2,723 research outputs found
Stationary Nonlinear Schr\"odinger Equation on Simplest Graphs: Boundary conditions and exact solutions
We treat the stationary (cubic) nonlinear Schr\"odinger equation (NSLE) on
simplest graphs. Formulation of the problem and exact analytical solutions of
NLSE are presented for star graphs consisting of three bonds. It is shown that
the method can be extended for the case of arbitrary number of bonds of star
graphs and for other simplest topologies such as tree and loop graphs. The case
of repulsive and attractive nonlinearities are treated separately
Quantum Fields on Star Graphs
We construct canonical quantum fields which propagate on a star graph
modeling a quantum wire. The construction uses a deformation of the algebra of
canonical commutation relations, encoding the interaction in the vertex of the
graph. We discuss in this framework the Casimir effect and derive the
correction to the Stefan-Boltzmann law induced by the vertex interaction. We
also generalize the algebraic setting for covering systems with integrable bulk
interactions and solve the quantum non-linear Schroedinger model on a star
graph.Comment: LaTex 23+1 pages, 4 figure
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs II: semiclassical approach
A semiclassical approach to the universal ergodic spectral statistics in
quantum star graphs is presented for all known ten symmetry classes of quantum
systems. The approach is based on periodic orbit theory, the exact
semiclassical trace formula for star graphs and on diagrammatic techniques. The
appropriate spectral form factors are calculated upto one order beyond the
diagonal and self-dual approximations. The results are in accordance with the
corresponding random-matrix theories which supports a properly generalized
Bohigas-Giannoni-Schmit conjecture.Comment: 15 Page
Fast solitons on star graphs
We define the Schr\"odinger equation with focusing, cubic nonlinearity on
one-vertex graphs. We prove global well-posedness in the energy domain and
conservation laws for some self-adjoint boundary conditions at the vertex, i.e.
Kirchhoff boundary condition and the so called and boundary
conditions. Moreover, in the same setting we study the collision of a fast
solitary wave with the vertex and we show that it splits in reflected and
transmitted components. The outgoing waves preserve a soliton character over a
time which depends on the logarithm of the velocity of the ingoing solitary
wave. Over the same timescale the reflection and transmission coefficients of
the outgoing waves coincide with the corresponding coefficients of the linear
problem. In the analysis of the problem we follow ideas borrowed from the
seminal paper \cite{[HMZ07]} about scattering of fast solitons by a delta
interaction on the line, by Holmer, Marzuola and Zworski; the present paper
represents an extension of their work to the case of graphs and, as a
byproduct, it shows how to extend the analysis of soliton scattering by other
point interactions on the line, interpreted as a degenerate graph.Comment: Sec. 2 revised; several misprints corrected; added references; 32
page
Universal spectral statistics in Wigner-Dyson, chiral and Andreev star graphs I: construction and numerical results
In a series of two papers we investigate the universal spectral statistics of
chaotic quantum systems in the ten known symmetry classes of quantum mechanics.
In this first paper we focus on the construction of appropriate ensembles of
star graphs in the ten symmetry classes. A generalization of the
Bohigas-Giannoni-Schmit conjecture is given that covers all these symmetry
classes. The conjecture is supported by numerical results that demonstrate the
fidelity of the spectral statistics of star graphs to the corresponding
Gaussian random-matrix theories.Comment: 15 page
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