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Yuriy Golovaty

Publications and source records attributed to Yuriy Golovaty.

At least 19 recordsLinked to original sources
arXiv2026

Spectral problems on open-book structures with singularly perturbed density: the limit operator

We investigate the spectral problem arising in the asymptotic analysis of vibrations of open-book structures with a mass density perturbed near the binding. The limiting problem is go\-ver\-ned by a non-self-adjoint block operator matrix coupling the macroscopic and microscopic components of the model. We describe the spectrum of this operator and completely characterize its eigenspaces and root subspaces. We further prove that generalized eigenvectors form chains of length at most two and derive an explicit criterion for the existence and number of Jordan blocks. This model provides a nontrivial example of a family of self-adjoint operators acting in varying Hilbert spaces whose limiting spectral behavior is described by a non-self-adjoint operator with a genuine Jordan structure.

math.SP↗
arXiv2026

Scattering in Quantum Graphs with Scale-Invariant Vertex Couplings: Resonances, Gaps and (Quasi-)Periodic Transmission

We study scattering on quantum graphs that consist of a channel with periodically attached resonators under scale-invariant vertex couplings. For this model, we derive explicit formulas for the transmission probability and analyse how it depends on the geometric and coupling parameters. Contrary to standard one-dimensional scattering, where the potential barrier becomes transparent at high energy, here the transmission probability does not approach unity; instead, it is periodic or quasi-periodic, with infinitely many energies of complete reflection and of perfect transmission persisting at arbitrarily high energy. We further show that the model exhibits strong transmission suppression near the anti-resonant frequencies, resulting in pronounced spectral gaps. The width and structure of these gaps depend on the number of resonators and the coupling parameters. As a result, such quantum graphs can be used to engineer transport properties and to tune spectral filtering.

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arXiv2025

Transmission resonances in scattering by $δ'$-like combs

We introduce a new exactly solvable model in quantum mechanics that describes the propagation of particles through a potential field created by regularly spaced $δ'$-type point interactions, which model the localized dipoles often observed in crystal structures. We refer to the corresponding potentials as $δ'_θ$-combs, where the parameter $θ$ represents the contrast of the resonant wave at zero energy and determines the interface conditions in the Hamiltonians. We explicitly calculate the scattering matrix for these systems and prove that the transmission probability exhibits sharp resonance peaks while rapidly decaying at other frequencies. Consequently, Hamiltonians with $δ'_θ$-comb potentials act as quantum filters, permitting tunnelling only for specific wave frequencies. Furthermore, for each $θ> 0$, we construct a family of regularized Hamiltonians approximating the ideal model and prove that their transmission probabilities have a similar structure, thereby confirming the physical realizability of the band-pass filtering effect.

math.SP↗
arXiv2025

On negative eigenvalues of 1D Schrödinger operators with $δ'$-like potentials

In this paper, we investigate negative eigenvalues of exactly solvable quantum models, particularly one-dimensional Hamiltonians with $δ'$-like potentials used to represent localized dipoles. These operators arise as norm resolvent limits of Schrödinger operators with suitably regularized potentials. Although the limiting operator is bounded below, we show that the approximating operators may possess a finite but arbitrarily large number of negative eigenvalues that diverge to $-\infty$ as the regularization parameter vanishes. This phenomenon illustrates a spectral instability of Schrödinger operators with $δ'$-like singularities.

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arXiv2025

On eigenvibrations of branched structures with heterogeneous mass density

We deal with a spectral problem for the Laplace-Beltrami operator posed on a stratified set $Ω$ which is composed of smooth surfaces joined along a line $γ$, the junction. Through this junction we impose the Kirchhoff-type vertex conditions, which imply the continuity of the solutions and some balance for normal derivatives, and Neumann conditions on the rest of the boundary of the surfaces. Assuming that the density is $O(\varepsilon^{-m})$ along small bands of width $O(\varepsilon)$, which collapse into the line $γ$ as $\varepsilon$ tends to zero, and it is $O(1)$ outside these bands, we address the asymptotic behavior, as $\varepsilon\to 0$, of the eigenvalues and of the corresponding eigenfunctions for a parameter $m\geq 1$. We also study the asymptotics for high frequencies when $m\in(1,2)$.

math.SP↗
arXiv2023

Quantum graphs: Coulomb-type potentials and exactly solvable models

We study the Schrödinger operators on a non-compact star graph with the Coulomb-type potentials having singularities at the vertex. The convergence of regularized Hamiltonians $H_\varepsilon$ with cut-off Coulomb potentials coupled with $(αδ+βδ')$-like ones is investigated.The 1D Coulomb potential and the $δ'$-potential are very sensitive to their regularization method. The conditions of the norm resolvent convergence of $H_\varepsilon$ depending on the regularization are established. The limit Hamiltonians give the Schrödinger operators with the Coulomb-type potentials a mathematically precise meaning, ensuring the correct choice of vertex conditions. We also describe all self-adjoint realizations of the formal Coulomb Hamiltonians on the star graph.

math.SP↗
arXiv2022

Membranes with thin and heavy inclusions: asymptotics of spectra

We study the asymptotic behaviour of eigenvalues and eigenfunctions of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which resonance frequencies of the membrane and thin inclusion coincide or closely situated is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator which is ultimately responsible for the asymptotics of eigenvalues and eigenfunctions is non-self-adjoint and possesses the Jordan chains of length $2$. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity.

math.SP↗
arXiv2020

2D Schrödinger operators with singular potentials concentrated near curves

We investigate the Schrödinger operators $H_\varepsilon=-Δ+W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $γ$. The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-Δ+W+a\partial_νδ_γ+bδ_γ$, where $δ_γ$ is Dirac's $δ$-function supported on $γ$ and $\partial_νδ_γ$ is its normal derivative on $γ$. Assuming that the operator $-Δ+W$ has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The transmission conditions on $γ$ for the eigenfunctions $u^+=αu^-$, $α\, \partial_νu^+-\partial_νu^-=βu^-$, which arise in the limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $γ$.

