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Resonant Rigidity for Schrödinger Operators in Even Dimensions

  • T. J. ChristiansenEmail author
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Abstract

This paper studies the resonances of Schrödinger operators with bounded, compactly supported, real-valued potentials on \({{\mathbb {R}}}^d\), where the dimension d is even. If the potential V is non-trivial and \(d\not =4\), then the meromorphic continuation of the resolvent of the Schrödinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if \(d=4\). We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schrödinger operators \(-\varDelta +V_1\) and \(-\varDelta +V_2\) have the same poles, \(V_1,\;V_2\in L^\infty _c({{\mathbb {R}}}^d;{{\mathbb {R}}})\), \(k\in {{\mathbb {N}}}\), and if \(V_1\in H^k({{\mathbb {R}}}^d;{{\mathbb {R}}})\), then \(V_2\in H^k\) as well. Moreover, we prove that certain sets of isoresonant potentials are compact. We also show that the poles of the resolvent for a smooth potential determine the heat coefficients and that the (resolvent) resonance sets of two potentials in \(L^\infty _c({{\mathbb {R}}}^d;{{\mathbb {R}}})\) cannot differ by a nonzero finite number of elements away from 0.

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Acknowledgements

It is a pleasure to thank Rafael Benguria, Peter Hislop, Antônio Sá Barreto, and Maciej Zworski for helpful conversations. The author is grateful to the Simons Foundation for its support through the Collaboration Grants for Mathematicians program.

