Schr�dinger operators with singular magnetic vector potentials

Leinfelder, Herbert; Simader, Christian G.

doi:10.1007/bf01258900

Cited by 191 publications

(124 citation statements)

References 15 publications

Supporting

Mentioning

123

Contrasting

Order By: Relevance

“…(1) (The operator nature of H ω ) Under the stated assumptions, H ω is essentially self adjoint on C ∞ 0 [51]. The random potential V ω (q) is non-negative and uniformly bounded by λb + .…”

Section: Appendix a Technical Commentsmentioning

confidence: 99%

Moment analysis for localization in random Schrödinger operators

et al. 2005

View full text Add to dashboard Cite

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.

show abstract

Section: Appendix a Technical Commentsmentioning

confidence: 99%

Moment analysis for localization in random Schrödinger operators

et al. 2005

View full text Add to dashboard Cite

show abstract

“…Then we can define H a,V as the operator defined by the closure of this quadratic form. This closure is well defined [22].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

Kondratiev

Maz′ya

Shubin

2005

Communications in Partial Differential Equations

View full text Add to dashboard Cite

We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with a positive scalar potential. They are expressed in terms of Wiener's capacity and the local energy of the magnetic field. The conditions for the discreteness of spectrum depend, in particular, on a functional parameter which is a decreasing function of one variable whose argument is the normalized local energy of the magnetic field. This function enters the negligibility condition of sets for the scalar potential. We give a description for the range of all admissible functions which is precise in a certain sense.

show abstract

“…Working under these minimal regularity conditions only, H. Leinfelder and C. Simader [52] were able to improve Kato's result by allowing V = V 1 + V 2 , where V 1 ≥ −c|x| 2 and V 2 is ∆-bounded with a relative bound a < 1. In particular, if V is semi-bounded below, then the above minimal regularity conditions on b and V are sufficient for essential self-adjointness (see also [23, Ch.…”

Section: Appendix C Stummel and Kato Classesmentioning

confidence: 99%

“…1]). A non-trivial technique of non-linear truncations is used in [52] to approximate functions from the maximal domain of the operator H b,V and its quadratic form by bounded functions.…”

Section: Appendix C Stummel and Kato Classesmentioning

confidence: 99%

Essential self-adjointness of Schrödinger-type operators on manifolds

2002

View full text Add to dashboard Cite

Abstract. We obtain several essential self-adjointness conditions for the Schrödinger type operator HV = D * D + V , where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV . These results generalize the theorems of E. C. Titchmarsh, D. B. Sears, F. S. Rofe-Beketov, I. M. Oleinik, M. A. Shubin and M. Lesch. We do not assume a priori that M is endowed with a complete Riemannian metric. This allows us to treat e.g. operators acting on bounded domains in R n with the Lebesgue measure. We also allow singular potentials V . In particular, we obtain a new self-adjointness condition for a Schrödinger operator on R n whose potential has the Coulomb-type singularity and is allowed to fall off to −∞ at infinity.For a specific case when the principal symbol of D * D is scalar, we establish more precise results for operators with singular potentials. The proofs are based on an extension of the Kato inequality which modifies and improves a result of

show abstract

Schr�dinger operators with singular magnetic vector potentials

Cited by 191 publications

References 15 publications

Moment analysis for localization in random Schrödinger operators

Moment analysis for localization in random Schrödinger operators

Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

Essential self-adjointness of Schrödinger-type operators on manifolds

Contact Info

Product

Resources

About