Trace distributions associated to the Schrödinger operator

Melin, Anders

doi:10.1007/bf02790222

Cited by 9 publications

(56 citation statements)

References 7 publications

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“…A similar result concerning perturbations by potentials (hence non-trapping) has been announced by Hitrik-Polterovitch [16], where Melin's regularization [21] is considered.…”

Section: If the Metric Is Non Trapping We Have A Complete Asymptotic supporting

confidence: 68%

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Asymptotic behavior of regularized scattering phases for long range perturbations

Bouclet¹

2002

Journées Équations Aux Dérivées Partielles

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We define scattering phases for Schrödinger operators on R d as limit of arguments of relative determinants. These phases can be defined for long range perturbations of the Laplacian and therefore they can replace the usual spectral shift function (SSF) of Birman-Krein's theory, which can be defined for only special short range perturbations (relatively trace class perturbations). We prove the existence of asymptotic expansions for these phases, which generalize results on the SSF.

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“…A similar result concerning perturbations by potentials (hence non-trapping) has been announced by Hitrik-Polterovitch [16], where Melin's regularization [21] is considered.…”

Section: If the Metric Is Non Trapping We Have A Complete Asymptotic supporting

confidence: 68%

“…Another way of regularizing trace formulae for long range Schrödinger operators, still based on Taylor's formula, has been considered by Melin in [21] where weak expansions are proved.…”

Section: Differential Operatorsmentioning

confidence: 99%

Asymptotic behavior of regularized scattering phases for long range perturbations

Bouclet¹

2002

Journées Équations Aux Dérivées Partielles

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“…This theorem is well known for p ¼ 1 [11,13,20,30,33,34] and was proved by the author for p ¼ 2 [4,5,7], but the result is completely new for pX3: A similar result has been announced by Hitrik-Polterovich in [23] with Melin's regularization [28] for a potential perturbation for which the nontrapping condition is always fulfilled. Now we quote our results for the semiclassical Schro¨dinger operator with a longrange potential.…”

Section: Article In Presssupporting

confidence: 56%

“…This is a rather usual process and we refer to the nice paper of Mu¨ller [29] on this subject (see also [8,12]). Finally we mention that another kind of regularization based on Taylor's formula too, has been considered by Melin in [28]. Recent results associated to the related regularized scattering phase, in connection with some heat invariants, have been announced by Hitrik-Polterovich [23].…”

Section: Article In Pressmentioning

confidence: 96%

Spectral distributions for long range perturbations

Bouclet¹

2004

Journal of Functional Analysis

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We study distributions which generalize the concept of spectral shift function, for pseudodifferential operators on R d : We call such distributions spectral distributions. Relations between relative scattering determinants and spectral distributions are established; they lead to the definition of regularized scattering phase. These relations are analogous to the usual one for the standard spectral shift function. We give several asymptotic properties in the high energy and semiclassical limits where both nontrapping and trapping cases are considered. In particular, we prove Breit-Wigner formulae for the regularized scattering phases, for semiclassical Schro¨dinger operators with long-range potentials. r

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“…We are principally concerned with the case of long range potentials. Here we use the general results of Melin [19] as our starting point. In [19], expansions and regularized traces of functions of H were studied, for long range perturbations V .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%