Letters in Mathematical Physics

, Volume 108, Issue 5, pp 1307–1322 | Cite as

Asymptotics of resonances for 1D Stark operators

  • Evgeny L. Korotyaev


We consider the Stark operator perturbed by a compactly supported potential on the real line. We determine the forbidden domain for resonances, asymptotics of resonances at high energy and asymptotics of the resonance counting function for large radius.


Stark operator Asymptotics of resonances The resonance counting function Forbidden domain for resonances 

Mathematics Subject Classification

34F15 (47E05) 



Our study was supported by the RSF Grant No. 15-11-30007.


  1. 1.
    Agler, J., Froese, R.: Existence of Stark ladder resonances. Commun. Math. Phys. 100, 161–171 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brown, B., Knowles, I., Weikard, R.: On the inverse resonance problem. J. Lond. Math. Soc. 68(2), 383–401 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Calogero, F., Degasperis, A.: Inverse spectral problem for the one-dimensional Schrödinger equation with an additional linear potential. Lettere Al Nuovo Cimento 23(4), 143–149 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Christiansen, T.: Resonances for steplike potentials: forward and inverse results. Trans. Am. Math. Soc. 358(5), 2071–2089 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Firsova, N.: Resonances of the perturbed Hill operator with exponentially decreasing extrinsic potential. Math. Notes 36(5), 854–861 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Froese, R.: Asymptotic distribution of resonances in one dimension. J. Differ. Equ. 137(2), 251–272 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gohberg, I., Krein, M.: Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian, Translations of Mathematical Monographs, Vol. 18 AMS, Providence, R.I. (1969)Google Scholar
  8. 8.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Translated from Russian by Scripta Technica, Inc. In: Jeffrey, A., Zwillinger, D. (eds.) Academic press (2007)Google Scholar
  9. 9.
    Herbst, I.: Dilation analyticity in constant electric field, I, The two body problem. Commun. Math. Phys. 64, 279–298 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Herbst, I.W., Howland, J.S.: The Stark ladder and other one-dimensional external field problems. Commun. Math. Phys. 80, 23–42 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Herbst, I., Mavi, R.: Can we trust the relationship between resonance poles and lifetimes? J. Phys. A 49(19), 195204–46 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hitrik, M.: Bounds on scattering poles in one dimension. Commun. Math. Phys. 208(2), 381–411 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jensen, A.: Commutator methods and asymptotic completeness for one-dimensional Stark effect Hamiltonians Schrödinger Operators, Aarhus 1985. Lecture Notes in Mathematics 1218, 151–166 (1986)Google Scholar
  14. 14.
    Jensen, A.: Asymptotic completeness for a new class of Stark effect Hamiltonians. Commun. Math. Phys. 107, 21–28 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jensen, A.: Perturbation results for Stark effect resonances. J. Reine Angew. Math 394, 168–179 (1989)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kachalov, A., Kurylev, Y.: The method of transformation operators in the inverse scattering problem. The one-dimensional Stark effect. J. Sov. Math. 57(3), 3111–3122 (1991)CrossRefzbMATHGoogle Scholar
  17. 17.
    Korotyaev, E.: Resonances for 1d Stark operators. J Spectr. Theor. 7(3), 699–732 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Korotyaev, E.: Inverse resonance scattering on the half line. Asymptot. Anal. 37(3–4), 215–226 (2004)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Korotyaev, E.: Stability for inverse resonance problem. Int. Math. Res. Not. 73, 3927–3936 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Korotyaev, E.: Inverse resonance scattering on the real line. Inverse Probl. 21(1), 325–341 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Korotyaev, E.: Resonance theory for perturbed Hill operator. Asymptot. Anal. 74(3–4), 199–227 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Korotyaev, E.: Estimates of 1D resonances in terms of potentials. J. d’Analyse Math. 130, 151–166 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Korotyaev, E., Schmidt, K.: On the resonances and eigenvalues for a 1D half-crystal with localized impurity. J. Reine Angew. Math. 670, 217–248 (2012)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kristensson, G.: The one-dimensional inverse scattering problem for an increasing potential. J. Math. Phys. 27(3), 804–815 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lin, Y., Qian, M., Zhang, Q.: Inverse scattering problem for one-dimensional Schrodinger operators related to the general Stark effect. Acta Math. Appl. Sin. 5(2), 116–136 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu, Y.: Scattering and spectral theory for Stark Hamiltonians in one dimension. Math. Scand. 72, 265–297 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Marletta, M., Shterenberg, R., Weikard, R.: On the inverse resonance problem for Schrödinger operators. Commun. Math. Phys. 295, 465–484 (2010)ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)zbMATHGoogle Scholar
  29. 29.
    Pozharskii, A.: On the nature of the Stark–Wannier spectrum. St. Petersb. Math. J. 16(3), 561–581 (2005)CrossRefzbMATHGoogle Scholar
  30. 30.
    Rejto, P.A., Sinha, K.: Absolute continuity for a 1-dimensional model of the Stark-Hamiltonian. Helv. Phys. Acta 49, 389–413 (1976)MathSciNetGoogle Scholar
  31. 31.
    Simon, B.: Resonances in one dimension and Fredholm determinants. J. Funct. Anal. 178(2), 396–420 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zworski, M.: Distribution of poles for scattering on the real line. J. Funct. Anal. 73, 277–296 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zworski, M., SIAM, : A remark on isopolar potentials. J. Math. Anal. 82(6), 1823–1826 (2002)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Saint-Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations