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Yuri Latushkin

Publications and source records attributed to Yuri Latushkin.

At least 19 recordsLinked to original sources
arXiv2026

The Damped Waves Equation and generalized Cosine and Sine families on Banach spaces

We study the abstract damped wave equation on a Banach space, allowing the damping coefficient to be unbounded. By recasting the equation as a first-order system and identifying conditions under which the associated block operator generates a $C_0$-group, we construct generalized cosine and sine families that represent mild and classical solutions, extending the classical undamped theory. We establish existence, uniqueness, regularity, invariant subspaces, growth rate, and trigonometric type identities for these families. Our setup applies to a broad class of damped wave, Klein--Gordon, and higher-order PDE examples, including cases where damping restores well-posedness that fails in the undamped equation.

math.AP↗
arXiv2026

The calculus of Duistermaat's triple index

In this paper we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago, this index has lately been the subject of renewed attention, due to its central role in eigenvalue interlacing problems on quantum graphs (and more abstractly for self-adjoint extensions of symmetric operators). Here we give an axiomatic characterization of the index that leads to elementary proofs of its fundamental properties. We also relate the index to other quantities often appearing in symplectic geometry, such as the Hörmander--Kashiwara--Wall index and the Maslov index. Among other things, this leads to a curious formula for the Morse index of a difference of Hermitian matrices.

math.SP↗
arXiv2026

Stability of Ginzburg-Landau pulses via Fredholm determinants of Birman-Schwinger operators

We introduce a numerical method to determine the stability of stationary pulse solutions of the complex Ginzburg-Landau equation. The method involves the computation of the point spectrum of the first-order linear differential operator with matrix-valued coefficients on the real line obtained by linearizing the Ginzburg-Landau equation about a stationary pulse. Applying a general theory of Gesztesy, Latushkin, and Makarov, we show that this point spectrum is given by the set of zeros of a 2-modified Fredholm determinant of a Hilbert-Schmidt, Birman-Schwinger operator. We establish conditions which guarantee that this operator is trace class. Applying results of Bornemann on the numerical computation of Fredholm determinants, we obtain a bound on the error between the regular Fredholm determinant of the trace class operator and its numerical approximation by a matrix determinant. We verify the new numerical Fredholm determinant method for computing the point spectrum of a Ginzburg-Landau pulse by exhibiting excellent agreement with previous methods. This new approach avoids the challenge of solving the numerically stiff system of equations for the matrix-valued Jost solutions, and it opens the way for the spectral analysis of breather solutions of nonlinear wave equations, for which an Evans function does not exist.

math.SP↗
arXiv2026

Sturm-Liouville problems on graphs with Robin boundary conditions

We study characteristic functions and describe asymptotics of the eigenvalues for the spectral Sturm-Liouville problem on graphs equipped with Robin-Kirhhoff boundary conditions. Also, we show how to recover the coefficients in the Robin conditions for the quantum graphs provided the shape of the graphs and some Robin eigenvalues are known.

math.SP↗
arXiv2025

Solutions of abstract wave equations, eigenvalues and resonances

We prove general representation formulas for strongly continuous cosine and sine operator families in terms of scattering resonances of their generators. This generalizes known results related to decay, growth and oscillatory behavior of solutions of abstract wave equations to a wide class of non-self-adjoint operators in Banach spaces. Inspired by the classical results on scattering resonances for Schrödinger operators with compactly supported potentials, we develop quite general abstract scheme of resonances that involves extensions of the resolvent operators from Banach to Frechet spaces. We split the solutions of the wave equations in two parts: The first part is related to finite rank operators induced by the eigenvalues and resonances while the second part involves a partial inversion of the Laplace transform whose exponential behavior is effectively controlled. Illustrations and applications cover a wide class of generators including the Schrödinger operators with non-symmetric complex matrix potentials, linearizations of nonlinear wave equations, Aharonov-Bohm and block-box Hamiltonians, etc.

math.FA↗
arXiv2025

The Duistermaat index and eigenvalue interlacing for self-adjoint extensions of a symmetric operator

