Results 1  10
of
16
Vacuum Energy and Closed Orbits in Quantum Graphs
 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS
, 2008
"... The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated selfadjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated selfadjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos–Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff and other scaleinvariant boundary conditions the sum over nonperiodic orbits in fact makes no contribution to the total energy, whereas for more general (frequencydependent) vertex scattering matrices it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be “indexed ” a posteriori by truly periodic orbits. For the scaleinvariant cases complete calculations have been done by both methods (2) and (3), with identical results. Indeed, applying the image method to the resolvent kernel provides an alternative derivation of the trace formula.
The BerryKeating Operator on compact quantum graphs with general selfadjoint realizations. Ulmer Seminare 2009
"... Abstract. The BerryKeating operator HBK: = −i~ xddx + ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The BerryKeating operator HBK: = −i~ xddx +
Trace formulae for quantum graphs
, 2007
"... Quantum graph models are based on the spectral theory of (differential) Laplace operators on metric graphs. We focus on compact graphs and survey various forms of trace formulae that relate Laplace spectra to periodic orbits on the graphs. Included are representations of the heat trace as well as of ..."
Abstract
 Add to MetaCart
(Show Context)
Quantum graph models are based on the spectral theory of (differential) Laplace operators on metric graphs. We focus on compact graphs and survey various forms of trace formulae that relate Laplace spectra to periodic orbits on the graphs. Included are representations of the heat trace as well as of the spectral density in terms of sums over periodic orbits. Finally, a general trace formula for any self adjoint realisation of the Laplacian on a compact, metric graph is given.
FINITE DEFICIENCY INDICES AND UNIFORM REMAINDER IN WEYL’S
, 2010
"... Von Neumann theory classifies all selfadjoint extensions of a given symmetric operator A in terms of the socalled deficiency indices. Since the choice of the selfadjoint condition reflects the physics that is underlying the problem, it is natural to ask how the spectra of two different ..."
Abstract
 Add to MetaCart
(Show Context)
Von Neumann theory classifies all selfadjoint extensions of a given symmetric operator A in terms of the socalled deficiency indices. Since the choice of the selfadjoint condition reflects the physics that is underlying the problem, it is natural to ask how the spectra of two different
Signed:
"... I, Joachim Friedrich Kerner, hereby declare that this thesis and the work presented in it is entirely my own. Where I have consulted the work of others, this is always clearly stated. ..."
Abstract
 Add to MetaCart
(Show Context)
I, Joachim Friedrich Kerner, hereby declare that this thesis and the work presented in it is entirely my own. Where I have consulted the work of others, this is always clearly stated.
2 lnλ+O λ
"... Abstract. We study the Schrödinger equation on an infinite metric graph where the Hamiltonian is given by a suitable onedimensional Dirichlet Laplacian. The metric structure is defined by assigning an interval In = [0, ln], n ∈ N, to each edge of the graph with ln = pin. The spectrum of this system ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We study the Schrödinger equation on an infinite metric graph where the Hamiltonian is given by a suitable onedimensional Dirichlet Laplacian. The metric structure is defined by assigning an interval In = [0, ln], n ∈ N, to each edge of the graph with ln = pin. The spectrum of this system is purely discrete with the eigenvalues given by λn = n2, n ∈ N, with multiplicities d(n), where d(n) denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet’s famous divisor problem and infer the nonstandard Weyl asymptotics N(λ) =