arXiv subjects

Konstantin A. Makarov

Publications and source records attributed to Konstantin A. Makarov.

34 records · Page 2Linked to original sources

arXiv2005

Evans Functions, Jost Functions, and Fredholm Determinants

The principal results of this paper consist of an intrinsic definition of the Evans function in terms of newly introduced generalized matrix-valued Jost solutions for general first-order matrix-valued differential equations on the real line, and a proof of the fact that the Evans function, a finite-dimensional determinant by construction, coincides with a modified Fredholm determinant associated with a Birman-Schwinger-type integral operator up to a nonvanishing factor.

math.DS↗

arXiv2005

The Threshold effects for the two-particle Hamiltonians on lattices

For a wide class of two-body energy operators $h(k)$ on the three-dimensional lattice $\bbZ^3$, $k$ being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values $k$, $k\ne 0$, the discrete spectrum of $h(k)$ below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian $h(0)$ corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

math-ph↗

arXiv2004

The singularly continuous spectrum and non-closed invariant subspaces

Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\mathfrak{H}_0$ is of codimension 1, we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ of $\mathbf{A}$ that are off-diagonal with respect to the decomposition $\mathfrak{H}= \mathfrak{H}_0\oplus\mathfrak{H}_1$. In particular, we prove the existence of a one-parameter family of dense non-closed invariant subspaces of the operator $\mathbf{A}+\mathbf{V}$ provided that this operator has a nonempty singularly continuous spectrum. We show that such subspaces are related to non-closable densely defined solutions of the operator Riccati equation associated with generalized eigenfunctions corresponding to the singularly continuous spectrum of $\mathbf{B}$.

math.SP↗

arXiv2003

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green's functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants. We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coefficient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant. Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Day's formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.

math.SP↗

arXiv2003

On the existence of solutions to the operator Riccati equation and the tanΘtheorem

Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d>0 be the distance between the spectra of A and C. We prove that under these assumptions the sharp value of the constant c in the condition ||B||<cd guaranteeing the existence of a (bounded) solution to the operator Riccati equation XA-CX+XBX=B^* is equal to \sqrt{2}. We also prove an extension of the Davis-Kahan \tanΘtheorem and provide a sharp estimate for the norm of the solution to the Riccati equation. If C is bounded, we prove, in addition, that the solution X is a strict contraction if B satisfies the condition ||B||<d, and that this condition is sharp.

math.SP↗

arXiv2003

A generalization of the $tan 2Θ$ Theorem

Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant subspace of $\mathbf{A}$. Assuming that $\sup\spec(A_0)\leq \inf\spec(A_1)$, where $A_0$ and $A_1$ are restrictions of $\mathbf{A}$ onto the subspaces $\mathfrak{H}_0$ and $\mathfrak{H}_1=\mathfrak{H}_0^\perp$, respectively, we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded self-adjoint perturbations $\mathbf{V}$ that are off-diagonal with respect to the decomposition $\mathfrak{H} = \mathfrak{H}_0\oplus\mathfrak{H}_1$. We obtain sharp two-sided estimates on the norm of the difference of the orthogonal projections onto invariant subspaces of the operators $\mathbf{A}$ and $\mathbf{B}=\mathbf{A}+\mathbf{V}$. These results extend the celebrated Davis-Kahan $\tan 2Θ$ Theorem. On this basis we also prove new existence and uniqueness theorems for contractive solutions to the operator Riccati equation, thus, extending recent results of Adamyan, Langer, and Tretter.

math.SP↗

arXiv2002

Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach

We introduce a new concept of unbounded solutions to the operator Riccati equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1 \end{pmatrix}$. We also provide a new characterization of the set of all contractive solutions under the assumption that the Riccati equation has a contractive solution associated with a spectral subspace of the operator $\mathbf{B}$. In this case we establish a criterion for the uniqueness of contractive solutions.

math.SP↗

arXiv2002

On a Subspace Perturbation Problem

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let $A$ and $V$ be bounded self-adjoint operators. Assume that the spectrum of $A$ consists of two disjoint parts $σ$ and $Σ$ such that $d=\text{dist}(σ, Σ)>0$. We show that the norm of the difference of the spectral projections $\EE_A(σ)$ and $\EE_{A+V}\big (\{λ| \dist(λ, σ)$ $<d/2\}\big)$ for $A$ and $A+V$ is less then one whenever either (i) $\|V\|<\frac{2}{2+π}d$ or (ii) $\|V\|<{1/2}d$ and certain assumptions on the mutual disposition of the sets $σ$ and $Σ$ are satisfied.

math.SP↗

arXiv2002

$\mathbf {SL_2(\bbR)}$, Exponential Herglotz Representations, and Spectral Averaging

We revisit the concept of spectral averaging and point out its origin in connection with one-parameter subgroups of $SL_2(\bbR)$ and the corresponding Möbius transformations. In particular, we identify exponential Herglotz representations as the basic ingredient for the absolute continuity of average spectral measures with respect to Lebesgue measure and the associated spectral shift function as the corresponding density for the averaged measure. As a by-product of our investigations we unify the treatment of rank-one perturbations of self-adjoint operators and that of self-adjoint extensions of symmetric operators with deficiency indices $(1,1)$. Moreover, we derive separate averaging results for absolutely continuous, singularly continuous, and pure point measures and conclude with an averaging result of the $κ$-continuous part (with respect to the $κ$-dimensional Hausdorff measure) of singularly continuous measures.

