165 research outputs found
An isoperimetric problem for leaky loops and related mean-chord inequalities
We consider a class of Hamiltonians in with attractive
interaction supported by piecewise smooth loops of a fixed
length , formally given by with .
It is shown that the ground state of this operator is locally maximized by a
circular . We also conjecture that this property holds globally and
show that the problem is related to an interesting family of geometric
inequalities concerning mean values of chords of .Comment: LaTeX, 16 page
Scattering by local deformations of a straight leaky wire
We consider a model of a leaky quantum wire with the Hamiltonian in , where is a compact
deformation of a straight line. The existence of wave operators is proven and
the S-matrix is found for the negative part of the spectrum. Moreover, we
conjecture that the scattering at negative energies becomes asymptotically
purely one-dimensional, being determined by the local geometry in the leading
order, if is a smooth curve and .Comment: Latex2e, 15 page
On the discrete spectrum of spin-orbit Hamiltonians with singular interactions
We give a variational proof of the existence of infinitely many bound states
below the continuous spectrum for spin-orbit Hamiltonians (including the Rashba
and Dresselhaus cases) perturbed by measure potentials thus extending the
results of J.Bruening, V.Geyler, K.Pankrashkin: J. Phys. A 40 (2007)
F113--F117.Comment: 10 pages; to appear in Russian Journal of Mathematical Physics
(memorial volume in honor of Vladimir Geyler). Results improved in this
versio
The absolutely continuous spectrum of one-dimensional Schr"odinger operators
This paper deals with general structural properties of one-dimensional
Schr"odinger operators with some absolutely continuous spectrum. The basic
result says that the omega limit points of the potential under the shift map
are reflectionless on the support of the absolutely continuous part of the
spectral measure. This implies an Oracle Theorem for such potentials and
Denisov-Rakhmanov type theorems.
In the discrete case, for Jacobi operators, these issues were discussed in my
recent paper [19]. The treatment of the continuous case in the present paper
depends on the same basic ideas.Comment: references added; a few very minor change
Boundary Conditions for Singular Perturbations of Self-Adjoint Operators
Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let
\tau:D(A)\to\X, X a Banach space, be a surjective linear map such that
\|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range}
(\tau')\cap\H' =\{0\}, we define a family of self-adjoint
operators which are extensions of the symmetric operator .
Any in the operator domain is characterized by a sort
of boundary conditions on its univocally defined regular component \phireg,
which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These
boundary conditions are written in terms of the map , playing the role of
a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension
parameter being a self-adjoint operator from X' to X. The self-adjoint
extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in
which is a convolution operator on LD, T a distribution with
compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and
Applications, vol. 13
Effect of warm to hot rolling on microstructure, texture and mechanical properties of an advanced medium-Mn Steel
The deformation microstructures and mechanical properties were studied in a medium-Mn austenitic steel subjected to warm-to-hot rollin
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