We discuss spectrum of a class of singular Schrödinger operator models known as leaky curves and show that if the interaction support has a periodic shape, its local perturbations can give rise to a discrete spectrum below the continuum threshold even if they are of `zero mean'.
AbstractBased on the symplectic Lie algebra $$\mathfrak {sp}(4)$$ sp ( 4 ) , we obtain two integrable hierarchies of $$\mathfrak {sp}(4)$$ sp ( 4 ) , and by using the trace identity, we give their Hamiltonian structures. Then, we use $$2\times 2$$ 2 × 2 Kronecker product, and construct integrable coupling systems of one soliton equation. Next, we consider two bases of Lie algebra $$\mathfrak {so}(5)$$ so ( 5 ) , and we get the corresponding two integrable hierarchies. Finally, we discuss the relation between the integrable hierarchies of two different bases associated with Lie algebra $$\mathfrak {so}(5)$$ so ( 5 ) .
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the corresponding eigenvalues. Also the inverse problem is approached, giving conditions for the existence of a differential operator from its eigenpolynomials and eigenvalues.
In this article we consider the one-dimensional Schrodinger operator L(Q) with a Hermitian periodic m by m matrix potential Q. We investigate the bands and gaps of the spectrum and prove that the main part of the positive real axis is overlapped by m bands. Moreover, we find a condition on the potential Q for which the number of gaps in the spectrum of L(Q) is finite.
We provide a detailed description of the maps associated with spectral interlacing, for rank one perturbations and bordering of symmetric and Hermitian matrices. The arguments rely on standard techniques of nonlinear analysis.
In the paper, we experimentally study the inverse problem with the resonant scattering determinant. We analyze the structure of characteristics of perturbed linear waves. Assuming there is the common part of potential perturbation propagating along the same strips, we estimate the common part of the perturbed wave, and its Fourier transform. We deduce the partial inverse uniqueness from the Nevanlinna type of representation theorem.
We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-Δ)$ in an open set $Ω\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue $λ_1(Ω)$ and compare them with previously known inequalities.
We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity.
We consider Sturm-Liouville operators with singular potentials from the class on star-type graph with cycle, which consist the edges with commensurable lengths. Asymptotic representation for eigenvalues for such operators is obtained. Recovering of the characteristic function the Sturm-Liouville operators with the singular potentials is considered.
In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $N\times N$ truncated Hilbert matrix for large values of $N$. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).
Two types of eigenvalue continuity are commonly used in the literature. However, their meanings and the conditions under which continuities are used are not always stated clearly. This can lead to some confusion and needs to be addressed. In this note, we revisit the Geršgorin disk theorem and clarify the issue concerning the proofs of the theorem by continuity.
We study the minimum of the essential spectrum of canonical systems $Ju'=-zHu$. Our results can be described as a generalized and more quantitative version of the characterization of systems with purely discrete spectrum, which was recently obtained by Romanov and Woracek [6]. Our key tool is oscillation theory.
In this paper, we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear Hamiltonian systems are equal, where both endpoints are in the limit circle cases. The proof is fundamental and is given for both continuous and discrete Hamiltonian systems. The method used in this paper also works for both endpoints are regular, or one endpoint is regular and the other is in the limit circle case.
The problem of finding eigenvalue estimates for the Schrödinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. We discuss these results and establish their counterparts for the operators on the combinatorial and metric graphs corresponding to the lattice Z^2.
The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on C. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H(x)=1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.
In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.