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Mark Malamud

Publications and source records attributed to Mark Malamud.

24 records · Page 2Linked to original sources
arXiv2012

On Titchmarsh-Weyl functions of first-order symmetric systems with arbitrary deficiency indices

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'(t)-B(t)y(t)=\D(t) f(t)$ on an interval $[a,b> $ with the regular endpoint $a$. The deficiency indices $n_\pm$ of the corresponding minimal relation $\Tmi$ may be arbitrary (possibly unequal). Our approach is based on the concept of a decomposing boundary triplet, which enables one to parametrize various classes of extensions of $\Tmi$ (self-adjoint, $m$-dissipative, etc.) in terms of boundary conditions imposed on regular and singular values of a function $y\in \dom \tma$ at the endpoints $a$ and $b$ respectively. In particular, we describe self-adjoint and $ł$-depending Nevanlinna boundary conditions which are analogs of separated ones for Hamiltonian systems. With a boundary value problem involving such conditions we associate the $m$-function $m(\cd)$, which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. In the simplest case of minimal (unequal) deficiency indices $n_\pm$ the $m$-function $m(\cd)$ coincides with the rectangular Titchmarsh-Weyl coefficient introduced by Hinton and Schneider. We parametrize all $m$-functions in terms of the Nevanlinna boundary parameter at the endpoint $b$ by means of the formula similar to the known Krein formula for resolvents. Application of these results to differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh-Weyl theory of such operators.

math.FA↗
arXiv2011

Sturm-Liouville boundary value problems with operator potentials and unitary equivalence

Consider the minimal Sturm-Liouville operator $A = A_{\rm min}$ generated by the differential expression $\mathcal{A} := -\frac{d^2}{dt^2} + T$ in the Hilbert space $L^2(\mathbb{R}_+,\mathcal{H})$ where $T = T^*\ge 0$ in $\mathcal{H}$. We investigate the absolutely continuous parts of different self-adjoint realizations of $\mathcal{A}$. In particular, we show that Dirichlet and Neumann realizations, $A^D$ and $A^N$, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if $\infσ_{ess}(T) = \infσ(T) \ge 0$, then the part $\widehat{A}^{ac}E_{\widehat{A}(σ(A^D))$ of any self-adjoint realization $\widehat{A}$ of $\mathcal{A}$ is unitarily equivalent to $A^D$. In addition, we prove that the absolutely continuous part $\widehat{A}^{ac}$ of any realization $\widehat{A}$ is unitarily equivalent to $A^D$ provided that the resolvent difference $(\widehat{A} - i)^{-1}- (A^D - i)^{-1}$ is compact. The abstract results are applied to elliptic differential expression in the half-space.

math-ph↗
arXiv2009

1--D Schrödinger operators with local interactions on a discrete set

Spectral properties of 1-D Schrödinger operators $\mathrm{H}_{X,α}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in X}α_nδ(x-x_n)$ with local point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty$ are well studied when $d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. We consider $\mathrm{H}_{X,α}$ in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of $\mathrm{H}_{X,α}$ like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators $\mathrm{H}_{X,α}$ to be self-adjoint, lower-semibounded, and discrete in the case $d_*=0$. The operators with $δ'$-type interactions are investigated too. The obtained results demonstrate that in the case $d_*=0$, as distinguished from the case $d_*>0$, the spectral properties of the operators with $δ$ and $δ'$-type interactions are substantially different.

math.SP↗
arXiv2006

Boundary relations and generalized resolvents of symmetric operators

The Kre\uın-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in $\dC_+$) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\uın-Naimark formula is investigated and explained.

math.SP↗
arXiv2005

Operator Holes and extensions of sectorial operators and dual pairs of contractions

A description of the set of $m$-sectorial extensions of a dual pair $\{A_1,A_2\}$ of nonnegative operators is obtained. Some classes of nonaccretive extensions of the dual pair $\{A_1,A_2\}$ are described too. Both problems are reduced to similar problems for a dual pair $\{T_1,T_2\}$ of nondensely defined symmetric contractions $T_j=(I-A_j)(I+A_j)^{-1},\ j\in\{1,2\}$. In turn these problems are reduced to the investigation of the corresponding operator "holes". A complete description of the set of all proper and improper extensions of a nonnegative operator is obtained too.

math.FA↗