arXiv subjects

Konstantin A. Makarov

Publications and source records attributed to Konstantin A. Makarov.

At least 19 recordsLinked to original sources

arXiv2026

Dissipation and c-Entropy in Nevanlinna-Pick Interpolation

We study interpolation L-systems realizing finite Nevanlinna-Pick data sets and analyze their structural and quantitative characteristics. Explicit formulas are derived for the c-entropy and dissipation coefficient, two intrinsic invariants that describe the dissipative structure of interpolation L-systems. These quantities depend only on the geometric placement of interpolation nodes in $\mathbb{C}_+$, attaining maximal finite values for purely imaginary nodes. The interpolation model $Θ_Δ$ and its unitary equivalents reveal that these invariants form a direct link between analytic interpolation data and the dynamical properties of L-systems. Particular attention is given to symmetric configurations, where the impedance function admits an explicit rational representation and a natural physical interpretation in terms of equivalent $LC$-networks.

math.FA↗

arXiv2026

Threshold Virtual States of a Jacobi operator

We prove that the set of parameters for which a virtual level appears at the edge of the continuous spectrum of a Jacobi matrix with a finite-rank diagonal perturbation constitutes an algebraic variety of codimension one. This variety partitions the parameter space into connected components, with their number determined by the size of the perturbation support. We also reveal a hierarchical structure underlying these critical varieties as the rank of the perturbation increases.

math.SP↗

arXiv2025

The c-Entropy of non-dissipative L-systems

In this paper, we extend the definition of c-entropy to canonical L-systems with non-dissipative state-space operators. We also introduce the concepts of dissipation and accumulation coefficients for such systems. In addition, we examine the coupling of these L-systems and derive closed form expressions for the corresponding c-entropy.

math.SP↗

arXiv2023

The L-system representation and c-entropy

Given a symmetric operator $\dot A$ with deficiency indices $(1,1)$ and its self-adjoint extension $A$ in a Hilbert space $\mathcal{H}$, we construct a (unique) L-system with the main operator in $\mathcal{H}$ such that its impedance mapping coincides with the Weyl-Titchmarsh function $M_{(\dot A, A)}(z)$ or its linear-fractional transformation $M_{(\dot A, A_α)}(z)$. Similar L-system constructions are provided for the Weyl-Titchmarsh function $aM_{(\dot A, A)}(z)$ with $a>0$. We also evaluate c-entropy and the main operator dissipation coefficient for the obtained L-systems.

math.SP↗

arXiv2017

Diagonalization of indefinite saddle point forms

We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes Operator from fluid dynamics.

math-ph↗

arXiv2017

The Tan $2 Θ$-Theorem in Fluid Dynamics

We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating the distance from the bottom of its spectrum to the origin and the length of the first positive gap.

math.SP↗

arXiv2016

On invariant graph subspaces

In this paper we discuss the problem of decomposition for unbounded $2\times 2$ operator matrices by a pair of complementary invariant graph subspaces. Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrices.

math.SP↗

arXiv2015

Convolutions with probability distributions, zeros of L-functions, and the least quadratic nonresidue

Let $d$ be a probability distribution. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{d^{*n}(x)}{n}=1,\qquad\text{where}\quad d^{*n}:=\underbrace{\,d*d*\cdots*d\,}_{n\text{ times}}. $$ For a compactly supported distribution $d$, we show that if $c>0$ is a given constant and the function $f(k):=\widehat d(k)-1$ does not vanish on the line $\{k\in{\mathbb C}:\Im\,k=-c\}$, where $\widehat d$ is the Fourier transform of $d$, then one has the asymptotic expansion $$ \sum_{n=1}^\infty\frac{d^{*n}(x)}{n}=\frac{1}{x}\bigg(1+\sum_k m(k) e^{-ikx}+O(e^{-c x})\bigg)\qquad (x\to +\infty), $$ where the sum is taken over those zeros $k$ of $f$ that lie in the strip $\{k\in{\mathbb C}:-c<\Im\,k<0\}$, $m(k)$ is the multiplicity of any such zero, and the implied constant depends only on $c$. For a given distribution $d$ of this type, we briefly describe the location of the zeros $k$ of $f$ in the lower half-plane $\{k\in{\mathbb C}:\Im\,k<0\}$. For an odd prime $p$, let $n_0(p)$ be the least natural number such that $(n|p)=-1$, where $(\cdot|p)$ is the Legendre symbol. As an application of our work on probability distributions, in this paper we generalize a well known result of Heath-Brown concerning the behavior of the Dirichlet $L$-function $L(s,(\cdot|p))$ under the assumption that the Burgess bound $n_0(p)\ll p^{1/(4\sqrt{e})+ε}$ cannot be improved.

