THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

2001 ◽  
Vol 64 (1) ◽  
pp. 125-143 ◽  
Author(s):  
MEIRONG ZHANG
Keyword(s):  
Dirichlet Problem ◽  
Rotation Number ◽  
Periodic Potential ◽  
Schrödinger Operators ◽  
Negative Integer ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Periodic Potentials ◽  
The One

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

scholarly journals Norm resolvent convergence of singularly scaled Schrödinger operators and δ′-potentials

2013 ◽  
Vol 143 (4) ◽  
pp. 791-816 ◽  
Author(s):  
Yu. D. Golovaty ◽  
R. O. Hryniv
Keyword(s):  
Schrödinger Operator ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
One Dimensional ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence ◽  
The One ◽  
Image Position

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

scholarly journals Schrödinger operators with dynamically defined potentials

2016 ◽  
Vol 37 (6) ◽  
pp. 1681-1764 ◽  
Author(s):  
DAVID DAMANIK
Keyword(s):  
General Theory ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Almost Periodic ◽  
One Dimensional ◽  
Quantum Dynamical ◽  
Periodic Potentials ◽  
Introductory Part ◽  
Dynamical Properties

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

GAP-LABELLING THEOREMS FOR SCHRÖDINGER OPERATORS ON THE SQUARE AND CUBIC LATTICE

1994 ◽  
Vol 06 (02) ◽  
pp. 319-342 ◽  
Author(s):  
ANDREAS VAN ELST
Keyword(s):  
Density Of States ◽  
Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Cubic Lattice ◽  
Higher Dimensional ◽  
K Theory

The spectra of Schrödinger operators on the square and cubic lattice are investigated by means of non-commutative topological K-theory. Using a general gap-labelling theorem, it is shown how to calculate the possible values of the integrated density of states on the gaps of the spectrum, provided some additional conditions hold. If the potential takes on only finitely many values, this reduces to the calculation of frequencies of patterns in the potential sequence. As an example, products of one-dimensional systems and potentials generated by higher-dimensional substitutions are considered.

scholarly journals Sparse Schrödinger Operators

1997 ◽  
Vol 09 (03) ◽  
pp. 315-341
Author(s):  
Claire Guille-Biel
Keyword(s):  
Dynamical System ◽  
Spectral Properties ◽  
Schrödinger Operators ◽  
Negative Integer ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Discrete Operators ◽  
Ergodic Dynamical System

We study spectral properties of a family [Formula: see text], indexed by a non-negative integer p, of one-dimensional discrete operators associated to an ergodic dynamical system (T,X,ℬ,μ) and defined for u in ℓ2(ℤ) and n in ℤ by [Formula: see text], where Vx(n)=f(Tnx) and f is a real-valued measurable bounded map on X. In some particular cases, we prove that the nature of the spectrum does not change with p. Applications include some classes of random and quasi-periodic substitutional potentials.

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