On the imaginary part of coupling resonance points

• Autorzy: Azamov Nurulla , Daniels Tom

Identyfikatory

DOI

10.7494/OpMath.2019.39.5.611

• Języki publikacji: EN

Abstrakty

EN

We prove for rank one perturbations that the imaginary part of a coupling resonance point is inversely proportional by a factor of —2 to the rate of change of the scattering phase, as a function of the coupling variable, evaluated at the real part of the resonance point. This equality is analogous to the Breit-Wigner formula from quantum scattering theory. For more general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of coupling resonance points, non-real and real.

Słowa kluczowe

EN

scattering matrix; scattering phase; resonance point; Breit-Wigner formula

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2019
• Tom: Vol. 39, no. 5
• Strony: 611--621
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Azamov Nurulla

• Flinders University College of Science and Engineering South Rd, Tonsley, SA 5042 Australia

autor

Daniels Tom

• Flinders University College of Science and Engineering South Rd, Tonsley, SA 5042 Australia

Bibliografia

• [1] M. Aizenman, S. Warzel, Random Operators: Disorder Effects on Quantum Spectra and Dynamics, Grad. Stud. Math., AMS, 2015.
• [2] M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry I-III, Math. Proc. Camb. Phil. Soc. 77-79 (1975-1976).
• [3] N.A. Azamov, Absolutely continuous and singular spectral shift functions, Dissertationes Math. 480 (2011), 1-102.
• [4] N.A. Azamov, Spectral flow inside essential spectrum, Dissertationes Math. 518 (2016), 1-156.
• [5] N.A. Azamov, Spectral flow and resonance index, Dissertationes Math. 528 (2017), 1-91.
• [6] N.A. Azamov, A.L. Carey, F.A. Sukochev, The spectral shift function and spectral flow, Commun. Math. Phys. 276 (2007) 1, 51-91.
• [7] N.A. Azamov, T.W. Daniels, Singular spectral shift function for resolvent comparable operators, Math. Nachr. (2019), early online publication; https://doi.org/10.1002/mana.201700293
• [8] M.Sh. Birman, M.G. Krem, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962), 475-478 [in Russian].
• [9] M.Sh. Birman, M.Z. Solomyak, Remarks on the spectral shift function, J. Soviet Math. 3 (1975) 4, 408-419.
• [10] M.Sh. Birman, D.R. Yafaev, The spectral shift function. The work of M. G. Krein and its further development, St. Petersburg Math. J. 4 (1993) 5, 833-870.
• [11] A. Bohm, Quantum Mechanics: Foundations and Applications, Second Edition, Texts and Monographs in Physics, Springer, 1986.
• [12] V. Bruneau, V. Petkov, Analytic continuation of the spectral shift function, Duke Math. J. 116 (2003) 3, 389-430.
• [13] S. Dyaltov, M. Zworski, Mathematical theory of scattering resonances, book in preparation; http://math.mit.edu/ dyatlov/res/
• [14] E. Getzler, The odd Chern character in cyclic homology and spectral flow, Topology 32 (1993), 489-507.
• [15] I.C. Gohberg, M.G. Krem, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., AMS, 1969.
• [16] I. Herbst, J. Rama, Instability of pre-existing resonances under a small constant electric field, Ann. Henri Poincare 16 (2015) 12, 2783-2835.
• [17] A. Jensen, K. Yajima, Instability of resonances under Stark perturbations, Ann. Henri Poincare (2018), to appear.
• [18] M.G. Krem, On the trace formula in perturbation theory, Mat. Sb. 33 (1953), 597-626 [in Russian].
• [19] J. Phillips, Self-adjoint Fredholm operators and spectral flow, Canad. Math. Bull. 39 (1996), 460-467.
• [20] B. Simon, Trace Ideals and Their Applications, 2nd ed., Math. Surveys Monogr., AMS, 2005.
• [21] J.R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Books on Engineering, Dover, 1972.
• [22] D.R. Yafaev, Mathematical Scattering Theory: General Theory, Trans. Math. Monographs 105, AMS, 1992.
• [23] M. Zworski, Mathematical study of scattering resonances, Bull. Math. Sci. 7 (2017) 1, 1-85.