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 $n$-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schr\"odinger operator to be $n$-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 $n$-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 $\mathcal B$ is regular if the constants belong to its space of associated functions and is symmetric if it is isometrically invariant under the map $F(z) \mapsto F(-z)$. Let $K_\mathcal{B}(z,w)$ be the reproducing kernel in $\mathcal B$ and $S_{\mathcal{B}}$ be the operator of multiplication by the independent variable with maximal domain in $\mathcal B$. Loosely speaking, we say that $\mathcal B$ has the $\ell_p$-oversampling property relative to a proper subspace $\mathcal A$ of it, with $p\in(2,\infty]$, if there exists $J_{\mathcal A\mathcal B}:\mathbb{C}\times\mathbb{C}\to\mathbb{C}$ such that $J(\cdot,w)\in\mathcal B$ for all $w\in\mathbb{C}$, \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 $S_{\mathcal B}^{\gamma}$ of $S_{\mathcal{B}}$. 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 $\ell_p$-oversampling property relative to a chain of de Branges subspaces of it.Comment: 18 page

A class of nn-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 $n$-entire class, which was defined in our previous work, for some $n$. 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