Schrödinger semigroups

Simon, Barry

doi:10.1090/s0273-0979-1982-15041-8

Cited by 1,017 publications

(882 citation statements)

References 130 publications

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“…Proofs can be found, for example, in refs. [65,67]. Under the assumptions A2 and A3, which imply bounds on the random potential V ω (q), the Schatten class bounds hold uniformly in ω.…”

Section: Appendix a Technical Commentsmentioning

confidence: 97%

Moment analysis for localization in random Schrödinger operators

et al. 2005

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Abstract. We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

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“…Proofs can be found, for example, in refs. [65,67]. Under the assumptions A2 and A3, which imply bounds on the random potential V ω (q), the Schatten class bounds hold uniformly in ω.…”

Section: Appendix a Technical Commentsmentioning

confidence: 97%

Moment analysis for localization in random Schrödinger operators

et al. 2005

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“…Some of them don't seem to be written up at all and I can't give a good reference for this section. For the continuous case you should look at the beautiful paper of Simon [204] and, again, at the book of Berezanskiȋ [27].…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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“…Therefore, the restriction E n XE n of any bounded self-adjoint operator X on L 2 (IR 2 ) to the eigenspace of the n-th Landau level is a Carleman integral operator [47,Corollary A.1.2]. Since E n (x, y) is smoothing and a projection, the integral kernel (x, y) → (E n XE n )(x, y) is jointly continuous in (x, y).…”

Section: Remarks 22mentioning

confidence: 99%

The fate of lifshits tails in magnetic fields

et al. 1995

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We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a non-negative algebraically decaying single-impurity potential we prove that the leading asymptotic behaviour for small energies is always given by the corresponding classical result in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eigenspace of any Landau level exhibits the same behaviour. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.

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Schrödinger semigroups

Cited by 1,017 publications

References 130 publications

Moment analysis for localization in random Schrödinger operators

Moment analysis for localization in random Schrödinger operators

Jacobi Operators and Completely Integrable Nonlinear Lattices

The fate of lifshits tails in magnetic fields

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