Eigenvalue interlacing is a useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts each eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as Weyl interlacing, Cauchy interlacing, Dirichlet--Neumann bracketing and so on. We prove a sharp version of the interlacing inequalities for ``finite-dimensional perturbations in boundary conditions'', expressed as bounds on the spectral shift between two self-adjoint extensions of a fixed symmetric operator with finite and equal defect numbers. The bounds are given in terms of the Duistermaat index, a topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the Lagrangian planes describe the self-adjoint extensions being compared, while the third corresponds to the Friedrichs extension, which acts as a reference point. Along the way several auxiliary results are established, including one-sided continuity properties of the Duistermaat triple index, smoothness of the (abstract) Cauchy data space without unique continuation-type assumptions, and a formula for the Morse index of an extension of a non-negative symmetric operator.

math.SP↗
arXiv2025

Resonances for the one dimensional Schrödinger operator with the matrix-valued complex square-well potential

We study the resonances of (generally, non-selfadjoint) Schrödinger operators with matrix-valued square-well potentials. We compute explicitly the Jost function and derive complex transcendental equations for the resonances. We prove several results concerning the distribution of resonances in the complex plane. We compute the multiplicity of resonances and prove a version of the Weyl Law for the number of resonances.

math-ph↗
arXiv2025

Numerical Fredholm determinants for matrix-valued kernels on the real line

We analyze a numerical method for computing Fredholm determinants of trace class and Hilbert Schmidt integral operators defined in terms of matrix-valued kernels on the entire real line. With this method, the Fredholm determinant is approximated by the determinant of a matrix constructed by truncating the kernel of the operator to a finite interval and then applying a quadrature rule. Under the assumption that the kernel decays exponentially, we derive an estimate relating the Fredholm determinant of the operator on the real line to that of its truncation to a finite interval. Then we derive a quadrature error estimate relating the Fredholm determinant of a matrix-valued kernel on a finite interval to its numerical approximation obtained via an adaptive composite Simpson's quadrature rule. These results extend the analysis of Bornemann which focused on Fredholm determinants of trace class operators defined by scalar-valued kernels on a finite interval. Numerical results are provided for a Birman-Schwinger operator that characterizes the stability of stationary solutions of nonlinear wave equations.

math.NA↗
arXiv2025

Recovering the shape of a quantum tree by scattering data

We consider a scattering problem generated by the Sturm-Liouville equation on a tree which consists of an equilateral compact subtree and a half-infinite lead attached to its root. We assume that the potential on the lead is identically zero while the potentials on the finite edges are real. We show how to find the shape of the tree using the S-function of the scattering problem and the eigenvalues of the operators associated with the compact tree.

math.FA↗
arXiv2024

A regularity condition under which integral operators with operator-valued kernels are trace class

We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is Hölder continuous with Hölder exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $\dim H < \infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.

math.FA↗
arXiv2024

First-order asymptotic perturbation theory for extensions of symmetric operators

This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness we obtain a new version of Krein formula for resolvent difference which facilitates asymptotic analysis of resolvent operators via first order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati-type differential equation and the first order asymptotic expansion for resolvents of self-adjoint extensions determined by smooth one-parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard-Rellich-type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self-adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig-Penney model, elliptic second order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.

math.SP↗
arXiv2024

Stability of fronts in the diffusive Rosenzweig-MacArthur model

We consider a diffusive Rosenzweig-MacArthur predator-prey model in the situation when the prey diffuses at the rate much smaller than that of the predator. In a certain parameter regime, the existence of fronts in the system is known: the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. The existence proof is based on application of the Geometric Singular Perturbation Theory with respect to two small parameters. This paper is focused on the stability of the fronts. We analyze the stability by means of energy estimates, exponential dichotomies, the Evans function calculation, and a technique that involves constructing the unstable augmented bundles. The energy estimates provide bounds on the unstable spectrum which depend on the small parameters of the system; the bounds are inversely proportional to these parameters. We further improve these estimates by showing that the eigenvalue problem is a small perturbation of some limiting (as the modulus of the eigenvalue parameter goes to infinity) system, and that the limiting system has exponential dichotomies. Persistence of the exponential dichotomies then leads to bounds uniform in the small parameters. The main novelty of this approach is related to the fact that the limit of the eigenvalue problem is not autonomous. We then use the concept of the unstable augmented bundles and by treating these as multiscale topological structures with respect to the same two small parameters consequently as in the existence proof, we show that the stability of the fronts is also governed by the scalar Fisher-KPP equation.