math.SP↗

arXiv2001

Graph Subspaces and the Spectral Shift Function

We extend the concept of Lifshits--Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Our main result is the following. Let \cH_0 and \cH_1 be separable Hilbert spaces, A_0 a self-adjoint operator in \cH_0, A_1 a self-adjoint operator in \cH_1, and B_{ij} a bounded operator from \cH_j to \cH_i, i=0,1, j=1-i, and B_10=B_01^*. Assume that the block operator matrix \bH=\bA+\bB=(A_0 & B_01 B_{10}& A_1) has reducing graph subspaces of the form {x_i\oplus Q_ji x_i: x_i\in\cH_i}, i=0,1, j=1-i, and Q_ji are Hilbert-Schmidt operators such that Q_ji=-Q_ij^*. If both (\bH-z\bI)^{-1}-(\bA-z\bI)^{-1} and \bB\bQ(\bA-z\bI)^{-1} are trace class operators in \cH for \Img(z)\neq 0, then the operators A_i+B_ij Q_ji and A_i, i=0,1, j=1-i, acting in the spaces \cH_i are resolvent comparable admissible operators. Moreover, the spectral shift function ξ(x,\bH,\bA) associated with the pair (\bH,\bA) admits the representation ξ(x,\bH,\bA)=ξ(x,A_0+B_01 Q_10,A_0)+ ξ(x,A_1+B_10 Q_01,A_1). We also obtain new representations for the solution to the operator Sylvester equation in the form of Stieltjes operator integrals and formulate sufficient criterion for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices.

math.SP↗

arXiv2001

Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators

Let $g(z,x)$ denote the diagonal Green's matrix of a self-adjoint $m\times m$ matrix-valued Schrödinger operator $H= -\f{d^2}{dx^2}I_m +Q(x)$ in $L^2 (\bbR)^{m}$, $m\in\bbN$. One of the principal results proven in this paper states that for a fixed $x_0\in\bbR$ and all $z\in\bbC_+$, $g(z,x_0)$ and $g^\prime (z,x_0)$ uniquely determine the matrix-valued $m\times m$ potential $Q(x)$ for a.e.~$x\in\bbR$. We also prove the following local version of this result. Let $g_j(z,x)$, $j=1,2$ be the diagonal Green's matrices of the self-adjoint Schrödinger operators $H_j=-\f{d^2}{dx^2}I_m +Q_j(x)$ in $L^2 (\bbR)^{m}$. Suppose that for fixed $a>0$ and $x_0\in\bbR$, $\|g_1(z,x_0)-g_2(z,x_0)\|_{\bbC^{m\times m}}+ \|g_1^\prime (z,x_0)-g_2^\prime (z,x_0)\|_{\bbC^{m\times m}} \underset{|z|\to\infty}{=}O\big(e^{-2\Im(z^{1/2})a}\big)$ for $z$ inside a cone along the imaginary axis with vertex zero and opening angle less than $π/2$, excluding the real axis. Then $Q_1(x)=Q_2(x)$ for a.e.~$x\in [x_0-a,x_0+a]$. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators.

math.SP↗

arXiv1999

The Xi Operator and its Relation to Krein's Spectral Shift Function

We explore connections between Krein's spectral shift function $ξ(λ,H_0,H)$ associated with the pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift operator $Ξ(J+K^*(H_0-λ-i0)^{-1}K)$ associated with the operator-valued Herglotz function $J+K^*(H_0-z)^{-1}K$, $\Im(z)>0$ in $\calH$, where $V=KJK^*$ and $J=\sgn(V)$. Our principal results include a new representation for $ξ(λ,H_0,H)$ in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections $(E_{J+A(λ)+tB(λ)}((-\infty,0)),E_J((-\infty,0)))$, $t\in\bbR$, where $A(λ)=\Re(K^*(H_0-λ-i0)^{-1}K)$, $B(λ)=\Im(K^*(H_0-λ-i0)^{-1}K)$ a.e. Moreover, introducing the new concept of a trindex for a pair of operators $(A,P)$ in $\calH$, where $A$ is bounded and $P$ is an orthogonal projection, we prove that $ξ(λ,H_0,H)$ coincides with the trindex associated with the pair $(Ξ(J+K^*(H_0-λ-i0)^{-1}K),Ξ(J))$. In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of $Ξ$-operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman-Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.

math.SP↗

arXiv1999

Some Applications of the Spectral Shift Operator

The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric operators.

math.SP↗

arXiv1999

The Spectral Shift Operator

We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein's trace formula and to the Birman-Solomyak spectral averaging formula are discussed.

math.SP↗

arXiv1998

Some Applications of Operator-Valued Herglotz Functions

We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for classes of Herglotz operators. Moreover, we study the concrete case of Schrödinger operators on a half-line and provide two illustrations of Livsic's result [44] on quasi-hermitian extensions in the special case of densely defined symmetric operators with deficiency indices (1,1).

math.FA↗

arXiv1997

An Addendum to Krein's Formula

We provide additional results in connection with Krein's formula, which describes the resolvent difference of two self-adjoint extensions A_1 and A_2 of a densely defined closed symmetric linear operator A with (possibly infinite) equal deficiency indices. In particular, we explicitly derive the linear fractional transformation relating the operator-valued Weyl-Titchmarsh M-functions M_1(z) and M_2(z) corresponding to A_1 and A_2.

funct-an↗