## 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.)
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### 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.
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