31 research outputs found
On dB spaces with nondensely defined multiplication operator and the existence of zero-free functions
In this work we consider de Branges spaces where the multiplication operator
by the independent variable is not densely defined. First, we study the
canonical selfadjoint extensions of the multiplication operator as a family of
rank-one perturbations from the viewpoint of the theory of de Branges spaces.
Then, on the basis of the obtained results, we provide new necessary and
sufficient conditions for a real, zero-free function to lie in a de Branges
space.Comment: 13 pages, no fugures. Theorem and remark have been added,
typographical erros correcte
Dispersion Estimates for One-Dimensional Schr\"odinger Equations with Singular Potentials
We derive a dispersion estimate for one-dimensional perturbed radial
Schr\"odinger operators. We also derive several new estimates for solutions of
the underlying differential equation and investigate the behavior of the Jost
function near the edge of the continuous spectrum.Comment: 26 page
Singular Schroedinger operators as self-adjoint extensions of n-entire operators
We investigate the connections between Weyl-Titchmarsh-Kodaira theory for
one-dimensional Schr\"odinger operators and the theory of -entire operators.
As our main result we find a necessary and sufficient condition for a
one-dimensional Schr\"odinger operator to be -entire in terms of square
integrability of derivatives (w.r.t. the spectral parameter) of the Weyl
solution. We also show that this is equivalent to the Weyl function being in a
generalized Herglotz-Nevanlinna class. As an application we show that perturbed
Bessel operators are -entire, improving the previously known conditions on
the perturbation.Comment: 14 page
On the spectral characterization of entire operators with deficiency indices (1,1)
For entire operators and entire operators in the generalized sense, we
provide characterizations based on the spectra of their selfadjoint extensions.
In order to obtain these spectral characterizations, we discuss the
representation of a simple, regular, closed symmetric operator with deficiency
indices (1,1) as a multiplication operator in a certain de Branges space.Comment: 22 pages; 1 section added; references added; typos corrected;
inaccuracy in a proof correcte
Oversampling on a class of symmetric regular de Branges spaces
A de Branges space is regular if the constants belong to its
space of associated functions and is symmetric if it is isometrically invariant
under the map . Let be the reproducing
kernel in and be the operator of multiplication
by the independent variable with maximal domain in .
Loosely speaking, we say that has the -oversampling
property relative to a proper subspace of it, with
, if there exists such that
for all , \begin{equation*} \sum_{\lambda\in\sigma(S_{\mathcal
B}^{\gamma})} \left(\frac{\lvert
J_{\mathcal{A}\mathcal{B}}(z,\lambda)\rvert}{K_\mathcal{B}(\lambda,\lambda)^{1/2}}\right)^{p/(p-1)}
<\infty, \quad\text{and}\quad F(z) = \sum_{\lambda\in\sigma(S_{\mathcal
B}^{\gamma})}
\frac{J_{\mathcal{A}\mathcal{B}}(z,\lambda)}{K_\mathcal{B}(\lambda,\lambda)}F(\lambda)
\quad (F\in\mathcal A), \end{equation*} for almost every self-adjoint extension
of . This definition is motivated by
the well-known oversampling property of Paley-Wiener spaces.
In this paper we provide sufficient conditions for a symmetric, regular de
Branges space to have the -oversampling property relative to a chain of
de Branges subspaces of it.Comment: 18 page
A class of -entire Schr\"odinger operators
We study singular Schr\"odinger operators on a finite interval as selfadjoint
extensions of a symmetric operator. We give sufficient conditions for the
symmetric operator to be in the -entire class, which was defined in our
previous work, for some . As a consequence of this classification, we obtain
a detailed spectral characterization for a wide class of radial Schr\"odinger
operators. The results given here make use of de Branges Hilbert space
techniques.Comment: 22 pages, no figures. Typographical errors corrected. References
added. The proof of Theorem 4.2 has been modifie
Entropy, fidelity, and double orthogonality for resonance states in two-electron quantum dots
Resonance states of a two-electron quantum dot are studied using a
variational expansion with both real basis-set functions and complex scaling
methods. The two-electron entanglement (linear entropy) is calculated as a
function of the electron repulsion at both sides of the critical value, where
the ground (bound) state becomes a resonance (unbound) state. The linear
entropy and fidelity and double orthogonality functions are compared as methods
for the determination of the real part of the energy of a resonance. The
complex linear entropy of a resonance state is introduced using complex scaling
formalism
Exponentially accurate semiclassical asymptotics of low-lying eigenvalues for 2×2 matrix Schrödinger operators
AbstractWe consider a simple molecular-type quantum system in which the nuclei have one degree of freedom and the electrons have two levels. The Hamiltonian has the form H(ɛ)=−ɛ42∂2∂y2+h(y),where h(y) is a 2×2 real symmetric matrix. Near a local minimum of an electron level E(y) that is not at a level crossing, we construct quasimodes that are exponentially accurate in the square of the Born–Oppenheimer parameter ɛ by optimal truncation of the Rayleigh–Schrödinger series. That is, we construct Eɛ and Ψɛ, such that ‖Ψɛ‖=O(1) and ‖(H(ɛ)−Eɛ)Ψɛ‖<Λexp(−Γ/ɛ2), where Γ>0
The class of n-entire operators
We introduce a classification of simple, regular, closed symmetric operators
with deficiency indices (1,1) according to a geometric criterion that extends
the classical notions of entire operators and entire operators in the
generalized sense due to M. G. Krein. We show that these classes of operators
have several distinctive properties, some of them related to the spectra of
their canonical selfadjoint extensions. In particular, we provide necessary and
sufficient conditions on the spectra of two canonical selfadjoint extensions of
an operator for it to belong to one of our classes. Our discussion is based on
some recent results in the theory of de Branges spaces.Comment: 33 pages. Typos corrected. Changes in the wording of Section 2.
References added. Examples added. arXiv admin note: text overlap with
arXiv:1104.476