Smooth Spectral Calculus

Ben‐Artzi, Matania

doi:10.1007/978-3-0348-0024-2_3

Cited by 7 publications

(5 citation statements)

References 70 publications

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“…We give a well-known type of sufficient criterion (cf. [Yaf92,BA11]), and sketch its proof in our context. Lemma 4.4.…”

Section: Setting Of the Smooth Methodsmentioning

confidence: 97%

“…Equally, one can start with the maps A(λ) = d dλ E(λ) and deduce the properties of the resolvents R(λ ± i0), see [BA11,Sec. 3].…”

Section: Setting Of the Smooth Methodsmentioning

confidence: 99%

“…We recall the basic facts about this setting in Sec. 4.1, mainly in the spirit of [KK71,BA11]; see also [Yaf92].…”

Section: Trace Class Perturbations and F -Boundednessmentioning

confidence: 99%

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High energy bounds on wave operators

Bostelmann,

Cadamuro,

Lechner

2019

Preprint

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In a general setting of scattering theory, we consider two self-adjoint operators H 0 and H 1 and investigate the behaviour of their wave operators W ± (H 1 , H 0 ) at asymptotic spectral values of H 0 and H 1 . Specifically, we analyse when (W ± (H 1 , H 0 ) − P ac 1 P ac 0 )f (H 0 ) < ∞, where P ac j is the projector onto the subspace of absolutely continuous spectrum of H j , and f is an unbounded function (f -boundedness). We provide sufficient criteria both in the case of trace-class perturbations V = H 1 − H 0 and within the general setting of the smooth method of scattering theory, where the high-energy behaviour of the boundary values of the resolvent of H 0 plays a major role. In particular, we establish f -boundedness for the perturbed polyharmonic operator and for Schrödinger operators with matrix-valued potentials. Applications of these results include the problem of quantum backflow.

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“…We give a well-known type of sufficient criterion (cf. [Yaf92,BA11]), and sketch its proof in our context. Lemma 4.4.…”

Section: Setting Of the Smooth Methodsmentioning

confidence: 97%

“…Equally, one can start with the maps A(λ) = d dλ E(λ) and deduce the properties of the resolvents R(λ ± i0), see [BA11,Sec. 3].…”

Section: Setting Of the Smooth Methodsmentioning

confidence: 99%

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High energy bounds on wave operators

Bostelmann,

Cadamuro,

Lechner

2019

Preprint

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“…Then one can use the tensor product structure (2.4) ofK to derive the limiting absorption principle for K from the fact that(∂ 2 t + λ ± i0) −1 := s-lim ǫց0 (∂ 2 t + λ ± iǫ) −1 , λ ∈ R \ {0},exists as a bounded operator from 〈t〉 −s L 2 (R) to 〈t〉 s L 2 (R) for s > 1 2 . See, for example,[3, Chap. 5] for results on the limiting absorption principle for operators of the form H = H 1 ⊗ 1 + 1 ⊗ H 2 .…”

mentioning

confidence: 99%

Feynman propagators on static spacetimes

Dereziński

Siemssen

2018

Rev. Math. Phys.

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Abstract. We consider the Klein-Gordon equation on a static spacetime and minimally coupled to a static electromagnetic potential. We show that it is essentially self-adjoint on C ∞ c . We discuss various distinguished inverses and bisolutions of the Klein-Gordon operator, focusing on the so-called Feynman propagator. We show that the Feynman propagator can be considered the boundary value of the resolvent of the Klein-Gordon operator, in the spirit of the limiting absorption principle known from the theory of Schrödinger operators. We also show that the Feynman propagator is the limit of the inverse of the Wick rotated Klein-Gordon operator.

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“…Recently, in [BA11] the author made use of the regularity of the projection-valued measure associated with a self-adjoint operator H on a Hilbert space H to derive the limiting absorption principle. The resolvent operator R(z) = (H − z) −1 is defined in the upper (lower) half plane C ± = {z : ± Imz > 0}.…”

Section: The Limiting Absorption Principle For Periodic Mediamentioning

confidence: 99%

Radiation conditions for periodic potentials

Ngoc¹

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The aim of the present thesis is to determine radiation conditions for higher-dimensional Schrödinger equations with periodic potentials. There is well-known result for Helmholtz equations. However to my knowledge, there have been only a few works on periodic Schrödinger equations.We employ Floquet Bloch theory to study periodic Schrödinger operators. The limiting absorption principal derives integral formulas for solutions of periodic Schrödinger equations. Then, we use the analytic perturbation theory and the stationary phase method to establish the radiation conditions. i 1493 "Mathematical Structures in Modern Quantum Physics" for their scientific support, especially Thang,

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Smooth Spectral Calculus

Cited by 7 publications

References 70 publications

High energy bounds on wave operators

High energy bounds on wave operators

Feynman propagators on static spacetimes

Radiation conditions for periodic potentials

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