123 research outputs found

### Wannier functions of elliptic one-gap potentials

Wannier functions of the one dimensional Schroedinger equation with elliptic
one gap potentials are explicitly constructed. Properties of these functions
are analytically and numerically investigated. In particular we derive an
expression for the amplitude of the Wannier function in the origin, a power
series expansion valid in the vicinity of the origin and an asymptotic
expansion characterizing the decay of the Wannier function at large distances.
Using these results we construct an approximate analytical expression of the
Wannier function which is valid in the whole spatial domain and is in good
agreement with numerical results.Comment: 24 pages, 5 figure

### Wannier functions for quasi-periodic finite-gap potentials

In this paper we consider Wannier functions of quasi-periodic g-gap ($g\geq
1$) potentials and investigate their main properties. In particular, we discuss
the problem of averaging underlying the definition of Wannier functions for
both periodic and quasi-periodic potentials and express Bloch functions and
quasi-momenta in terms of hyperelliptic $\sigma$ functions. Using this approach
we derive a power series expansion of the Wannier function for quasi-periodic
potentials valid at $|x|\simeq 0$ and an asymptotic expansion valid at large
distance. These functions are important for a number of applied problems

### Notes on Euclidean Wilson loops and Riemann Theta functions

The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal
area surfaces in AdS5 space. In this paper we consider the case of Euclidean
flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3
space. Using known mathematical results for such minimal area surfaces we
describe an infinite parameter family of analytic solutions for closed Wilson
loops. The solutions are given in terms of Riemann theta functions and the
validity of the equations of motion is proven based on the trisecant identity.
The world-sheet has the topology of a disk and the renormalized area is written
as a finite, one-dimensional contour integral over the world-sheet boundary. An
example is discussed in detail with plots of the corresponding surfaces.
Further, for each Wilson loops we explicitly construct a one parameter family
of deformations that preserve the area. The parameter is the so called spectral
parameter. Finally, for genus three we find a map between these Wilson loops
and closed curves inside the Riemann surface.Comment: 35 pages, 7 figures, pdflatex. V2: References added. Typos corrected.
Some points clarifie

### Spectral singularities in PT-symmetric periodic finite-gap systems

The origin of spectral singularities in finite-gap singly periodic
PT-symmetric quantum systems is investigated. We show that they emerge from a
limit of band-edge states in a doubly periodic finite gap system when the
imaginary period tends to infinity. In this limit, the energy gaps are
contracted and disappear, every pair of band states of the same periodicity at
the edges of a gap coalesces and transforms into a singlet state in the
continuum. As a result, these spectral singularities turn out to be analogous
to those in the non-periodic systems, where they appear as zero-width
resonances. Under the change of topology from a non-compact into a compact one,
spectral singularities in the class of periodic systems we study are
transformed into exceptional points. The specific degeneration related to the
presence of finite number of spectral singularities and exceptional points is
shown to be coherently reflected by a hidden, bosonized nonlinear
supersymmetry.Comment: 16 pages, 3 figures; a difference between spectral singularities and
exceptional points specified, the version to appear in PR

### Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type

We discuss a family of multi-term addition formulae for Weierstrass functions
on specialized curves of genus one and two with many automorphisms. In the
genus one case we find new addition formulae for the equianharmonic and
lemniscate cases, and in genus two we find some new addition formulae for a
number of curves, including the Burnside curve.Comment: 19 pages. We have extended the Introduction, corrected some typos and
tidied up some proofs, and inserted extra material on genus 3 curve

### Rotation Numbers, Boundary Forces and Gap labelling

We review the Johnson-Moser rotation number and the $K_0$-theoretical gap
labelling of Bellissard for one-dimensional Schr\"odinger operators. We compare
them with two further gap-labels, one being related to the motion of Dirichlet
eigenvalues, the other being a $K_1$-theoretical gap label. We argue that the
latter provides a natural generalisation of the Johnson-Moser rotation number
to higher dimensions.Comment: 10 pages, version accepted for publicatio

### Exact solutions for a class of integrable Henon-Heiles-type systems

We study the exact solutions of a class of integrable Henon-Heiles-type
systems (according to the analysis of Bountis et al. (1982)). These solutions
are expressed in terms of two-dimensional Kleinian functions. Special periodic
solutions are expressed in terms of the well-known Weierstrass function. We
extend some of our results to a generalized Henon-Heiles-type system with n+1
degrees of freedom.Comment: RevTeX4-1, 13 pages, Submitted to J. Math. Phy

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