Commutators, C0-semigroups and resolvent estimates

Georgescu, Vladimir; Gérard, Christian; Møller, Jacob Schach

doi:10.1016/j.jfa.2004.03.004

Cited by 54 publications

(102 citation statements)

References 22 publications

(73 reference statements)

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“…We refer to [DJ1], [DJ2], [O], [GGS1], [GGS2], and to the book [ABG], for recent different implementations of this method.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

Another Return of “Return to Equilibrium”

Fröhlich

Merkli

2004

Commun. Math. Phys.

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The property of "return to equilibrium" is established for a class of quantum-mechanical models describing interactions of a (toy) atom with black-body radiation, or of a spin with a heat bath of scalar bosons, under the assumption that the interaction strength is sufficiently weak. For models describing the first class of systems, our upper bound on the interaction strength is independent of the temperature T , (with 0 < T ≤ T 0 < ∞), while, for the spin-boson model, it tends to zero logarithmically, as T → 0. Our result holds for interaction form factors with physically realistic infrared behaviour.Three key ingredients of our analysis are: a suitable concrete form of the Araki-Woods representation of the radiation field, Mourre's positive commutator method combined with a recent virial theorem, and a norm bound on the difference between the equilibrium states of the interacting and the non-interacting system (which, for the system of an atom coupled to black-body radiation, is valid for all temperatures T ≥ 0, assuming only that the interaction strength is sufficiently weak).

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“…We refer to [DJ1], [DJ2], [O], [GGS1], [GGS2], and to the book [ABG], for recent different implementations of this method.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

Another Return of “Return to Equilibrium”

Fröhlich

Merkli

2004

Commun. Math. Phys.

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“…Such a partition of unity was used previously in [1] (see [9] for details). Hence, we construct a conjugate operator A by following the idea of Hübner and Spohn [35] (see also [23,24]). As in [35], the operator A is only maximal symmetric, and generates a C 0 -semigroup of isometries.…”

Section: Spec(hmentioning

confidence: 99%

“…Therefore, we need to use Singular Mourre theory with non self-adjoint conjugate operator. Such extensions of the usual conjugate operator theory [38,3] considered in [35] were later extended in [45] and in [23,24].…”

Section: Spec(hmentioning

confidence: 99%

Spectral Properties for Hamiltonians of Weak Interactions

Barbaroux

Faupin

Guillot

2016

Operator Theory: Advances and Applications

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We present recent results on the spectral theory for Hamiltonians of the weak decay. We discuss rigorous results on self-adjointness, location of the essential spectrum, existence of a ground state, purely absolutely continuous spectrum and limiting absorption principles. The last two properties heavily rely on the so-called Mourre Theory, which is used, depending on the Hamiltonian we study, either in its standard form, or in a more general framework using non self-adjoint conjugate operators.

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“…As pointed out in [13], the C 1 assumption of regularity is essential in the Mourre theory and should be checked carefully. See [20] for a different approach.…”

Section: The Mourre Estimatementioning

confidence: 99%

The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds

Golenia

Moroianu

2011

Trans. Amer. Math. Soc.

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Abstract. We describe the spectrum of the k-form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k and k − 1 de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case.In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials V such that the Schrödinger operator has compact resolvent, although V tends to −∞ in most of the infinity. We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension four whose cusps are rational homology spheres.

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Commutators, C0-semigroups and resolvent estimates

Cited by 54 publications

References 22 publications

Another Return of “Return to Equilibrium”

Another Return of “Return to Equilibrium”

Spectral Properties for Hamiltonians of Weak Interactions

The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds

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