Schrödinger operators in the twentieth century

Simon, Barry

doi:10.1063/1.533321

Cited by 118 publications

(93 citation statements)

References 227 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The continuity in the case of the presence of magnetic vector potentials was stated in [54] as an open problem; only several years ago Simon's results were extended in [7,8] to magnetic Schrödinger operators on domains in Euclidean space with vector and scalar potentials of Kato's type. In the both cases, the proof used certain probablistic technique.…”

Section: Introductionmentioning

confidence: 99%

“…A huge literature is dedicated to the study of properties of H A,U in its dependence on A and U , see the recent reviews [48,54,55]. An essential feature of the quantummechanical operators in comparision to the differential operator theory is admitting singular potentials [19], although the operator itself preserves some properties like regularity of solutions [29].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

Brüning

Geyler

Pankrashkin

2007

Ann. Henri Poincaré

View full text Add to dashboard Cite

For Schrödinger operators (including those with magnetic fields) with singular scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann-Schwinger equation. IntroductionThe analysis of Schrödinger operators occupies a central place in quantum mechanics. Suitably normalized, over the configuration space R n these operators have the formwhere A is the magnetic vector potential and U is an electric potential. A huge literature is dedicated to the study of properties of H A,U in its dependence on A and U , see the recent reviews [48,54,55]. An essential feature of the quantummechanical operators in comparision to the differential operator theory is admitting singular potentials [19], although the operator itself preserves some properties like regularity of solutions [29].

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

Brüning

Geyler

Pankrashkin

2007

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…A comprehensive review of self-adjointness results for one-dimensional operators can be found in N. Dunford and J. Schwartz [25]. About the multidimensional case (for operators on R n ) the reader may consult M. Reed [26], and also review papers by H. Kalf, U.-W Schminke, J. Walter and R. Wüst [43], H. Kalf [41] and B. Simon [83]. We tried not to repeat these sources unless this was relevant for the main text of our paper.…”

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

“…However, the presence of a non-trivial singular continuous part of ρ for some potentials was only guessed. More attention to the subject was attracted when Simon [28] asked the following question.…”

Section: Then the Support Of The (Possible) Singular Part Of ρ Has Hamentioning

confidence: 99%

On the singular spectrum of Schrödinger operators with decaying potential

Denisov

Kupin

2004

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the socalled Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal.

show abstract

Schrödinger operators in the twentieth century

Abstract: This paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics.

Cited by 118 publications

References 227 publications

Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

Continuity Properties of Integral Kernels Associated with Schrödinger Operators on Manifolds

Essential self-adjointness of Schrödinger-type operators on manifolds

On the singular spectrum of Schrödinger operators with decaying potential

Contact Info

Product

Resources

About