References

  1. 1.
    Autin, A.: Isoresonant complex-valued potentials and symmetries. Can. J. Math. 63(4), 721–754 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bañuelos, R., Sá Barreto, A.: On the heat trace of Schrödinger operators. Commun. Partial Differ. Equ. 20(11–12), 2153–2164 (1995)zbMATHGoogle Scholar
  3. 3.
    Benguria, R., Yarur, C.: Sharp condition on the decay of the potential for the absence of a zero-energy ground state of the Schrödinger equation. J. Phys. A 23(9), 1513–1518 (1990)ADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    Boas Jr., R.P.: Entire Functions. Academic Press, New York (1954)zbMATHGoogle Scholar
  5. 5.
    Bollé, D., Danneels, C., Osburn, T.A.: Local and global spectral shift functions in \({\mathbb{R}}^2\). J. Math. Phys. 30(2), 420–432 (1989)ADSMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bollé, D., Gesztesy, F., Danneels, C.: Threshold scattering in two dimensions. Ann. Inst. H. Poincaré Phys. Théor. 48(2), 175–204 (1988)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brüning, J.: On the compactness of isospectral potentials. Commun. Partial Differ. Equ. 9(7), 687–698 (1984)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen, L.-H.: A sub-logarithmic lower bound for resonance counting function in two-dimensional potential scattering. Rep. Math. Phys. 65(2), 157–164 (2010)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Christiansen, T.: Schrödinger operators with complex-valued potentials and no resonances. Duke Math. J. 133(2), 313–323 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Christiansen, T.: Isophasal, isopolar, and isospectral Schrödinger operators and elementary complex analysis. Am. J. Math. 130(1), 49–58 (2008)zbMATHGoogle Scholar
  11. 11.
    Christiansen, T.J.: Lower bounds for resonance counting functions for obstacle scattering in even dimensions. Am. J. Math. 139(3), 617–640 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Christiansen, T.J., Hislop, P.D.: Maximal order of growth for the resonance counting functions for generic potentials in even dimensions. Indiana Univ. Math. J. 59(2), 621–660 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Christiansen, T.J., Hislop, P.D.: Some remarks on resonances in even-dimensional Euclidean scattering. Trans. Am. Math. Soc. 368(2), 1361–1385 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Colin de Verdière, Y.: Une formule de traces pour l’opérateur de Schrödinger dans \({\mathbb{R}}^3\). Ann. Sci. École Norm. Sup. (4) 14(1), 27–39 (1981)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Donnelly, H.: Compactness of isospectral potentials. Trans. Am. Math. Soc. 357(5), 1717–1730 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dyatlov, S., Zworski, M.: Mathematical theory of scattering resonances. http://math.mit.edu/~dyatlov/res/res_20170323.pdf. Accessed 10 July 2018
  17. 17.
    Froese, R.: Asymptotic distribution of resonances in one dimension. J. Differ. Equ. 137(2), 251–272 (1997)ADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Froese, R.: Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions. Can. J. Math. 50(3), 538–546 (1998)zbMATHGoogle Scholar
  19. 19.
    Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence, RI (1969)zbMATHGoogle Scholar
  20. 20.
    Guillopé, L.: Asymptotique de la phase de diffusion pour l’opérateur de Schrödinger dans \({\mathbb{R}}^n\). Bony-Sjöstrand-Meyer seminar, 1984–1985, Exp. No. 5, École Polytech., Palaiseau (1985)Google Scholar
  21. 21.
    Hislop, P.D., Wolf, R.: Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three. Preprint arXiv:1803.02172
  22. 22.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin (1990)Google Scholar
  23. 23.
    Intissar, A.: A polynomial bound on the number of the scattering poles for a potential in even-dimensional spaces \({\mathbb{R}}^n\). Commun. Partial Differ. Equ. 11(4), 367–396 (1986)zbMATHGoogle Scholar
  24. 24.
    Isozaki, H., Korotyaev, E.L.: New trace formulas in terms of resonances for three-dimensional Schrödinger operators. Russ. J. Math. Phys. 25(1), 27–43 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in \(L^2({\mathbb{R}}^m),\; m \ge 5\). Duke Math. J. 47(1), 57–80 (1980)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in \(L^2({\mathbb{R}}^4)\). J. Math. Anal. Appl. 101(2), 397–422 (1984)MathSciNetGoogle Scholar
  27. 27.
    Jensen, A.: High Energy Asymptotics for the Total Scattering Phase in Potential Scattering Theory. Functional-analytic methods for partial differential equations (Tokyo, 1989). Lecture Notes in Mathematics, vol. 1450, pp. 187–195. Springer, Berlin (1990)Google Scholar
  28. 28.
    Komech, A., Kopylova, E.: Dispersion Decay and Scattering Theory. Wiley, Hoboken, NJ (2012)zbMATHGoogle Scholar
  29. 29.
    Korotyaev, E.: Inverse resonance scattering on the half line. Asymptot. Anal. 37(3–4), 215–226 (2004)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Korotyaev, E.: Stability for inverse resonance problem. Int. Math. Res. Not. 73, 3927–3936 (2004)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Korotyaev, E.: Inverse resonance scattering on the real line. Inverse Probl. 21(1), 325–341 (2005)ADSMathSciNetzbMATHGoogle Scholar
  32. 32.
    Levin, B.Ja: Distribution of Zeros of Entire Functions. American Mathematical Society, Providence, RI (1964)zbMATHGoogle Scholar
  33. 33.
    Müller, J., Strohmaier, A.: The theory of Hahn-meromorphic functions, a holomorphic Fredholm theorem, and its applications. Anal. PDE 7(3), 745–770 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Popov, G.S.: Asymptotic behaviour of the scattering phase for the Schrödinger operator. C. R. Acad. Bulgare Sci. 35(7), 885–888 (1982)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Sá Barreto, A.: Lower bounds for the number of resonances in even-dimensional potential scattering. J. Funct. Anal. 169(1), 314–323 (1999)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Sá Barreto, A.: Remarks on the distribution of resonances in odd dimensional Euclidean scattering. Asymptot. Anal. 27(2), 161–170 (2001)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Simon, B.: Resonances in one dimension and Fredholm determinants. J. Funct. Anal. 178(2), 396–420 (2000)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Sjöstrand, J., Zworski, M.: Complex scaling and the distribution of scattering poles. J. Am. Math. Soc. 4(4), 729–769 (1991)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Smith, H.: On the trace of Schrödinger heat kernels and regularity of potentials. Preprint arXiv:1809.05614
  40. 40.
    Smith, H., Zworski, M.: Heat traces and existence of scattering resonances for bounded potentials. Ann. Inst. Fourier (Grenoble) 66(2), 455–475 (2016)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Vodev, G.: Sharp bounds on the number of scattering poles in even-dimensional spaces. Duke Math. J. 74(1), 1–17 (1994)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Vodev, G.: Sharp bounds on the number of scattering poles in the two-dimensional case. Math. Nachr. 170, 287–297 (1994)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Yafaev, D.: Scattering Theory: Some Old and New Problems. Lecture Notes in Mathematics, vol. 1735. Springer, Berlin (2000)zbMATHGoogle Scholar
  44. 44.
    Yafaev, D.: The Schrödinger operator: perturbation determinants, the spectral shift function, trace identities, and more. Funktsional. Anal. i Prilozhen. 41(3), 60–83, 96 (2007). (translation in Funct. Anal. Appl. 41(3), 217–236, 2007)MathSciNetGoogle Scholar
  45. 45.
    Yafaev, D.: Mathematical Scattering Theory. Analytic Theory. Mathematical Surveys and Monographs, vol. 158. American Mathematical Society, Providence, RI (2010)zbMATHGoogle Scholar
  46. 46.
    Zworski, M.: Distribution of poles for scattering on the real line. J. Funct. Anal. 73(2), 277–296 (1987)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Zworski, M.: Poisson formula for resonances in even dimensions. Asian J. Math. 2(3), 609–617 (1998)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Zworski, M.: A remark on isopolar potentials. SIAM J. Math. Anal. 32(6), 1324–1326 (2001)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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