106 research outputs found
Point Interactions: PT-Hermiticity and Reality of the Spectrum
General point interactions for the second derivative operator in one
dimension are studied. In particular, -self-adjoint
point interactions with the support at the origin and at points are
considered. The spectrum of such non-Hermitian operators is investigated and
conditions when the spectrum is pure real are presented. The results are
compared with those for standard self-adjoint point interactions.Comment: 17 page
Many Body Problems with "Spin"-Related Contact Interactions
We study quantum mechanical systems with "spin"-related contact interactions
in one dimension. The boundary conditions describing the contact interactions
are dependent on the spin states of the particles. In particular we investigate
the integrability of -body systems with -interactions and point spin
couplings. Bethe ansatz solutions, bound states and scattering matrices are
explicitly given. The cases of generalized separated boundary condition and
some Hamiltonian operators corresponding to special spin related boundary
conditions are also discussed.Comment: 13 pages, Late
Connection Conditions and the Spectral Family under Singular Potentials
To describe a quantum system whose potential is divergent at one point, one
must provide proper connection conditions for the wave functions at the
singularity. Generalizing the scheme used for point interactions in one
dimension, we present a set of connection conditions which are well-defined
even if the wave functions and/or their derivatives are divergent at the
singularity. Our generalized scheme covers the entire U(2) family of
quantizations (self-adjoint Hamiltonians) admitted for the singular system. We
use this scheme to examine the spectra of the Coulomb potential and the harmonic oscillator with square inverse potential , and thereby provide a general perspective for these
models which have previously been treated with restrictive connection
conditions resulting in conflicting spectra. We further show that, for any
parity invariant singular potentials , the spectrum is determined
solely by the eigenvalues of the characteristic matrix .Comment: TeX, 18 page
The regulated four parameter one dimensional point interaction
The general four parameter point interaction in one dimensional quantum
mechanics is regulated. It allows the exact solution, but not the perturbative
one. We conjecture that this is due to the interaction not being asymptotically
free. We then propose a different breakup of unperturbed theory and
interaction, which now is asymptotically free but leads to the same physics.
The corresponding regulated potential can be solved both exactly and
perturbatively, in agreement with the conjecture.Comment: 17 pages, no figures, Tex fil
A family of diameter-based eigenvalue bounds for quantum graphs
We establish a sharp lower bound on the first non-trivial eigenvalue of the
Laplacian on a metric graph equipped with natural (i.e., continuity and
Kirchhoff) vertex conditions in terms of the diameter and the total length of
the graph. This extends a result of, and resolves an open problem from, [J. B.
Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17
(2016), 2439--2473, Section 7.2], and also complements an analogous lower bound
for the corresponding eigenvalue of the combinatorial Laplacian on a discrete
graph. We also give a family of corresponding lower bounds for the higher
eigenvalues under the assumption that the total length of the graph is
sufficiently large compared with its diameter. These inequalities are sharp in
the case of trees.Comment: Substantial revision of v1. The main result, originally for the first
eigenvalue, has been generalised to the higher ones. The title has been
changed and the proofs substantially reorganised to reflect the new result,
and a section containing concluding remarks has been adde
Wave equation with concentrated nonlinearities
In this paper we address the problem of wave dynamics in presence of
concentrated nonlinearities. Given a vector field on an open subset of
\CO^n and a discrete set Y\subset\RE^3 with elements, we define a
nonlinear operator on L^2(\RE^3) which coincides with the free
Laplacian when restricted to regular functions vanishing at , and which
reduces to the usual Laplacian with point interactions placed at when
is linear and is represented by an Hermitean matrix. We then consider the
nonlinear wave equation and study the
corresponding Cauchy problem, giving an existence and uniqueness result in the
case is Lipschitz. The solution of such a problem is explicitly expressed
in terms of the solutions of two Cauchy problem: one relative to a free wave
equation and the other relative to an inhomogeneous ordinary differential
equation with delay and principal part . Main properties of
the solution are given and, when is a singleton, the mechanism and details
of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and
General, special issue on Singular Interactions in Quantum Mechanics:
Solvable Model
Spectral properties on a circle with a singularity
We investigate the spectral and symmetry properties of a quantum particle
moving on a circle with a pointlike singularity (or point interaction). We find
that, within the U(2) family of the quantum mechanically allowed distinct
singularities, a U(1) equivalence (of duality-type) exists, and accordingly the
space of distinct spectra is U(1) x [SU(2)/U(1)], topologically a filled torus.
We explore the relationship of special subfamilies of the U(2) family to
corresponding symmetries, and identify the singularities that admit an N = 2
supersymmetry. Subfamilies that are distinguished in the spectral properties or
the WKB exactness are also pointed out. The spectral and symmetry properties
are also studied in the context of the circle with two singularities, which
provides a useful scheme to discuss the symmetry properties on a general basis.Comment: TeX, 26 pages. v2: one reference added and two update
Wavefunctions, Green's functions and expectation values in terms of spectral determinants
We derive semiclassical approximations for wavefunctions, Green's functions
and expectation values for classically chaotic quantum systems. Our method
consists of applying singular and regular perturbations to quantum
Hamiltonians. The wavefunctions, Green's functions and expectation values of
the unperturbed Hamiltonian are expressed in terms of the spectral determinant
of the perturbed Hamiltonian. Semiclassical resummation methods for spectral
determinants are applied and yield approximations in terms of a finite number
of classical trajectories. The final formulas have a simple form. In contrast
to Poincare surface of section methods, the resummation is done in terms of the
periods of the trajectories.Comment: 18 pages, no figure
A new proof of the Vorono\"i summation formula
We present a short alternative proof of the Vorono\"i summation formula which
plays an important role in Dirichlet's divisor problem and has recently found
an application in physics as a trace formula for a Schr\"odinger operator on a
non-compact quantum graph \mathfrak{G} [S. Egger n\'e Endres and F. Steiner, J.
Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new
proof of a non-trivial identity for a particular Lambert series which involves
the divisor function d(n) and is identical with the trace of the Euclidean wave
group of the Laplacian on the infinite graph \mathfrak{G}.Comment: Enlarged version of the published article J. Phys. A: Math. Theor. 44
(2011) 225302 (11pp
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