The Spectral Shift Function and the Invariance Principle

Pushnitski, Alexander

doi:10.1006/jfan.2001.3751

Cited by 38 publications

(48 citation statements)

References 17 publications

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“…In this subsection we summarize several results due to A. Pushnitski on the representation of the SSF for a pair of lower-bounded self-adjoint operators (see [8]- [10] …”

Section: 2mentioning

confidence: 99%

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On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

Fernández

Raikov

2004

Ann. Henri Poincaré

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We consider the three-dimensional Schrödinger operators H 0 and H ± where H 0 = (i∇+A) 2 −b, A is a magnetic potential generating a constant magnetic field of strength b > 0, and H ± = H 0 ± V where V ≥ 0 decays fast enough at infinity. Then, A. Pushnitski's representation of the spectral shift function (SSF) for the pair of operators H ± , H 0 is well defined for energies E = 2qb, q ∈ Z + . We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels 2qb, q ∈ Z + . Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Résumé. On considère les opérateurs de Schrödinger tridimensionnelsAlors, la représentation obtenue par A. Pushnitski de la fonction du décalage spectral pour les opérateurs H ± , H 0 est bien définie pour desénergies E = 2qb, q ∈ Z + . Onétudie le comportement du représentant associé de la classe d'équivalence déterminée par la fonction du décalage spectral, au voisinage des niveaux de Landau 2bq, q ∈ Z + . En réduisant l'analyseà l'investigation de l'asymptotique des valeurs propres d'une famille d'opérateurs de Toeplitz compacts, onétablit une relation entre le type des singularités de la fonction du décalage spectral aux niveaux de Landau et la vitesse de la décroissance de Và l'infini.

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“…In this subsection we summarize several results due to A. Pushnitski on the representation of the SSF for a pair of lower-bounded self-adjoint operators (see [8]- [10] …”

Section: 2mentioning

confidence: 99%

“…The representation of the SSF described in the above theorem has been generalized to non-sign-definite perturbations in [6] in the case of trace-class perturbations, and in [10] in the case of relatively trace-class perturbations. These generalizations are based on the concept of the index of orthogonal projections.…”

Section: 2mentioning

confidence: 99%

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

Fernández

Raikov

2004

Ann. Henri Poincaré

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“…[16] for the proof in the case of trace class perturbations; see also [22,23]). For the concept of the Fredholm index for a pair of orthogonal projections we refer to [1].…”

Section: Introductionmentioning

confidence: 99%

The Birman–Schwinger principle in von Neumann algebras of finite type

Kostrykin¹,

Makarov²,

Skripka³

2007

Journal of Functional Analysis

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We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman-Schwinger principle in this setting. As an application of this result, revisiting the Birman-Krein formula in the abstract scattering theory, we represent the de la HarpeSkandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.

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“…In [10] this representation was obtained by Gesztesy and Makarov for the case of no-signdefinite perturbations (0.1). Then the Gesztesy Makarov formula (see (1.7)) was extended by Pushnitski [18] to the case when V is not necessarily of the trace class. However, one of the main conditions on H 0 and H(:) in [18] was that there exists a function f, such that f (H(:))& f (H 0 ) is a nuclear operator.…”

mentioning

confidence: 99%

“…Then the Gesztesy Makarov formula (see (1.7)) was extended by Pushnitski [18] to the case when V is not necessarily of the trace class. However, one of the main conditions on H 0 and H(:) in [18] was that there exists a function f, such that f (H(:))& f (H 0 ) is a nuclear operator. In our paper the r.h.s.…”

mentioning

confidence: 99%

Spectral Shift Function in the Large Coupling Constant Limit

Safronov

2001

Journal of Functional Analysis

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The Spectral Shift Function and the Invariance Principle

Cited by 38 publications

References 17 publications

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

The Birman–Schwinger principle in von Neumann algebras of finite type

Spectral Shift Function in the Large Coupling Constant Limit

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