×

The determinant of the scattering matrix and its relation to the number of eigenvalues. (English) Zbl 0382.47005


MSC:

47A40 Scattering theory of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agmon, S, Spectral properties of Schrödinger operators and scattering theory, Ann. scuola norm. sup. Pisa cl. sci., 2, 151-218, (1975) · Zbl 0315.47007
[2] {\scW. O. Amrein, J. M. Jauch, and K. B. Sinha}, “Quantum Scattering Theory,” Academic Press, New York, to appear. · Zbl 0376.47001
[3] Beregi, P; Zakhar’ev, B.N; Niyazgulov, S.A, Levinson’s theorem, Soviet J. particles nucl., 4, 217-230, (1973)
[4] Buslaev, V.S, Trace formulas for Schrödinger’s operator in three-space, Soviet math. dokl., 7, 295-297, (1962) · Zbl 0118.09302
[5] Buslaev, V.S, Spectral identities and the trace formula in the Friedrichs model, Problemy math. phys. (leningrad), 4, 43-53, (1969)
[6] Callaway, J, Quantum theory of the solid state, (1974), Academic Press New York, Part B
[7] Dashen, R; Kane, G.L, Counting hadron states, Phys. rev. D, 11, 136-139, (1975)
[8] Dreyfus, T, On the number of bound states and the determinant of the scattering matrix, ()
[9] Faddeev, L.D, On the Friedrichs model in the theory of perturbations of a continuous spectrum, Amer. math. soc. transl., 62, 177-203, (1967) · Zbl 0183.41902
[10] Gokhberg, I.C; Krein, M.G, Introduction to the theory of linear nonselfadjoint operators, (1969), Amer. Math. Soc Providence, R.I · Zbl 0181.13504
[11] Jauch, J.M, On the relation between scattering phase and bound states, Helv. phys. acta, 30, 143-156, (1957) · Zbl 0098.43003
[12] Jauch, J.M, Theory of the scattering operator, Helv. phys. acta, 31, 127-158, (1958) · Zbl 0081.43304
[13] Kato, T, Wave operators and similarity for some non-selfadjoint operators, Math. ann., 162, 258-279, (1966) · Zbl 0139.31203
[14] Kato, T; Kuroda, S.T, The abstract theory of scattering, Rocky mountain J. math., 1, 127-171, (1971) · Zbl 0241.47005
[15] Konno, R; Kuroda, S.T, On the finiteness of perturbed eigenvalues, J. fac. sci. univ. Tokyo, 13, 55-63, (1966) · Zbl 0149.10203
[16] Kuroda, S.T, An abstract stationary approach to perturbation of continuous spectra and scattering theory, J. analyse math., 20, 57-117, (1967) · Zbl 0153.16903
[17] Levinson, N, On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase, Mat.-fys. medd. danske vid. selsk., 25/9, 1-29, (1949) · Zbl 0032.20702
[18] Martin, A, On the validity of Levinson’s theorem for non-local interactions, Nuovo cimento, 7, 607-627, (1958) · Zbl 0080.22601
[19] Newton, R.G, Non-central potentials: the generalized Levinson theorem and the structure of the spectrum, J. math. phys., 18, 1348-1357, (1977)
[20] Simon, B, Quantum mechanics for Hamiltonians defined as quadratic forms, (1971), Princeton Univ. Press Princeton, N.J · Zbl 0232.47053
[21] Simon, B, Notes on infinite determinants of Hilbert space operators, Advances in math., 24, 244-273, (1977) · Zbl 0353.47008
[22] Smirnow, W.I, Lehrgang der höheren Mathematik, (1962), Deutscher Verlag der Wissenschaften Berlin, GDR · Zbl 0105.04101
[23] Taylor, J.R, Scattering theory, (1972), Wiley New York
[24] Thomas, L.E, Time dependent approach to scattering from impurities in a crystal, Comm. math. phys., 33, 335-343, (1973)
[25] Wollenberg, M, Levinsontheorem, instabile eigenwerte und streuquerschnittmaxima, (1975), preprint, Berlin · Zbl 0384.47010
[26] Dreyfus, T, Levinson’s theorem for non-local interactions, J. phys. A. math. gen., 9, L187-L191, (1976)
[27] {\scT. Dreyfus}, The number of states bound by non-central potentials, Helv. Phys. Acta, to appear.
[28] {\scT. Dreyfus}, The number of states bound by impurities in crystals, in preparation.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.