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In quantum mechanics based on a Hilbert space $H$ wth quantum evolution semigroup of unitary operators $U(t)$, $t\in \mathbf{R}$, Lax and Phillips formulated an abstract formulation of scattering theory, based on two distinguished subspaces of states $\mathcal{D}_-$ of ingoing states and $\mathcal{D}_+$ of outgoing states satisfying certain system of axioms.
Apart from quantum mechanics, abstract scattering theory has other interesting special cases, applying to the theory of automorphic functions (due to an idea of Israel Gelfand, later realized by B. S. Pavlov and L. D. Faddeev), zeta function analysis and so on.
Peter D. Lax, Ralph S. Phillips, Scattering theory, Bull. Amer. Math. Soc. Volume 70, Number 1 (1964), 130-142 euclid MR0167868; Scattering theory (book), Academic Press 1990, 2nd ed. gBooks; Scattering theory for automorphic functions, Annals of Mathematics Studies 87, Princeton Univ. Press 1976
B. S. Pavlov, L. D. Faddeev, Scattering theory and automorphic functions, Zapiski Sem. LOMI 27 (1972) 161-193
Extension of the method which Lax and Phillips used for automorphic forms ($SL(2,\mathbf{R})$-case) to the case of more general harmonic analysis on more general Riemannian symmetric spaces is studied in
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