Transition Decomposition of Quantum Mechanical Evolution

Abstract

We show that the existence of the family of self-adjoint Lyapunov operators introduced in [J. Math. Phys. 51, 022104 (2010)] allows for the decomposition of the state of a quantum mechanical system into two parts: A past time asymptote, which is asymptotic to the state of the system at t goes to minus infinity and vanishes at t goes to plus infinity, and a future time asymptote, which is asymptotic to the state of the system at t goes to plus infinity and vanishes at t goes to minus infinity. We demonstrate the usefulness of this decomposition for the description of resonance phenomena by considering the resonance scattering of a particle off a square barrier potential. We show that the past time asymptote captures the behavior of the resonance. In particular, it exhibits the expected exponential decay law and spatial probability distribution.Comment: Accepted for publication in Int. J. Theor. Phy

Manufacturing time operators: Covariance, selection criteria, and examples

“…[8] an operatorL was called a Lyapunov operator if for any normalized |ψ and |ψ t ≡ e −iĤt/ |ψ , the expectation value ψ t |L|ψ t is monotonically decreasing to 0 as t → ∞ and goes to 1 for t → −∞. Ref.…”

“…Ref. [8] considered the case of a HamiltonianĤ with purely continuous eigenvalues ranging from 0 to infinity and degeneracy parameter j. The particular Lyapunov operator suggested there can be written aŝ …”

“…In Section VI the results are applied to Lyapunov operators which were considered in Ref. [8]. It is shown that the expression given there is a special case, and the general form as well as possible uniqueness conditions are presented.…”

A modified Lax-Phillips scattering theory for quantum mechanics

“…In addition, it is shown in Refs. [S2,SSMH1,SSMH2] that Ran M + is dense in H ac . Similarly, under the same assumptions, there exists a self-adjoint, contractive, injective and non-negative backward Lyapunov operator M − : H ac → H ac for the quantum evolution (2) we are led to define for a quantum mechanical scattering problem a family of operators {Z app (t)} t≥0 : H ac → H ac , to which we refer as the approximate Lax-Phillips semigroup, via the definition…”

“…[S1] it has been shown in Refs. [S2,SSMH1,SSMH2] that if a quantum mechanical scattering problem satisfies the assumptions that: (a) The absolutely continuous spectrum of the unperturbed and perturbed Hamiltonians is σ ac (H 0 ) = σ ac (H) = R + ,(b) The multiplicity of the a.c. spectrum of H is uniform, (c) The incoming and outgoing Møller wave operators Ω ± (H 0 , H) exist and are complete; then, if H ac is the subspace of H corresponding to the a.c. spectrum of H, there exists a self-adjoint, contractive, injective and non-negative forward Lyapunov operator M + : H ac → H ac for the quantum evolution, i.e., for any ψ ∈ H ac we have…”