51 research outputs found

### Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory

In this paper we present a general framework for solving the stationary
nonlinear Schr\"odinger equation (NLSE) on a network of one-dimensional wires
modelled by a metric graph with suitable matching conditions at the vertices. A
formal solution is given that expresses the wave function and its derivative at
one end of an edge (wire) nonlinearly in terms of the values at the other end.
For the cubic NLSE this nonlinear transfer operation can be expressed
explicitly in terms of Jacobi elliptic functions. Its application reduces the
problem of solving the corresponding set of coupled ordinary nonlinear
differential equations to a finite set of nonlinear algebraic equations. For
sufficiently small amplitudes we use canonical perturbation theory which makes
it possible to extract the leading nonlinear corrections over large distances.Comment: 26 page

### Quantum Graphs via Exercises

Studying the spectral theory of Schroedinger operator on metric graphs (also
known as quantum graphs) is advantageous on its own as well as to demonstrate
key concepts of general spectral theory. There are some excellent references
for this study such as a mathematically oriented book by Berkolaiko and
Kuchment, a review with applications to theoretical physicsby Gnutzmann and
Smilansky, and elementary lecture notes by Berkolaiko. Here, we provide a set
of questions and exercises which can accompany the reading of these references
or an elementary course on quantum graphs. The exercises are taken from courses
on quantum graphs which were taught by the authors

### Topological Resonances in Scattering on Networks (Graphs)

We report on a hitherto unnoticed type of resonances occurring in scattering
from networks (quantum graphs) which are due to the complex connectivity of the
graph - its topology. We consider generic open graphs and show that any cycle
leads to narrow resonances which do not fit in any of the prominent paradigms
for narrow resonances (classical barriers, localization due to disorder,
chaotic scattering). We call these resonances `topological' to emphasize their
origin in the non-trivial connectivity. Topological resonances have a clear and
unique signature which is apparent in the statistics of the resonance
parameters (such as e.g., the width, the delay time or the wave-function
intensity in the graph). We discuss this phenomenon by providing analytical
arguments supported by numerical simulation, and identify the features of the
above distributions which depend on genuine topological quantities such as the
length of the shortest cycle (girth). These signatures cannot be explained
using any of the other paradigms for narrow resonances. Finally, we propose an
experimental setting where the topological resonances could be demonstrated,
and study the stability of the relevant distribution functions to moderate
dissipation

### Resolving isospectral "drums" by counting nodal domains

Several types of systems were put forward during the past decades to show
that there exist {\it isospectral} systems which are {\it metrically}
different. One important class consists of Laplace Beltrami operators for pairs
of flat tori in $\mathbb{R}^n$ with $n\geq 4$. We propose that the spectral
ambiguity can be resolved by comparing the nodal sequences (the numbers of
nodal domains of eigenfunctions, arranged by increasing eigenvalues). In the
case of isospectral flat tori in four dimensions - where a 4-parameters family
of isospectral pairs is known- we provide heuristic arguments supported by
numerical simulations to support the conjecture that the isospectrality is
resolved by the nodal count. Thus - one can {\it count} the shape of a drum (if
it is designed as a flat torus in four dimensions...).Comment: 13 pages, 3 figure

### Electron-hole coherent states for the Bogoliubov-de Gennes equation

We construct a new set of generalized coherent states, the electron-hole
coherent states, for a (quasi-)spin particle on the infinite line. The
definition is inspired by applications to the Bogoliubov-de Gennes equations
where the quasi-spin refers to electron- and hole-like components of electronic
excitations in a superconductor. Electron-hole coherent states generally
entangle the space and the quasi-spin degrees of freedom. We show that the
electron-hole coherent states allow obtaining a resolution of unity and form
minimum uncertainty states for position and velocity where the velocity
operator is defined using the Bogoliubov-de Gennes Hamiltonian. The usefulness
and the limitations of electron-hole coherent states and the phase space
representations built from them are discussed in terms of basic applications to
the Bogoliubov-de Gennes equation such as Andreev reflection.Comment: 18 page

### Sigma models for quantum chaotic dynamics

We review the construction of the supersymmetric sigma model for unitary
maps, using the color- flavor transformation. We then illustrate applications
by three case studies in quantum chaos. In two of these cases, general Floquet
maps and quantum graphs, we show that universal spectral fluctuations arise
provided the pertinent classical dynamics are fully chaotic (ergodic and with
decay rates sufficiently gapped away from zero). In the third case, the kicked
rotor, we show how the existence of arbitrarily long-lived modes of excitation
(diffusion) precludes universal fluctuations and entails quantum localization

### A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs

We study implications of unitarity for pseudo-orbit expansions of the
spectral determinants of quantum maps and quantum graphs. In particular, we
advocate to group pseudo-orbits into sub-determinants. We show explicitly that
the cancellation of long orbits is elegantly described on this level and that
unitarity can be built in using a simple sub-determinant identity which has a
non-trivial interpretation in terms of pseudo-orbits. This identity yields much
more detailed relations between pseudo orbits of different length than known
previously. We reformulate Newton identities and the spectral density in terms
of sub-determinant expansions and point out the implications of the
sub-determinant identity for these expressions. We analyse furthermore the
effect of the identity on spectral correlation functions such as the
auto-correlation and parametric cross correlation functions of the spectral
determinant and the spectral form factor.Comment: 25 pages, one figur

### Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016)]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs

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