math.NT↗

arXiv2015

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, ${\rm ind}(D_A)$, of the operator $D_A = (d/dt) + A$ on $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(A f)(t) = A(t) f(t)$ for a.e. $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$, via the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ associated with the pair $(A_+, A_-)$ of asymptotic operators $A_{\pm}=A(\pm\infty)$ on the separable complex Hilbert space $\mathcal{H}$ in the case when $A(t)$ is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator $A_-$. We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function $ξ(\, \cdot \,;A_+,A_-)$ for the pair $(A_+, A_-)$, and the corresponding spectral shift function $ξ(\, \cdot \,;H_2,H_1)$ for the pair of operators $(H_2,H_1)=(D_A {D_A}^*, {D_A}^* D_A)$ in this relative trace class context. This formula is then used to identify the Fredholm index of $D_A$ with $ξ(0;A_+,A_-)$. In addition, we prove that ${\rm ind}(D_A)$ coincides with the spectral flow SpFlow$(\{A(t)\}_{t=-\infty}^\infty)$ of the family $\{A(t)\}_{t\in\mathbb{R}}$ and also relate it to the (Fredholm) perturbation determinant for the pair $(A_+, A_-)$. We also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.

math.SP↗

arXiv2014

On a perturbation determinant for accumulative operators

For a purely imaginary sign-definite perturbation of a self-adjoint operator, we obtain exponential representations for the perturbation determinant in both upper and lower half-planes and derive respective trace formulas.

math.SP↗

arXiv2013

Reducing graph subspaces and strong solutions to operator Riccati equations

The problem of block diagonalization for diagonally dominant symmetric block operator matrices with self-adjoint diagonal entries is considered. We show that a reasonable block diagonalization with respect to a reducing graph subspace requires a related skew-symmetric operator to be a strong solution to the associated Riccati equation. Under mild additional regularity conditions, we also establish that this skew-symmetric operator is a strong solution to the Riccati equation if and only if the graph subspace is reducing for the given operator matrix. These regularity conditions are shown to be automatically fulfilled whenever the corresponding relative bound of the off-diagonal part is sufficiently small. This extends the results by Albeverio, Makarov, and Motovilov in [Canad. J. Math. Vol. \textbf{55}, 2003, 449--503], where the off-diagonal part is required to be bounded.

math.SP↗

arXiv2012

Representation Theorems for Indefinite Quadratic Forms Revisited

The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.

math.FA↗

arXiv2010

Metric properties of the set of orthogonal projections and their applications to operator perturbation theory

We prove that the set of orthogonal projections on a Hilbert space equipped with the length metric is $\frac\pi2$-geodesic. As an application, we consider the problem of variation of spectral subspaces for bounded linear self-adjoint operators and obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unpertubed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No. 1, 77 -- 89] and [Proc. Amer. Math. Soc., 131, 3469 -- 3476] are sharpened.

math.SP↗

arXiv2010

The TAN $2Θ$ Theorem for Indefinite Quadratic Forms

A version of the Davis-Kahan Tan $2Θ$ theorem [SIAM J. Numer. Anal. \textbf{7} (1970), 1 -- 46] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a recent result by Motovilov and Selin [Integr. Equat. Oper. Theory \textbf{56} (2006), 511 -- 542].

math.SP↗

arXiv2008

Essential Closures and AC Spectra for Reflectionless CMV, Jacobi, and Schrödinger Operators Revisited

We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application of the notion of the essential closure of subsets of $\bbR$ we revisit the fact that CMV, Jacobi, and Schrödinger operators, reflectionless on a set E of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure of the set E (with uniform multiplicity two on E). Though this result in the case of Schrödinger and Jacobi operators is known to experts, we feel it nicely illustrates the concept and usefulness of essential closures in the spectral theory of classes of reflectionless differential and difference operators.

math.SP↗

arXiv2007

Perturbation of spectra and spectral subspaces

We consider the problem of variation of spectral subspaces for linear self-adjoint operators under off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections.

math.SP↗

arXiv2006

The Birman-Schwinger principle in von Neumann algebras of finite type

We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman-Schwinger principle in this setting. As an application of this result, revisiting the Birman-Krein formula in the abstract scattering theory, we represent the de la Harpe-Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.

math.SP↗

arXiv2006

On Krein's Example

In his 1953 paper [Matem.~Sbornik \textbf{33} (1953), 597 -- 626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval $[-1,1]$ and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm.

math.SP↗