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New trace formulas in terms of resonances for three-dimensional Schrödinger operators. (English) Zbl 1391.81082

Summary: We consider the Schrödinger operator \(-\Delta+V (x)\) in \(L^{2}(\mathbb{R}^{3})\) with a real shortrange (integrable) potential \(V\). Using the associated Fredholm determinant, we present new trace formulas, in particular, on expressed in terms of resonances and eigenvalues only. We also derive expressions of the Dirichlet integral, and the scattering phase. The proof is based on a change of view the point for the above mentioned problems from that of operator theory to that of complex analytic (entire) function theory.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35J10 Schrödinger operator, Schrödinger equation
35B34 Resonance in context of PDEs
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[1] Bafiuelos, R.; Sa Barreto, A., On the heat trace of Schrödinger operators, Comm. Partial. Differential Equations, 20, 2153-2164, (1995) · Zbl 0843.35016 · doi:10.1080/03605309508821166
[2] Birman, M. S.; Yafaev, D., The spectral shift function. the papers of M. G. Krein and their further development, St. Petersburg Math. J., 4, 833-870, (1993)
[3] Buslaev, V., The trace formulas and certain asymptotic estimates of the kernel of the resolvent for the schröinger operator in three-dimensional space, 82-101, (1966)
[4] Colin de Verdière, Y., Une formule de traces pour l’opérateur de schröinger dans R\^{}{3}, Ann. Sci. Scola Norm. Sup. (4), 14, 27-39, (1981) · Zbl 0482.35068
[5] Deift, P.; Killip, R., On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys., 203, 341-347, (1999) · Zbl 0934.34075 · doi:10.1007/s002200050615
[6] Faddeev, L.; Zakharov, V., The Korteweg-de Vries equation Is a completely integrable Hamiltonian system, Funktsional Anal. i Prilozhen., 5, 18-27, (1971)
[7] Froese, R., Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions, Canad. J. Math., 50, 538-546, (1998) · Zbl 0918.47005 · doi:10.4153/CJM-1998-029-0
[8] Ginibre, J.; Moulin, M., Hilbert space approach to the quantum mechanical three-body problem, Ann. Inst. H. Poincaré Sect. A (N.S.), 21, 97-145, (1974) · Zbl 0311.47003
[9] I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonself-adjoint Operators (Translated from the Russian, Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc., Providence, R.I. 1969). · Zbl 0181.13503
[10] Iantchenko, A.; Korotyaev, E., No article title, Resonances for 1D Massless Dirac Operators J Diff. Eq., 256, 3038-3066, (2014)
[11] P. Koosis, The Logarithmic Integral I (Cambridge Univ. Press, Cambridge-London-New York, 1988). · Zbl 0665.30038 · doi:10.1017/CBO9780511566196
[12] Kargaev, P.; Korotyaev, E., Effective masses and conformal mapping, Comm. Math. Phys., 169, 597-625, (1995) · Zbl 0828.34076 · doi:10.1007/BF02099314
[13] Kargaev, P.; Korotyaev, E., Inverse problem for the Hill operator, the direct approach, Invent. Math., 129, 567-593, (1997) · Zbl 0878.34011 · doi:10.1007/s002220050173
[14] Korotyaev, E., Inverse resonance scattering on the half line, Asymptot. Anal., 37, 215-226, (2004) · Zbl 1064.34007
[15] Korotyaev, E., The estimates of periodic potentials in terms of effective masses, Comm. Math. Phys., 183, 383-400, (1997) · Zbl 0870.34080 · doi:10.1007/BF02506412
[16] Korotyaev, E.; Pushnitski, A., Trace formulae and high energy asymptotics for the Stark operator, Comm. Partial Differential Equations, 28, 817-842, (2003) · Zbl 1106.35305 · doi:10.1081/PDE-120020498
[17] Korotyaev, E., Resonances for 1D Stark operators, J. Spectral Theory, 7, 699-732, (2017) · Zbl 1460.34071 · doi:10.4171/JST/175
[18] E. Korotyaev, Asymptotics of Resonances for 1D Stark Operators, to be published in Lett. Math. Phys.. · Zbl 0843.35016
[19] Korotyaev, E.; Pushnitski, A., A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian, J. Funct. Anal., 217, 221-248, (2004) · Zbl 1069.35048 · doi:10.1016/j.jfa.2004.03.003
[20] Laptev, A.; Weidl, T., Sharp Lieb-thiring inequalities in high dimensions, Acta Math., 184, 87-111, (2000) · Zbl 1142.35531 · doi:10.1007/BF02392782
[21] Lax, P.; Phillips, R., Decaying modes for the wave equation in the exterior of an obstacle, Comm. Pure Appl. Math., 22, 737-787, (1969) · Zbl 0181.38201 · doi:10.1002/cpa.3160220603
[22] B. Levin, Distribution of Zeros of Entire Functions (AMS, Providence RI, 1964). · Zbl 0152.06703 · doi:10.1090/mmono/005
[23] Menzala, G.; Schonbek, T., Scattering frequencies for the wave equation with a potential term, J. Funct. Anal., 55, 297-322, (1984) · Zbl 0536.35060 · doi:10.1016/0022-1236(84)90002-8
[24] Marchenko, V.; Ostrovski, I., A characterization of the spectrum of the Hill operator, Math. USSR Sbornik, 26, 493-554, (1975) · Zbl 0343.34016 · doi:10.1070/SM1975v026n04ABEH002493
[25] Melrose, R., Polynomial bound on the number of scattering poles, J. Funct. Anal., 53, 287-303, (1983) · Zbl 0535.35067 · doi:10.1016/0022-1236(83)90036-8
[26] Newton, R., Noncentral potentials: the generalized Levinson theorem and the structure of the spectrum, J. Math. Phys., 18, 1348-1357, (1977) · doi:10.1063/1.523428
[27] Olmedilla, E., Inverse scattering transform for general matrix schrodinger operators and the related symplectic structure, Inverse Problems, I, 2, 19-236, (1985) · Zbl 0614.35076
[28] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. III: Scattering Theory (Academic Press, New York, 1979). · Zbl 0405.47007
[29] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators (Academic Press, New York, 1978). · Zbl 0401.47001
[30] Robert, D., Semiclassical asymptotics for the spectral shift function, (1999), Providence RI · Zbl 0922.35108
[31] Sa Barreto, A.; Zworski, M., Existence of resonances in potential scattering, Comm. Pure Appl. Math., 49, 1271-128, (1996) · Zbl 0877.35087 · doi:10.1002/(SICI)1097-0312(199612)49:12<1271::AID-CPA2>3.0.CO;2-7
[32] Sjöstrand, J.; Zworski, M., Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc., 4, 729-769, (1991) · Zbl 0752.35046 · doi:10.2307/2939287
[33] Zworski, M., Distribution of poles for scattering on the real line, J. Funct. Anal., 73, 277-296, (1987) · Zbl 0662.34033 · doi:10.1016/0022-1236(87)90069-3
[34] Zworski, M., Sharp polynomial bounds on the number of scattering poles, Duke Math. J., 59, 311-323, (1989) · Zbl 0705.35099 · doi:10.1215/S0012-7094-89-05913-9
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