A. Minkin
S.G.Krein's conjecture concerning Birkhoff-regularity of dissipative differential operators has been proved in the even order case. As a byproduct an existence of the limit of characteristic matrix as in the lower half-plane has been established. Up to multiplication by a nonvanishing matrix this limit coincides with the ratio of the matrices of regularity determinants.
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We consider Schrödinger operators of the form $H_R = - d^2/ d x^2 + q + i γχ_{[0,R]}$ for large $R>0$, where $q \in L^1(0,\infty)$ and $γ> 0$. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large $R$.
We prove that a dissipative operator in Krein space possesses a maximal non-negative invariant subspace provided that this operator admits matrix representation with respect to canonical decomposition of the space and the right upper entry of the operator matrix is relatively compact with respect to the right lower entry. Under additional condition when the left upper and left lower operators are bounded (the so-called "Langer condition") this result was proved (in the increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. The Langer condition is replaced in the paper by a m…
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 subg…
Dissipative eigenvalue problems for a singular quantum Dirac system
本文在特定Hilbert空间中研究耗散奇异q-Dirac算子，通过构造自伴膨胀等方法得到其散射矩阵、特征函数，并证明相关向量系的完备性。
We consider an integral dissipative operator in its Brodskii-Livshits triangular representation. The main question we are concerned with is similarity of the operator to a normal one. We obtain necessary as well as sufficient conditions for the similarity. The study is based on functional model technique.
In this article we develop a functional model for a general maximal dissipative operator. We construct the selfadjoint dilation of such operators. Unlike previous functional models, our model is given explicitly in terms of parameters of the original operator, making it more useful in concrete applications. For our construction we introduce an abstract framework for working with a maximal dissipative operator and its anti-dissipative adjoint and make use of the Straus characteristic function in our setting. Explicit formulae are given for the selfadjoint dilation, its resolvent, a core and the…
Complete Non-Selfadjointness of Extensions of Symmetric Operators with Bounded Dissipative Perturbations
本文通过边界三元组建立抽象框架，研究对称算子加有界非负扰动形成的耗散算子的极大扩张的完全非自伴性，并以区间上的耗散薛定谔算子为例。
The purpose of this paper is to obtain an integral representation for the difference $f(L_1)-f(L_2)$ of functions of maximal dissipative operators. This representation in terms of double operator integrals will allow us to establish Lipschitz type estimates for functions of maximal dissipative operators. We also consider a similar problem for quasicommutators, i.e., operators of the form $f(L_1)R-Rf(L_2)$.
In this note we investigate complete non-selfadjointness for all maximally dissipative extensions of a Schrödinger operator on a half-line with dissipative bounded potential and dissipative boundary condition. We show that all maximally dissipative extensions that preserve the differential expression are completely non-selfadjoint. However, it is possible for maximally dissipative extensions to have a one-dimensional reducing subspace on which the operator is selfadjoint. We give a characterisation of these extensions and the corresponding subspaces and present a specific example.
It is shown that positivity in $(0,1)\times (0,1)$ of Green function of positively defined fourth-order ordinary differential operator (with separated boundary conditions) is a criterium of sign-regularity of this operator.
We generalize our results of \cite{AP2} and \cite{AP3} to the case of maximal dissipative operators. We obtain sharp conditions on a function analytic in the upper half-plane to be operator Lipschitz. We also show that a Hölder function of order $\a$, $0<\a<1$, that is analytic in the upper half-plane must be operator Hölder of order $\a$. Then we generalize these results to higher order operator differences. We obtain sharp conditions for the existence of operator derivatives and express operator derivatives in terms of multiple operator integrals with respect to semi-spectral measures. Final…
We consider the damped Schrödinger semigroup $e^{-it \frac{d^2}{dx^2}}$ on the tadpole graph ${\mathcal R}$. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of $L^2({\mathcal R})$, we prove that the corresponding energy decay exponentially.
We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $\partial_ν u - γ(x) \partial_t u = 0$ on the boundary $Γ$ and $γ(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The eigenvalues $λ_k$ of $G$ with ${\rm Re}\: λ_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{λ_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $\min_{x\in Γ} γ(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $…
Spectral theory of infinite dimensional dissipative Hamiltonian systems
本文研究无限维耗散哈密顿系统中算子束的谱理论，引入三种奇异性概念，分析其与微分代数方程解唯一性的关系，并揭示在耗散结构化算子束中唯一性与奇异性紧密相关。
The paper investigates spectral properties of multi-interval Sturm-Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the trace class and for the systems of root functions to be complete are found. Results of paper are new for one-interval boundary value problems as well.
Real-valued spectral shift functions for contractions and dissipative operators
本文针对收缩算子对与耗散算子对，公布了可积实值谱移函数存在性的充分条件，补充了此前迹类差收缩算子可积谱移函数存在性相关研究的实值性结论。
Real spectral shift functions for pairs of contractions and pairs of dissipative operators
本文研究了收缩算子对和耗散算子对的实谱移函数条件，推导了其存在性，并应用于耗散Schrödinger算子。
Let $f$ be a function in the inhomogeneous analytic Besov space $B_{\infty,1}^1$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, we define the function $f(L,M)$ of $L$ and $M$ as a densely defined linear operator. We prove for $p\in[1,2]$ that if $(L_1,M_1)$ and $(L_2,M_2)$ are pairs of not necessarily commuting maximal dissipative operators such that both differences $L_1-L_2$ and $M_1-M_2$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ than for an arbitrary function $f$ in the inhomogeneous analytic Besov space $B_{\infty,1}^1$, the operator diff…
For Maxwell operators $(E,H) \to (i ε^{-1} \nabla\times H, -i μ^{-1} \nabla \times E)$ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted for the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a tran…
Functions of dissipative operators under relatively bounded and relatively trace class perturbations
本文研究耗散算子函数在相对有界与相对迹类扰动下的行为，引入解析相对算子Lipschitz函数类，证明其是使迹公式成立的最大函数类，并给出谱位移函数的可积性不等式。
We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the o…
We prove uniform resolvent estimates for an abstract operator given by a dissipative perturbation of a self-adjoint operator in the sense of forms. For this we adapt the commutators method of Mourre. We also obtain the limiting absorption principle and uniform estimates for the derivatives of the resolvent. This abstract work is motivated by the Schr{ö}dinger and wave equations on a wave guide with dissipation at the boundary.
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