arXiv subjects
Publications and source records attributed to K. A. Makarov.
We extend the invariance principle for a characteristic function of a dissipative operator with respect to the group of affine transformations of the real axis preserving the orientation to the case of general $SL_2(\bbR)$ transformations.
We prove the uniqueness theorem for the solutions to the restricted Weyl commutation relations braiding unitary groups and semi-groups of contractions that are close to unitaries. We also discuss related mathematical problems of continuous monitoring of quantum systems and provide rigorous foundations for the exponential decay phenomenon of a resonant state in quantum mechanics.
We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine group $\cG$, the group of affine transformations of the real axis preserving the orientation. If $A$ is a symmetric operator, the group $G$ induces an action/flow on the operator unit ball of contractive transformations from $\Ker (A^*-iI)$ to $\Ker (A^*+iI)$. We establish several fixed point theorems for this flow. In the case of one-parameter continuous subgroups of linear transformations, self-adjoint (maximal dissipative) operators associated with the fixed points of the flow give rise to solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of non-unitarily equivalent representations. In the case of deficiency indices $(1,1)$, our general results can be strengthened to the level of an alternative.
We study the connection between the Livšic class of functions $s(z)$ that are the characteristic functions of densely defined symmetric operators $\dot A$ with deficiency indices $(1, 1)$, the characteristic functions $S(z)$ (the Möbius transform of $s(z)$) of a maximal dissipative extension $T$ of $\dot A$ (determined by the von Neumann parameter $κ$ of the extension relative to an appropriate basis in the deficiency subspaces) and the transfer functions $W_Θ(z)$ of a conservative L-system $Θ$ with the main operator $T$. It is shown that under a natural hypothesis $S(z)$ and $W_Θ(z)$ are reciprocal to each other. In particular, when $κ=0$, $W_Θ(z)=\frac{1}{S(z)}=-\frac{1}{s(z)}$. It is established that the impedance function of a conservative L-system with the main operator $T$ coincides with the function from the Donoghue class if and only if the von Neumann parameter vanishes ($κ=0$). Moreover, we introduce the generalized Donoghue class and obtain the criteria for an impedance function to belong to this class. All results are illustrated by a number of examples.
We discuss the classes $\fC$, $\fM$, and $\fS$ of analytic functions that can be realized as the Livšic characteristic functions of a symmetric densely defined operator $\dot A$ with deficiency indices $(1,1)$, the Weyl-Titchmarsh functions associated with the pair $(\dot A, A)$ where $A$ is a self-adjoint extension of $\dot A$, and the characteristic function of a maximal dissipative extension $\widehat A$ of $\dot A$, respectively. We show that the class $\fM$ is a convex set, both of the classes $\fS$ and $\fC$ are closed under multiplication and, moreover, $\fC\subset \fS$ is a double sided ideal in the sense that $\fS\cdot \fC=\fC\cdot \fS\subset \fS$. The goal of this paper is to obtain these analytic results by providing explicit constructions for the corresponding operator realizations. In particular, we introduce the concept of an operator coupling of two unbounded maximal dissipative operators and establish an analog of the Livšic-Potapov multiplication theorem [14] for the operators associated with the function classes $\fC$ and $\fS$. We also establish that the modulus of the von Neumann parameter characterizing the domain of $\widehat A$ is a multiplicative functional with respect to the operator coupling.
We discuss convergence properties of the spectral shift functions associated with a pair of Schrodinger operators with Dirichlet boundary conditions at the end points of a finite interval (0, r) as the length of interval approaches infinity.
We introduce the concept of a μ-scale invariant operator with respect to unitary transformation in a separable complex Hilbert space. We show that if a nonnegative densely defined symmetric operator is μ-scale invariant for some μ>0, then both the Friedrichs and the Krein-von Neumann extensions are also μ-scale invariant.
We prove a generalization of the well-known theorems by Borg and Hochstadt for periodic self-adjoint Schrödinger operators without a spectral gap, respectively, one gap in their spectrum, in the matrix-valued context. Our extension of the theorems of Borg and Hochstadt replaces the periodicity condition of the potential by the more general property of being reflectionless (the resulting potentials then automatically turn out to be periodic and we recover Després' matrix version of Borg's result). In addition, we assume the spectra to have uniform maximum multiplicity (a condition automatically fulfilled in the scalar context considered by Borg and Hochstadt). Moreover, the connection with the stationary matrix KdV hierarchy is established. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-valued Schrödinger operators.
Let $H_0$ and $V(s)$ be self-adjoint, $V,V'$ continuously differentiable in trace norm with $V''(s)\geq 0$ for $s\in (s_1,s_2)$, and denote by $\{E_{H(s)}(λ)\}_{λ\in\bbR}$ the family of spectral projections of $H(s)=H_0+V(s)$. Then we prove for given $μ\in\bbR$, that $s\longmapsto \tr\big (V'(s)E_{H(s)}((-\infty, μ))\big) $ is a nonincreasing function with respect to $s$, extending a result of Birman and Solomyak. Moreover, denoting by $ζ(μ,s)=\int_{-\infty}^μdλξ(λ,H_0,H(s))$ the integrated spectral shift function for the pair $(H_0,H(s))$, we prove concavity of $ζ(μ,s)$ with respect to $s$, extending previous results by Geisler, Kostrykin, and Schrader. Our proofs employ operator-valued Herglotz functions and establish the latter as an effective tool in this context.