24 research outputs found
RESONANCES ON HEDGEHOG MANIFOLDS
We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate this on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a ‘hedgehog’ manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis
Spectral determinant for the damped wave equation on an interval
We evaluate the spectral determinant for the damped wave equation on an
interval of length with Dirichlet boundary conditions, proving that it does
not depend on the damping. This is achieved by analysing the square of the
damped wave operator using the general result by Burghelea, Friedlander, and
Kappeler on the determinant for a differential operator with matrix
coefficients.Comment: 11 page
The role of the branch cut of the logarithm in the definition of the spectral determinant for non-selfadjoint operators
The spectral determinant is usually defined using the spectral zeta function
that is meromorphically continued to zero. In this definition, the complex
logarithms of the eigenvalues appear. Hence the notion of the spectral
determinant depends on the way how one chooses the branch cut in the definition
of the logarithm. We give results for the non-self-adjoint operators that state
when the determinant can and cannot be defined and how its value differs
depending on the choice of the branch cut.Comment: 11 pages, 2 figure
Non-Weyl resonance asymptotics for quantum graphs in a magnetic field
We study asymptotical behaviour of resonances for a quantum graph consisting
of a finite internal part and external leads placed into a magnetic field, in
particular, the question whether their number follows the Weyl law. We prove
that the presence of a magnetic field cannot change a non-Weyl asymptotics into
a Weyl one and vice versa. On the other hand, we present examples demonstrating
that for some non-Weyl graphs the ``effective size'' of the graph, and
therefore the resonance asymptotics, can be affected by the magnetic field
Resonances from perturbations of quantum graphs with rationally related edges
We discuss quantum graphs consisting of a compact part and semiinfinite
leads. Such a system may have embedded eigenvalues if some edge lengths in the
compact part are rationally related. If such a relation is perturbed these
eigenvalues may turn into resonances; we analyze this effect both generally and
in simple examples.Comment: LaTeX source file with 10 pdf figures, 24 pages; a replaced version
with minor improvements, to appear in J. Phys. A: Math. Theo
Magnetic square lattice with vertex coupling of a preferred orientation
We analyze a square lattice graph in a magnetic field assuming that the
vertex coupling is of a particular type violating the time reversal invariance.
Calculating the spectrum numerically for rational values of the flux per
plaquette we show how the two effects compete; at the high energies it is the
magnetic field which dominates restoring asymptotically the familiar
Hofstadter's butterfly pattern.Comment: 19 pages, 6 figure
Non-Weyl asymptotics for quantum graphs with general coupling conditions
Inspired by a recent result of Davies and Pushnitski, we study resonance
asymptotics of quantum graphs with general coupling conditions at the vertices.
We derive a criterion for the asymptotics to be of a non-Weyl character. We
show that for balanced vertices with permutation-invariant couplings the
asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions,
while for graphs without permutation numerous examples of non-Weyl behaviour
can be constructed. Furthermore, we present an insight helping to understand
what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance
point of view. Finally, we demonstrate a generalization to quantum graphs with
nonequal edge weights.Comment: minor changes, to appear in Pierre Duclos memorial issue of J. Phys.
A: Math. Theo