Resolvents of self-adjoint extensions with mixed boundary conditions

Self-adjoint Extensions of Restrictions

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator $S$ obtained by restricting the self-adjoint operator $A:\D(A)\subseteq\H\to\H$ to the dense, closed with respect to the graph norm, subspace $\N\subset \D(A)$. Neither the knowledge of $S^*$ nor of the deficiency spaces of $S$ is required. Typically $A$ is a differential operator and $\N$ is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle $π:\E(\fh)\to¶(\fh)$, where $¶(\fh)$ denotes the set of orthogonal…

Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the "free" operator with domain $H^{2}(\mathbb{R}^{n})$; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditio…

张量提升的多变量函数学超越交换性和有界性

Tensor-Lifted Multivariate Functional Calculus Beyond Commutativity and Boundedness

本文提出一种多变量函数学框架，结合谱和代数信息，处理非交换、非自共轭和无界算子，通过张量提升和双重收敛理论，保留Nilpotent结构并提供收敛保证。

自伴半群的个体与一致最终正性

Individual and Uniform Eventual Positivity of Self-Adjoint Semigroups

本文构造了一个生成元具紧预解式的实自伴半群，其个体最终强正但任意大时间算子非正，证明L²空间自伴半群的个体与一致最终正性不等价。

Generalization of Lax Equivalence Theorem on Unbounded Self-adjoint Operators with Applications to Schrödinger Operators

Define $ A $ a unbounded self-adjoint operator on Hilbert space $ X $. Let $ \{ A_n \} $ be its resolvent approximation sequence with closed range $ \mathcal{R}(A_n) (n \in \mathrm{N}) $, that is, $ A_n (n \in \mathrm{N}) $ are all self-adjoint on Hilbert space $ X $ and \begin{equation*} \hbox{ \raise-2mm\hbox{$\textstyle s-\lim \atop \scriptstyle {n \to \infty}$}} R_λ(A_n) = R_λ(A)\quad (λ\in \mathrm{C} \setminus \mathrm{R}), \ \textrm{where} \ R_ λ(A) := (λI-A)^{-1}. \end{equation*} The Moore-Penrose inverse $ A^\dagger_n \in \mathcal{B}(X) $ is a natural approximation to the Moore-Penrose …

谱作用泛函在$\tau$-紧预解式情形下的逼近

Approximation of the spectral action functional in the case of $τ$-compact resolvents

本文针对半有限冯·诺依曼代数中具有$\tau$-紧预解式的自伴算子，建立了谱作用泛函泰勒逼近余项的估计，显著推广了现有结果并扩展到非紧支集函数情形。

Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

Given the symmetric operator $A_N$ obtained by restricting the self-adjoint operator $A$ to $N$, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint $A_N^*$ and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of $A_N$ and their resolvents.

自伴算子的二进预解式表示：传播子展开、谱测度和zeta函数

Dyadic Resolvent Representations of Self-Adjoint Operators: Propagator Expansions, Spectral Measures, and Zeta Functions

研究自伴算子\(A\)的二进预解式表示，通过将底层算子视为服从Landen加倍递归的单参数尺度族等方法，得到相关级数及特殊值，推导方程基本解的二进表示，还能从二进表示重建\(A\)的谱数据，且二进样本唯一确定\(A\)。

Self-Adjoint Extensions by Additive Perturbations

Let $A_\N$ be the symmetric operator given by the restriction of $A$ to $\N$, where $A$ is a self-adjoint operator on the Hilbert space $\H$ and $\N$ is a linear dense set which is closed with respect to the graph norm on $D(A)$, the operator domain of $A$. We show that any self-adjoint extension $A_Θ$ of $A_\N$ such that $D(A_Θ)\cap D(A)=\N$ can be additively decomposed by the sum $A_Θ=\A+T_Θ$, where both the operators $\A$ and $T_Θ$ take values in the strong dual of $D(A)$. The operator $\A$ is the closed extension of $A$ to the whole $\H$ whereas $T_Θ$ is explicitly written in terms of a (a…

