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Alexander K. Motovilov

Publications and source records attributed to Alexander K. Motovilov.

31 records · Page 2Linked to original sources
arXiv2007

Resonances of Multichannel Systems

We describe structure of the T-matrices, scattering matrices, and Green functions on unphysical energy sheets in multichannel scattering problems with binary channels and in the three-body problem. Based on the explicit representations obtained for the values of T- and S-matrices on the unphysical sheets, we prove that the resonances belonging to an unphysical sheet are just those energies where the correspondingly truncated scattering matrix, taken in the physical sheet, has eigenvalue zero. We show, in addition, that eigenvectors of the truncated scattering matrix associated with its zero eigenvalue are formed of the breakup amplitudes for the respective resonant states.

quant-ph↗
arXiv2006

Some sharp norm estimates in the subspace perturbation problem

We discuss the spectral subspace perturbation problem for a self-adjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an \textit{a priori} sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, \textit{a priori}, $\tanΘ$ Theorem. We also extend the Davis--Kahan $\tan 2Θ$ Theorem to the case of some unbounded perturbations.

math.SP↗
arXiv2005

The a priori tanθtheorem for eigenvectors

Let $A$ be a self-adjoint operator on a Hilbert space $\fH$. Assume that the spectrum of $A$ consists of two disjoint components $σ_0$ and $σ_1$ such that the convex hull of the set $σ_0$ does not intersect the set $σ_1$. Let $V$ be a bounded self-adjoint operator on $\fH$ off-diagonal with respect to the orthogonal decomposition $\fH=\fH_0\oplus\fH_1$ where $\fH_0$ and $\fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $σ_0$ and $σ_1$, respectively. It is known that if $\|V\|<\sqrt{2}d$ where $d=\dist(σ_0,σ_1)>0$ then the perturbation $V$ does not close the gaps between $σ_0$ and $σ_1$. Assuming that $f$ is an eigenvector of the perturbed operator $A+V$ associated with its eigenvalue in the interval $(\min(σ_0)-d,\max(σ_0)+d)$ we prove that under the condition $\|V\|<\sqrt{2}d$ the (acute) angle $θ$ between $f$ and the orthogonal projection of $f$ onto $\fH_0$ satisfies the bound $\tanθ\leq\frac{\|V\|}{d}$ and this bound is sharp.

math.SP↗
arXiv2005

Explicit Representations for the T-Matrix on Unphysical Energy Sheets and Resonances in Two- and Three-Body Systems

We discuss the structure of the two- and three-body T-matrices, scattering matrices, and resolvents continued to the unphysical energy sheets. Our conclusions arise due to the representations that have been found for analytically continued momentum-space kernels of the T-operators. These representations are explicitly written only in terms of the physical-sheet kernels of the T-matrix itself. One of advantages of the representations in the three-body case is that they show which portions of the physical-sheet three-body scattering matrix are ``responsible'' for the resonances associated with a particular unphysical sheet. A resonance appears to be the energy where the correspondingly truncated scattering matrix (taken on the physical sheet) has eigenvalue zero. We also mention applications of this approach to some specific three-body systems, based on the Faddeev differential equations.

physics.atm-clus↗
arXiv2005

Operator integrals with respect to a spectral measure and solutions to some operator equations

We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the entry A is separated from the spectrum of C, and D is a bounded operator, we obtain a representation for the strong solution X to the Sylvester equation XA-CX=D in the form of an operator Stieltjes integral with respect to the spectral measure of C. By using this result we then establish sufficient conditions for the existence of a strong solution to the operator Riccati equation YA-CY+YBY=D where B is another bounded operator.

math.SP↗
arXiv2004

Perturbation of finite-lattice spectral levels by nearby nuclear resonances

We consider finite linear or cyclic crystalline structures with molecular cells having narrow pre-threshold nuclear resonance. We prove that if the real part of such a nuclear resonance lies within the energy band (the convex hull of the energy levels) of the crystalline structure arising of a separated molecular level, then there exist molecular crystalline states that decay exponentially in time and the decay rate $Γ_R^{(m)}$ of these states in the main order is described by the formula $Γ_R^{(m)}\cong 4\frac{\mathop{\rm Re} a}{Γ_R^{(n)}}$ where $a$ is the value of the residue of the molecular channel transfer function at the nuclear resonance point and $Γ_R^{(n)}$ is the nuclear resonance width.

cond-mat.other↗
arXiv2004

Scattering length of the helium atom - helium dimer collision

We present our recent results on the scattering length of ^4He-^4He_2 collisions. These investigations are based on the hard-core version of the Faddeev differential equations. As compared to our previous calculations of the same quantity, a much more refined grid is employed, providing an improvement of about 10%. Our results are compared with other ab initio, and with model calculations.

physics.atm-clus↗
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↗
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↗