37 research outputs found
Strong-coupling asymptotic expansion for Schr\"odinger operators with a singular interaction supported by a curve in
We investigate a class of generalized Schr\"{o}dinger operators in
with a singular interaction supported by a smooth curve
. We find a strong-coupling asymptotic expansion of the discrete
spectrum in case when is a loop or an infinite bent curve which is
asymptotically straight. It is given in terms of an auxiliary one-dimensional
Schr\"{o}dinger operator with a potential determined by the curvature of
. In the same way we obtain an asymptotics of spectral bands for a
periodic curve. In particular, the spectrum is shown to have open gaps in this
case if is not a straight line and the singular interaction is strong
enough.Comment: LaTeX 2e, 30 pages; minor improvements, to appear in Rev. Math. Phy
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
Schroedinger operators with singular interactions: a model of tunneling resonances
We discuss a generalized Schr\"odinger operator in , with an attractive singular interaction supported by a
-dimensional hyperplane and a finite family of points. It can be
regarded as a model of a leaky quantum wire and a family of quantum dots if
, or surface waves in presence of a finite number of impurities if .
We analyze the discrete spectrum, and furthermore, we show that the resonance
problem in this setting can be explicitly solved; by Birman-Schwinger method it
is cast into a form similar to the Friedrichs model.Comment: LaTeX2e, 34 page
N-(2-Hydroxyphenyl)-1-[3-(2-oxo-2,3-dihydro-1Hbenzimidazol-1-yl)propyl]piperidine-4-Carboxamide (D2AAK4), a Multi-Target Ligand of Aminergic GPCRs, as a Potential Antipsychotic
N-(2-hydroxyphenyl)-1-[3-(2-oxo-2,3-dihydro-1H-benzimidazol -1-yl)propyl]piperidine-4-carboxamide (D2AAK4) is a multitarget ligand of aminergic G protein-coupled receptors (GPCRs) identified in structure-based virtual screening. Here we present detailed in vitro, in silico and in vivo investigations of this virtual hit. D2AAK4 has an atypical antipsychotic profile and low affinity to off-targets. It interacts with aminergic GPCRs, forming an electrostatic interaction between its protonatable nitrogen atom and the conserved Asp 3.32 of the receptors. At the dose of 100 mg/kg D2AAK4 decreases amphetamine-induced hyperactivity predictive of antipsychotic activity, improves memory consolidation in passive avoidance test and has anxiogenic properties in elevated plus maze test (EPM). Further optimization of the virtual hit D2AAK4 will be aimed to balance its multitarget profile and to obtain analogs with anxiolytic activity.The research was performed under OPUS grant from National Science Center (NCN, Poland), grant number 2017/27/B/NZ7/01767 (to A.A.K). Calculations were partially performed under a computational grant by Interdisciplinary Center for Mathematical and Computational Modeling (ICM), Warsaw, Poland, grant number G30-18 (to A.A.K.), under resources and licenses from CSC, Finland (to A.A.K). In vitro pharmacology assays were performed with support from the Spanish Ministry of Economy and Competitiveness (MINECO) (grant number SAF2014-57138-C2-1-R to M.C.). A.G.S. acknowledges funding from XUNTA de Galicia (Spain)S
Asymptotic behaviour of the spectrum of a waveguide with distant perturbations
We consider the waveguide modelled by a -dimensional infinite tube. The
operator we study is the Dirichlet Laplacian perturbed by two distant
perturbations. The perturbations are described by arbitrary abstract operators
''localized'' in a certain sense, and the distance between their ''supports''
tends to infinity. We study the asymptotic behaviour of the discrete spectrum
of such system. The main results are a convergence theorem and the asymptotics
expansions for the eigenvalues. The asymptotic behaviour of the associated
eigenfunctions is described as well. We also provide some particular examples
of the distant perturbations. The examples are the potential, second order
differential operator, magnetic Schroedinger operator, curved and deformed
waveguide, delta interaction, and integral operator
Bound states due to a strong interaction supported by a curved surface
We study the Schr\"odinger operator in
with a interaction supported by an infinite non-planar
surface which is smooth, admits a global normal parameterization with
a uniformly elliptic metric. We show that if is asymptotically planar
in a suitable sense and is sufficiently large this operator has a
non-empty discrete spectrum and derive an asymptotic expansion of the
eigenvalues in terms of a ``two-dimensional'' comparison operator determined by
the geometry of the surface . [A revised version, to appear in J. Phys.
A]Comment: LaTeX 2e, 21 page
Spectra of soft ring graphs
We discuss of a ring-shaped soft quantum wire modeled by interaction
supported by the ring of a generally nonconstant coupling strength. We derive
condition which determines the discrete spectrum of such systems, and analyze
the dependence of eigenvalues and eigenfunctions on the coupling and ring
geometry. In particular, we illustrate that a random component in the coupling
leads to a localization. The discrete spectrum is investigated also in the
situation when the ring is placed into a homogeneous magnetic field or threaded
by an Aharonov-Bohm flux and the system exhibits persistent currents.Comment: LaTeX 2e, 17 pages, with 10 ps figure
Schrödinger operators with δ and δ′-potentials supported on hypersurfaces
Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
We present an introduction to the framework of strongly local Dirichlet forms
and discuss connections between the existence of certain generalized
eigenfunctions and spectral properties within this framework. The range of
applications is illustrated by a list of examples