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arXiv2020

On spectrum of strings with $δ'$-like perturbations of mass density

We study the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem for the Sturm-Liouville operator with general boundary conditions and the weight function perturbed by the so-called $δ'$-like sequence $\varepsilon^{-2}h(x/\varepsilon)$. The eigenvalue problem is realized as a family of non-self-adjoint matrix operators acting on the same Hilbert space and the norm resolvent convergence of this family is established. We also prove the Hausdorff convergence of the perturbed spectra.

math.SP↗
arXiv2020

Eigenvalues of Schrödinger operators near thresholds: two term approximation

We consider one dimensional Schrödinger operators $H_λ=-\frac{d^2}{dx^2}+U+ λV_λ$ with nonlinear dependence on the parameter $λ$ and study the small $λ$ behaviour of eigenvalues. The potentials $U$ and $V_λ$ are real-valued bounded functions of compact support. Under some assumptions on $U$ and $V_λ$, we prove the existence of a negative eigenvalue that is absorbed at the bottom of the continuous spectrum as $λ\to 0$. We also construct two term asymptotic formulas for the threshold eigenvalues.

math.SP↗
arXiv2020

On coupling constant thresholds in one dimension

The threshold behaviour of negative eigenvalues for Schrödinger operators of the type $$ H_λ=-\frac{d^2}{dx^2}+U(x)+λα_λV(α_λx) $$ is considered. The potentials $U$ and $V$ are real-valued bounded functions of compact support, $λ$ is a positive parameter, and positive sequence $α_λ$ has a finite or infinite limit as $λ\to 0$. Under certain conditions on the potentials there exists a bound state of $H_λ$ which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence $α_λ$, asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.

math.SP↗
arXiv2019

Schrödinger operators with Coulomb-like potentials

We study the convergence of 1D Schrödinger ope\-rators $H_\varepsilon$ with the potentials which are regularizations of a class of pseudo-potentials having in particular the form $$ αδ'(x)+βδ(x)+γ/|x|\quad\text{or}\quad αδ'(x)+βδ(x)+γ/x. $$ The limit behaviour of $H_\varepsilon$ in the norm resolvent topology, as $\varepsilon\to 0$, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for $δ'$-like potential. All possible limits are described in terms of point interactions at the origin. As a consequence of the convergence results, different kinds of $L^\infty(\mathbb{R})$-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.

math.SP↗
arXiv2019

Approximating points of a Banach space by points of an operator image

Answering one problem that has its origins in quantum mechanics, we prove that for any sequence $(A_n)_{n\in\mathbb N}$ of convex nowhere dense sets in a Banach space $X$ and any sequence $(\varepsilon_n)_{n=1}^\infty$ of positive real numbers with $\lim_{n\to\infty}\varepsilon_n=0$, the set $A=\{x\in X:\forall n\in\mathbb N\;\exists a\in A_n\;\;\|x-a\|< \varepsilon_n\}$ is nowhere dense in $X$.

math.FA↗
arXiv2018

Some remarks on 1D Schrödinger operators with localized magnetic and electric potentials

One-dimensional Schrödinger operators with singular perturbed magnetic and electric potentials are considered. We study the strong resolvent convergence of two families of the operators with potentials shrinking to a point. Localized $δ$-like magnetic fields are combined with $δ\,'$-like perturbations of the electric potentials as well as localized rank-two perturbations. The limit results obtained heavily depend on zero-energy resonances of the electric potentials. In particular, the approximation for a wide class of point interactions in one dimension is obtained.

math.SP↗
arXiv2018

Schrödinger operators with singular rank-two perturbations and point interactions

Norm resolvent approximation for a wide class of point interactions in one dimension is constructed. To analyse the limit behaviour of Schrödinger operators with localized singular rank-two perturbations coupled with δ-like potentials as the support of perturbation shrinks to a point, we show that the set of limit operators is quite rich. Depending on parameters of the perturbation, the limit operators are described by both the connected and separated boundary conditions. In particular an approximation for a four-parametric subfamily of all the connected point interactions is built. We give examples of the singular perturbed Schrödinger operators without localized gauge fields, which converge to point interactions with the non-trivial phase parameter. We also construct an approximation for the point interactions that are described by different types of the separated boundary conditions such as the Robin-Dirichlet, the Neumann-Neumann or the Robin-Robin types.

math.SP↗
arXiv2017

Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

We are interested in evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.

math.AP↗
arXiv2016

Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations

In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous cells, whose proliferation is controled by growth factors diffusing in the extracellular medium and binding to the cell surface. We extend the results on the existence of nonhomogenous stationary solutions obtained in the previous work of one of the authors to a general Hill-type production function and full parameter set. Using spectral analysis and perturbation theory we derive conditions for the linearized stability of such spatial patterns.

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