10 research outputs found
Norm resolvent convergence of singularly scaled Schr\"odinger operators and \delta'-potentials
For a real-valued function V from the Faddeev-Marchenko class, we prove the
norm resolvent convergence, as \epsilon goes to 0, of a family S_\epsilon of
one-dimensional Schr\"odinger operators on the line of the form S_\epsilon:=
-D^2 + \epsilon^{-2} V(x/\epsilon). Under certain conditions the family of
potentials converges in the sense of distributions to the first derivative of
the Dirac delta-function, and then the limit of S_\epsilon might be considered
as a "physically motivated" interpretation of the one-dimensional Schr\"odinger
operator with potential \delta'.Comment: 30 pages, 2 figure; submitted to Proceedings of the Royal Society of
Edinburg
Controlling a resonant transmission across the -potential: the inverse problem
Recently, the non-zero transmission of a quantum particle through the
one-dimensional singular potential given in the form of the derivative of
Dirac's delta function, , with , being a
potential strength constant, has been discussed by several authors. The
transmission occurs at certain discrete values of forming a resonance
set . For
this potential has been shown to be a perfectly reflecting wall. However, this
resonant transmission takes place only in the case when the regularization of
the distribution is constructed in a specific way. Otherwise, the
-potential is fully non-transparent. Moreover, when the transmission
is non-zero, the structure of a resonant set depends on a regularizing sequence
that tends to in the sense of
distributions as . Therefore, from a practical point of
view, it would be interesting to have an inverse solution, i.e. for a given
to construct such a regularizing sequence
that the -potential at this value is
transparent. If such a procedure is possible, then this value
has to belong to a corresponding resonance set. The present paper is devoted to
solving this problem and, as a result, the family of regularizing sequences is
constructed by tuning adjustable parameters in the equations that provide a
resonance transmission across the -potential.Comment: 21 pages, 4 figures. Corrections to the published version added;
http://iopscience.iop.org/1751-8121/44/37/37530
On -like potential scattering on star graphs
We discuss the potential scattering on the noncompact star graph. The
Schr\"{o}dinger operator with the short-range potential localizing in a
neighborhood of the graph vertex is considered. We study the asymptotic
behavior the corresponding scattering matrix in the zero-range limit. It has
been known for a long time that in dimension 1 there is no non-trivial
Hamiltonian with the distributional potential , i.e., the
potential acts as a totally reflecting wall. Several authors have, in recent
years, studied the scattering properties of the regularizing potentials
\alpha\eps^{-2}Q(x/\eps) approximating the first derivative of the Dirac
delta function. A non-zero transmission through the regularized potential has
been shown to exist as \eps\to0. We extend these results to star graphs with
the point interaction, which is an analogue of potential on the line.
We prove that generically such a potential on the graph is opaque. We also show
that there exists a countable set of resonant intensities for which a partial
transmission through the potential occurs. This set of resonances is referred
to as the resonant set and is determined as the spectrum of an auxiliary
Sturm-Liouville problem associated with on the graph.Comment: 16 pages, 2 figure