in the quantum many-body problem. Nonexistence of very negative ions. Commun. Math. Phys. 85(2), 309–324 (1982)

    Article  MATH  Google Scholar 

  44. Simon, B.: Large time behavior of the \(L^p\) norm of Schrödinger semigroups. J. Funct. Anal. 40(1), 66–83 (1981). https://doi.org/10.1016/0022-1236(81)90073-2

  45. Simon, B.: Schrödinger semigroups. Bull. Amer. Math. Soc. (New Series) 7(3), 447–526 (1982). https://doi.org/bams/1183549767

  46. Teschl, G.: Mathematical Methods in Quantum Mechanics. Graduate Studies in Mathematics, vol. 157, 2nd edn. American Mathematical Society, Providence (2014). https://doi.org/10.1090/gsm/157

  47. Uchiyama, J.: Finiteness of the number of discrete eigenvalues of the schrödinger operator for a three particle system. Publ. Res. Inst. Math. Sci. 5(1), 51–63 (1969). https://doi.org/10.2977/prims/1195194752

    Article  MathSciNet  MATH  Google Scholar 

  48. von Neumann, J., Wigner, E.P.: Über merkwürdige diskrete Eigenwerte. In: The Collected Works of Eugene Paul Wigner, pp. 291–293. Springer, Berlin (1993)

    Google Scholar 

  49. Yafaev, D.R.: The virtual level of the Schrödinger equation. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 51, 203–216, 220 (1975)

    Google Scholar 

  50. Yafaev, D.R.: The low energy scattering for slowly decreasing potentials. Commun. Math. Phys. 85(2), 177–196 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhislin, G.M.: Discussion of the spectrum of schrödinger operators for systems of many particles. Trudy Moskovskogo matematiceskogo obscestva 9, 81–120 (1960)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 258734477—SFB 1173 (Dirk Hundertmark). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement MDFT No. 725528) (Michal Jex). Michal Jex also received financial support from the Ministry of Education, Youth and Sport of the Czech Republic under the Grant No. RVO 14000. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC StG MaMBoQ, grant agreement No. 802901) (Markus Lange).

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute for Analysis, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Dirk Hundertmark

  2. Department of Mathematics, Altgeld Hall, University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Dirk Hundertmark

  3. Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Michal Jex

  4. CEREMADE, Université Paris-Dauphine, PSL Research University, Paris, France

    Michal Jex

  5. Institute for AI-Safety and Security, German Aerospace Center (DLR), Sankt Augustin & Ulm, Germany

    Markus Lange

Authors
  1. Dirk Hundertmark
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michal Jex
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Markus Lange
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michal Jex .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Michele Correggi

  2. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Marco Falconi

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Hundertmark, D., Jex, M., Lange, M. (2023). Quantum Systems at the Brink. In: Correggi, M., Falconi, M. (eds) Quantum Mathematics I. INdAM 2022. Springer INdAM Series, vol 57. Springer, Singapore. https://doi.org/10.1007/978-981-99-5894-8_10

Download citation

Publish with us

Policies and ethics

Search

Navigation