scholarly journals Spectral characterization and Schrödinger operator of space-like submanifolds

2015 ◽  
Vol 23 (2) ◽  
pp. 241-257
Author(s):  
Shichang Shu ◽  
Tianmin Zhu
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Spectral Characterization ◽  
Schrodinger Operators ◽  
First Eigenvalue ◽  
Schrodinger Operator ◽  
De Sitter ◽  
Totally Umbilical ◽  
The First Eigenvalue ◽  
Spectral Characterizations

Abstract In this paper, we would like to study space-like submanifolds in a de Sitter spaces Spn+p(1). We define and discuss three Schrödinger operators LH, LR, LR/H and obtain some spectral characterizations of totally umbilical space-like submanifolds in terms of the first eigenvalue of the Schrödinger operators LH, LR and LR/H respectively.

THE ESTIMATES OF RIESZ TRANSFORMS ASSOCIATED TO SCHRÖDINGER OPERATORS

2016 ◽  
Vol 101 (3) ◽  
pp. 290-309 ◽  
Author(s):  
QINGQUAN DENG ◽  
YONG DING ◽  
XIAOHUA YAO
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Reverse Hölder Class ◽  
Holder Class ◽  
Small Norm ◽  
Reverse Hölder

Let$H=-\unicode[STIX]{x1D6E5}+V$be a Schrödinger operator with some general signed potential$V$. This paper is mainly devoted to establishing the$L^{q}$-boundedness of the Riesz transform$\unicode[STIX]{x1D6FB}H^{-1/2}$for$q>2$. We mainly prove that under certain conditions on$V$, the Riesz transform$\unicode[STIX]{x1D6FB}H^{-1/2}$is bounded on$L^{q}$for all$q\in [2,p_{0})$with a given$2<p_{0}<n$. As an application, the main result can be applied to the operator$H=-\unicode[STIX]{x1D6E5}+V_{+}-V_{-}$, where$V_{+}$belongs to the reverse Hölder class$B_{\unicode[STIX]{x1D703}}$and$V_{-}\in L^{n/2,\infty }$with a small norm. In particular, if$V_{-}=-\unicode[STIX]{x1D6FE}|x|^{-2}$for some positive number$\unicode[STIX]{x1D6FE}$,$\unicode[STIX]{x1D6FB}H^{-1/2}$is bounded on$L^{q}$for all$q\in [2,n/2)$and$n>4$.

Non-existence of eigenvalues of Schrödinger operators

1977 ◽  
Vol 79 (1-2) ◽  
pp. 157-172 ◽  
Author(s):  
H. Kalf
Keyword(s):  
Schrödinger Operator ◽  
Exterior Domain ◽  
Virial Theorem ◽  
Schrödinger Operators ◽  
Quantum Mechanical ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Image Position

SynopsisThe paper provides conditions which enstlre that the Schrödinger operatordefined on an exterior domain has no eigenvalues on a certain half-ray. These conditions are in terms of weak local assumptions onThe proof uses Kato's ideas [16] in conjunction with the physicists' “commutator proof” of the quantum mechanical virial theorem.

scholarly journals A theorem on “localized” self-adjointness of Shrödinger operators withLLOC1-potentials

1982 ◽  
Vol 5 (3) ◽  
pp. 545-552 ◽  
Author(s):  
Hans L. Cycon
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
The Self ◽  
Schrodinger Operators ◽  
Schrodinger Operator

We prove a result which concludes the self-adjointness of a Schrödinger operator from the self-adjointness of the associated “localized” Schrödinger operators havingLLOC1-Potentials.

NORM RESOLVENT CONVERGENCE TO MAGNETIC SCHRÖDINGER OPERATORS WITH POINT INTERACTIONS

2001 ◽  
Vol 13 (04) ◽  
pp. 465-511 ◽  
Author(s):  
HIDEO TAMURA
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Two Dimensions ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Point Interactions ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence

The Schrödinger operator with δ-like magnetic field at the origin in two dimensions is not essentially self-adjoint. It has the deficiency indices (2, 2) and each self-adjoint extension is realized as a differential operator with some boundary conditions at the origin. We here consider Schrödinger operators with magnetic fields of small support and study the norm resolvent convergence to Schrödinger operator with δ-like magnetic field. We are concerned with the boundary conditions realized in the limit when the support shrinks. The results obtained heavily depend on the total flux of magnetic field and on the resonance space at zero energy, and the proof is based on the analysis at low energy for resolvents of Schrödinger operators with magnetic potentials slowly falling off at infinity.

The first eigenvalue for a quasilinear Schrödinger operator and its application

2017 ◽  
Vol 97 (4) ◽  
pp. 499-512 ◽  
Author(s):  
Olimpio Hiroshi Miyagaki ◽  
Sandra Imaculada Moreira ◽  
Ricardo Ruviaro
Keyword(s):  
Schrödinger Operator ◽  
First Eigenvalue ◽  
Schrodinger Operator ◽  
The First Eigenvalue

scholarly journals The Boundedness of Marcinkiewicz Integrals Associated with Schrödinger Operator on Morrey Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Dongxiang Chen ◽  
Fangting Jin
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Morrey Spaces ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Marcinkiewicz Integrals ◽  
Reverse Hölder Class ◽  
Hölder Class ◽  
Holder Class ◽  
Reverse Hölder

LetL=-Δ+Vbe a Schrödinger operator, whereVbelongs to some reverse Hölder class. The authors establish the boundedness of Marcinkiewicz integrals associated with Schrödinger operators and their commutators on Morrey spaces.

Upper Bounds for the Resonance Counting Function of Schrödinger Operators in Odd Dimensions

1998 ◽  
Vol 50 (3) ◽  
pp. 538-546 ◽  
Author(s):  
Richard Froese
Keyword(s):  
Simple Proof ◽  
Schrödinger Operator ◽  
Upper Bound ◽  
Counting Function ◽  
Schrödinger Operators ◽  
Upper Bounds ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Odd Dimensions

AbstractThe purpose of this note is to provide a simple proof of the sharp polynomial upper bound for the resonance counting function of a Schrödinger operator in odd dimensions. At the same time we generalize the result to the class of superexponentially decreasing potentials.

scholarly journals A generalization related to Schrödinger operators with a singular potential

2002 ◽  
Vol 29 (10) ◽  
pp. 609-611 ◽  
Author(s):  
Toka Diagana
Keyword(s):  
Schrödinger Operator ◽  
Singular Potential ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Open Question

The purpose of this note is to generalize a result related to the Schrödinger operatorL=−Δ+Q, whereQis a singular potential. Indeed, we show thatD(L)={0}inL2(ℝd)ford≥4. This fact answers to an open question that we formulated.

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