scholarly journals Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues I: The odd dimensional case

2015 ◽  
Vol 269 (3) ◽  
pp. 633-682 ◽  
Author(s):  
Michael Goldberg ◽  
William R. Green
Keyword(s):  
Schrödinger Operators ◽  
Dimensional Case ◽  
Schrodinger Operators ◽  
Higher Dimensional

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.

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 A Herman–Avila–Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic cocycles

2014 ◽  
Vol 35 (5) ◽  
pp. 1582-1591 ◽  
Author(s):  
CHRISTIAN SADEL
Keyword(s):  
Lyapunov Exponents ◽  
Unitary Group ◽  
Symplectic Group ◽  
Hermitian Form ◽  
Schrödinger Operators ◽  
Parameter Family ◽  
Schrodinger Operators ◽  
Type Formula ◽  
Higher Dimensional

A Herman–Avila–Bochi type formula is obtained for the average sum of the top$d$Lyapunov exponents over a one-parameter family of$\mathbb{G}$-cocycles, where$\mathbb{G}$is the group that leaves a certain, non-degenerate Hermitian form of signature$(c,d)$invariant. The generic example of such a group is the pseudo-unitary group$\text{U}(c,d)$or, in the case$c=d$, the Hermitian-symplectic group$\text{HSp}(2d)$which naturally appears for cocycles related to Schrödinger operators. In the case$d=1$, the formula for$\text{HSp}(2d)$cocycles reduces to the Herman–Avila–Bochi formula for$\text{SL}(2,\mathbb{R})$cocycles.

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