• Documents
  • Authors
  • Tables
  • Log in
  • Sign up
  • MetaCart
  • Donate

CiteSeerX logo

Advanced Search Include Citations
Advanced Search Include Citations

Schrödinger operators in the twentieth century (0)

by B Simon
Venue:J. Math. Phys
Add To MetaCart

Tools

Sorted by:
Results 1 - 10 of 37
Next 10 →

Schrödinger operators in the twenty-first century

by B Simon , 2000
"... ..."
Abstract - Cited by 36 (1 self) - Add to MetaCart
Abstract not found
(Show Context)

Citation Context

...n a reasonable time scale and so they involve an element of prediction. We have seen remarkable progress in the past fifty years in our understanding of Schrodinger operators, as I discussed in Simon =-=[1]-=-. In this companion piece, I present fifteen open problems. In 1984, I presented a list of open problem in Mathematical Physics, including thirteen in Schrodinger operators. Depending on how you count...

ON ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SCHRÖDINGER EQUATIONS WITH SINGULAR DIPOLE–TYPE POTENTIALS

by Veronica Felli, Elsa M. Marchini, Susanna Terracini , 2007
"... Abstract. Asymptotics of solutions to Schrödinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the lin ..."
Abstract - Cited by 28 (20 self) - Add to MetaCart
Abstract. Asymptotics of solutions to Schrödinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered. Dedicated to Prof. Norman Dancer on the occasion of his 60th birthday.

Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials

by A. Laptev, S. Naboko, O. Safronov - Comm. Math. Phys
"... ABSTRACT. The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidi-mensional Schrödinger operator is essentially supported by [0,∞). Our main theorem states that this property is preserved for slowly decaying potentials provi ..."
Abstract - Cited by 18 (1 self) - Add to MetaCart
ABSTRACT. The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidi-mensional Schrödinger operator is essentially supported by [0,∞). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables. 1.

Evolution of a Model Quantum System under Time Periodic Forcing

by O. Costin, R. D. Costin, J. L. Lebowitz, A. Rokhlenko - Conditions for Complete Ionization, Comm. Math. Phys. 221, n , 2001
"... Abstract: We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the s ..."
Abstract - Cited by 17 (5 self) - Add to MetaCart
Abstract: We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as t →∞. This is irrespective of the magnitude or frequency (resonant or not) of η(t). There are however exceptional, very non-generic η(t), that do not lead to full ionization, which include rather simple explicit periodic functions. For these η(t) the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.
(Show Context)

Citation Context

..., belonging to some Hilbert space H, H0 and H1 are Hermitian operators and Eq. (1) is to be solved subject to some initial condition ψ0. Such questions about the solutions of (1) belong to what Simon=-= [1] cal-=-ls “second level foundation” problems of quantum mechanics. They are of particular practical interest for the ionization of atoms and/or dissociation of molecules, in the case when H0 has both a d...

Recent developments in quantum mechanics with magnetic fields

by László Erdős - Proc. of Symposia in Pure Math. Vol 76 Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday Part , 2006
"... We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon. ..."
Abstract - Cited by 16 (0 self) - Add to MetaCart
We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon.
(Show Context)

Citation Context

...differ from their classical counterparts in many aspects. The mathematical theory of this operator is the most developed and most extensive in mathematical physics: the best recent review is by Simon =-=[152]-=-. As a next approximation, classical magnetic fields are included in the theory, but spins are neglected. The kinetic energy operator is modified from p2 to (p + A) 2 by the minimal substitution rule:...

A Szegő Condition for a Multidimensional Schrödinger Operator

by A. Laptev, S. Naboko, O. Safronov , 2002
"... We consider spectral properties of a Schrödinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szegő type condition. ..."
Abstract - Cited by 13 (0 self) - Add to MetaCart
We consider spectral properties of a Schrödinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szegő type condition.

Essential self-adjointness of Schrödinger type operators on manifolds

by Maxim Braverman, Ognjen Milatovic, Mikhail Shubin - RUSS. MATH. SURVEYS , 2002
"... 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 bu ..."
Abstract - Cited by 13 (7 self) - Add to MetaCart
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
(Show Context)

Citation Context

...C.1. Stummel classes. Uniform Stummel classes on R n were introduced by F. Stummel [84]. More details about Stummel classes and proofs can be found in Sect. 1.2 in [23], Ch. 5 and 9 in [72], and also =-=[2, 83, 84]-=-.sESSENTIAL SELF-ADJOINTNESS OF SCHRÖDINGER TYPE OPERATORS ON MANIFOLDS 41 The (uniform) Stummel class Sn consists of measurable real-valued functions V on Rn , such that � � lim sup |x − y| r↓0 x |x−...

Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials

by Pascal Auscher, Besma Ben Ali , 2006
"... We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of − ∆ + ..."
Abstract - Cited by 8 (3 self) - Add to MetaCart
We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of − ∆ + V and their gradients.

Analytic criteria in the qualitative spectral analysis of the . . .

by Vladimir Maz'ya , 2007
"... ..."
Abstract - Cited by 7 (0 self) - Add to MetaCart
Abstract not found

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

by Thierry Gallay, Romain Joly , 2007
"... ..."
Abstract - Cited by 7 (1 self) - Add to MetaCart
Abstract not found
Powered by: Apache Solr
  • About CiteSeerX
  • Submit and Index Documents
  • Privacy Policy
  • Help
  • Data
  • Source
  • Contact Us

Developed at and hosted by The College of Information Sciences and Technology

© 2007-2018 The Pennsylvania State University