math.AP↗
arXiv2024

Fredholm determinants, continued fractions, Jost and Evans functions for a Jacobi matrix associated with the 2D-Euler equations

For a second order difference equation that arises in the study of stability of unidirectional (generalized Kolmogorov) flows for the Euler equations of ideal fluids on the two dimensional torus, we relate the following five functions of the spectral parameter: the Fredholm determinants of the Birman-Schwinger operator pencils associated with the second order equation and the equivalent system of the first order equations; the Jost function constructed by means of the Jost solutions of the second order equation; the Evans function constructed by means of the matrix valued Jost solutions of the first order system, and, finally, to backward and forward continued fractions associated with the second order difference equation. We prove that all five functions are equal, and that their zeros are the discrete eigenvalues of the second order difference equation. We use this to improve an instability result for a generalization of the Kolmogorov (unidirectional) flow for the Euler equation on the 2D torus.

math.SP↗
arXiv2023

Uniform bounds of families of analytic semigroups and Lyapunov linear stability of planar fronts

We study families of analytic semigroups, acting in a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction-diffusion equations, and the bidomain equation, important in electrophysiology.

math.SP↗
arXiv2023

Hamiltonian spectral flows, the Maslov index, and the stability of standing waves in the nonlinear Schrödinger equation

We use the Maslov index to study the spectrum of a class of linear Hamiltonian differential operators. We provide a lower bound on the number of positive real eigenvalues, which includes a contribution to the Maslov index from a non-regular crossing. A close study of the eigenvalue curves, which represent the evolution of the eigenvalues as the domain is shrunk or expanded, yields formulas for their concavity at the non-regular crossing in terms of the corresponding Jordan chains. This, along with homotopy techniques, enables the computation of the Maslov index at such a crossing. We apply our theory to study the spectral (in)stability of standing waves in the nonlinear Schrödinger equation on a compact spatial interval. We derive new stability results in the spirit of the Jones--Grillakis instability theorem and the Vakhitov--Kolokolov criterion, both originally formulated on the real line. A fundamental difference upon passing from the real line to the compact interval is the loss of translational invariance, in which case the zero eigenvalue of the linearised operator is geometrically simple. Consequently, the stability results differ depending on the boundary conditions satisfied by the wave. We compare our lower bound to existing results involving constrained eigenvalue counts, finding a direct relationship between the correction factors found therein and the objects of our analysis, including the second-order Maslov crossing form.

math.SP↗
arXiv2022

Fredholm determinants, Evans functions and Maslov indices for partial differential equations

The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin-to-Dirichlet operator pencils. In the self-adjoint case we relate our construction to the Maslov index, another well known tool in the spectral theory of differential operators. This gives new insight into the Maslov index and allows us to obtain crucial monotonicity results by complex analytic methods.

math.SP↗
arXiv2021

Resolvent expansions for self-adjoint operators via boundary triplets

In this paper we develop certain aspects of perturbation theory for self-adjoint operators subject to small variations of their domains. We use the abstract theory of boundary triplets to quantify such perturbations and give the second order asymptotic analysis for resolvents, spectral projections, discrete eigenvalues of the corresponding self-adjoint operators. In particular, we derive explicit formulas for the first variation and the Hessian of the eigenvalue curves bifurcating from a discrete eigenvalue of an unperturbed operator. An application is given to a matrix valued Robin Laplacian and more general Robin-type self-adjoint extensions.

math.SP↗