Theory of Simple Scattering and Eigenfunction Expansions

Kato, Tosio; Kuroda, S. T.

doi:10.1007/978-3-642-48272-4_5

Cited by 24 publications

(40 citation statements)

References 18 publications

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“…Therefore, from relation (2.1) and Weyl's theorem on the invariance of the essential spectrum of bounded operators under compact perturbations (e.g., [6]), it is clear that the spectrum of V κT is pure point for any κ ∈ R and any T > 0. • If the vector φ is not cyclic, the whole family of unitary operators (V κT ) κ∈R, T >0 is simultaneously reduced by the orthogonal subspaces H 0 and H ⊥ 0 (e.g., [7,Paragraph 5]), where…”

Section: Remarkmentioning

confidence: 99%

Singular continuous Floquet operator for periodic quantum systems

Bourget

2005

Journal of Mathematical Analysis and Applications

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Consider the Floquet operator of a time independent quantum system, acting on a separable Hilbert space, periodically perturbed by a rank one kick: e −iH 0 T e −iκT |φ φ| where T is the period, κ the coupling constant, and H 0 is a pure point self-adjoint operator, bounded from below. Under some hypotheses on the vector φ, cyclic w.r.t. H 0 we prove the following:• If the gaps between the eigenvalues (λ n ) are such that λ n+1 − λ n Cn −γ for some γ ∈ ]0, 1[ and C > 0, then the Floquet operator of the perturbed system is purely singular continuous T -a.e. • If H 0 is the Hamiltonian of the one-dimensional rotator on L 2 (R/T 0 Z) and the ratio 2πT /T 2 0 is irrational, then the Floquet operator is purely singular continuous as soon as κT = 0 (2π).We also establish an integral formula for the family (e −iH 0 T e −iκT |φ φ| ) T >0, κ∈R .

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Section: Remarkmentioning

confidence: 99%

Singular continuous Floquet operator for periodic quantum systems

Bourget

2005

Journal of Mathematical Analysis and Applications

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“…• Remarks 2.4. In [11], another proof of Theorem 2.3 is given, using results of Kato and Kuroda [10], via the techniques of Ben-Artzi and Devinatz [4] (limiting absorption principles, i.e. existence of boundary values at the real axis of the resolvent of H in certain weighted Sobolev spaces).…”

Section: Scattering Theorymentioning

confidence: 99%

Spectral analysis of perturbed multiplication operators occurring in polymerisation chemistry

Kokholm

1989

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisWe consider a mathematical model for the motion of a marked monomer in a system of reacting polymers at equilibrium. A well-posed integro-differential initial value problem for the probability of finding the marked monomer in a molecule of a given length is formulated. We prove exponential convergence of the probability to a unique equilibrium distribution. A quite complete spectral analysis is carried out for a self adjoint operator, which is a perturbation of a multiplication operator by an integral operator and is related to the generator of the time evolution.

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“…There are various approaches and an extensive literature on this subject, see for example [1], [3], [9], [13], [17], [18], [22], [25], [27], and [35]. There are various approaches and an extensive literature on this subject, see for example [1], [3], [9], [13], [17], [18], [22], [25], [27], and [35].…”

Section: Andp/r > {-{I -A/a/)mentioning

confidence: 99%

Perturbation of translation invariant positivity preserving semigroups on 𝐿²(𝑅^{𝑁})

Herbst¹,

Sloan²

1978

Trans. Amer. Math. Soc.

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The theory of singular local perturbations of translation invariant positivity preserving semigroups on L2(R", d"x) is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups, e~'H, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of -A.

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Theory of Simple Scattering and Eigenfunction Expansions

Cited by 24 publications

References 18 publications

Singular continuous Floquet operator for periodic quantum systems

Singular continuous Floquet operator for periodic quantum systems

Spectral analysis of perturbed multiplication operators occurring in polymerisation chemistry

Perturbation of translation invariant positivity preserving semigroups on 𝐿²(𝑅^{𝑁})

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