38 research outputs found
A Perturbative Approach to the Tunneling Phenomena
The double-well potential is a good example, where we can compute the
splitting in the bound state energy of the system due to the tunneling effect
with various methods, namely WKB or instanton calculations. All these methods
are non-perturbative and there is a common belief that it is difficult to find
the splitting in the energy due to the barrier penetration from a perturbative
analysis. However, we will illustrate by explicit examples containing singular
potentials (e.g., Dirac delta potentials supported by points and curves and
their relativistic extensions)that it is possible to find the splitting in the
bound state energies by developing some kind of perturbation method.Comment: 24 pages, 4 figure
Generating Axially Symmetric Minimal Hyper-surfaces in R^{1,3}
It is shown that, somewhat similar to the case of classical Baecklund
transformations for surfaces of constant negative curvature, infinitely many
axially symmetric minimal hypersurfaces in 4-dimensional Minkowski-space can be
obtained, in a non-trivial way, from any given one by combining the scaling
symmetries of the equations in light cone coordinates with a non-obvious
symmetry (the analogue of Bianchis original transformation) - which can be
shown to be involutive/correspond to a space-reflection.Comment: 7 page
Relativistic Lee Model and its Resolvent Analysis
We reexamine the relativistic 2+1 dimensional Lee model in light-front
coordinates on flat space and on a space-time with a spatial section given by a
compact manifold in the usual canonical formalism. The simpler 2+1 dimension is
chosen because renormalization is needed only for the mass difference but not
required for the coupling constant and the wavefunction. The model is
constructed non-perturbatively based on the resolvent formulation [1]. The
bound state spectrum is studied through its ``principal operator" and bounds
for the ground state energy are obtained. We show that the formal expression
found indeed defines the resolvent of a self-adjoint operator--the Hamiltonian
of the interacting system. Moreover, we prove an essential result that the
principal operator corresponds to a self-adjoint holomorphic family of type-A
in the sense of Kato.Comment: 35 page
Comment on "Scattering of light by a parity-time-symmetric dipole beyond the first Born approximation"
In [J. A. Rebou\c{c}as and P. A. Brand\~{a}o, Phys. Rev. A 104, 063514
(2021)] the authors compute the scattering amplitude for a
-symmetric double-delta-function potential in three
dimensions by invoking the far-zone approximation and summing the resulting
Born series. We show that the analysis of this paper suffers from a basic
error. Therefore its results are inconclusive. We give an exact closed-form
expression for the scattering amplitude of this potential.Comment: 2 page