Summary
An example of a scattering system with the following properties is constructed: 1) the unperturbed Hamiltonian possesses no bound states, 2) the wave operators exist and 3) the scattering operator is not unitary.
Riassunto
Si costruisce un esempio di un sistema disperdente avente le seguenti proprietà: 1) i suoi gli Hamiltoniani non perturbati non hanno stati limiti; 2) esistono gli operatori d’onda; 3) l’operatore di scattering non è unitario.
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References
S. T. Kuroda:Nuovo Cimento,12, 431 (1959).
Notations such asE 1,L 2(E 1),L 2(0, ∞) etc., will be used in a similar sense. For those notions in the theory of Hilbert space which are used in the following, see (1) and the literature cited in (1).
T. Kato:Trans. Amer. Math. Soc.,70, 195 (1951).
See (1). ℜ(A) denotes the range of an operatorA.
SinceH 12 is an ordinary differential operator, this is a consequence of the theory of eigenfunction expansions. See,e.g., (7), especially §§ 5.2 and 5.3.
E. C. Titchmarsh:Eigenfunction Expansions Associated with Second-Order Differential Equations (Oxford, 1946).
T. Kato:Journ. Math. Soc. Japan,9, 239 (1957).
R. Schatten:A Theory of Cross Spaces, « Ann. Math. Studies » (Princeton, 1950).
F. Riesz andB. von Sz.-Nagy:Functional Analysis (New York, 1955).
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Kato, T., Kuroda, S.T. A remark on the unitarity property of the scattering operator. Nuovo Cim 14, 1102–1107 (1959). https://doi.org/10.1007/BF02728185
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DOI: https://doi.org/10.1007/BF02728185