通过伴随线性算子求解具有混合边界条件的静态线性弹性应力问题

Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

本文从几何与泛函分析视角研究混合边界条件下静态线性弹性应力问题，通过构造一对无界伴随算子，将应力解表示为基本子空间仿射平移的交集，并利用位移边界条件问题的算子性质进行分析。

On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $\mathfrak H$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $\mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $\widetilde A =…

伴随对的广义边界三元组及其在非自伴Schrödinger算子中的应用

Generalized boundary triples for adjoint pairs with applications to non-self-adjoint Schrödinger operators

本文将广义边界三元组及Weyl函数推广至算子伴随对，给出诱导非空预解集闭算子的边界参数准则，并应用于具紧致边界的Lipschitz域上带复Lp势的Schrödinger算子。

Removal of the Energy Dependence from the Resolvent-like Energy-Dependent Interactions

The spectral problem $(A + V(z))ψ=zψ$ is considered with $A$, a self-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation $V(z)$ is assumed to depend on the energy $z$ as resolvent of another self-adjoint operator $A':$ $V(z)=-B(A'-z)^{-1}B^{*}$. It is supposed that operator $B$ has a finite Hilbert-Schmidt norm and spectra of operators $A$ and $A'$ are separated. The conditions are formulated when the perturbation $V(z)$ may be replaced with an energy-independent ``potential'' $W$ such that the Hamiltonian $H=A +W$ has the same spectrum (more exactly a part of spectrum) and …

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

Let $J$ and $R$ be anti-commuting fundamental symmetries in a Hilbert space $\mathfrak{H}$. The operators $J$ and $R$ can be interpreted as basis (generating) elements of the complex Clifford algebra ${\mathcal C}l_2(J,R):={span}\{I, J, R, iJR\}$. An arbitrary non-trivial fundamental symmetry from ${\mathcal C}l_2(J,R)$ is determined by the formula $J_{\vecα}=α_{1}J+α_{2}R+α_{3}iJR$, where ${\vecα}\in\mathbb{S}^2$. Let $S$ be a symmetric operator that commutes with ${\mathcal C}l_2(J,R)$. The purpose of this paper is to study the sets $Σ_{J_{\vecα}}$ ($\forall{\vecα}\in\mathbb{S}^2$) of self-a…

The quantum information manifold for epsilon-bounded forms

Let H be a self-adjoint operator bounded below by 1, and let V be a small form perturbation such that RVS has finite norm, where R is the resolvent at zero to the power 1/2 +epsilon, and S is the resolvent to the power 1/2-epsilon. Here, epsilon lies between 0 and 1/2. If the Gibbs state defined by H is sufficiently regular, we show that the free energy is an analytic function of V in the sense of Frechet, and that the family of density operators defined in this way is an analytic manifold modelled on a Banach space.

Formally self-adjoint quasi-differential operators and boundary value problems

We develop the machinery of boundary triplets for one-dimensional operators generated by formally self-adjoint quasi-differential expression of arbitrary order on a finite interval. The technique are then used to describe all maximal dissipative, accumulative and self-adjoint extensions of the associated minimal operator and its generalized resolvents in terms of the boundary conditions. Some specific classes are considered in greater detail.

Computing spectral measures of self-adjoint operators

Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high-orders of convergence in terms of a smoothing parameter for computing spectral measures of general differential, integral, and lattice operators. Explicit pointwise and $L^p$-error bounds are derived in terms of the local regularity of the measure. We provide numerical examples, including a partial differential operator, a magnetic tight-binding model of graphene, and compute one thousand eigenvalues …

Self-adjoint Jacobi operators in the limit circle case

We consider symmetric Jacobi operators with recurrence coefficients such that the corresponding difference equation is in the limit circle case. Equivalently, this means that the associated moment problem is indeterminate. Our main goal is to find a representation for the resolvents of self-adjoint realizations $J$ of such Jacobi operators. This representation implies the classical Nevanlinna formula for the Cauchy-Stieltjes transforms of the spectral measures of the operators $J$. We also efficiently describe domains of the operators $J$ in terms of boundary conditions at infinity.

Higher-order spectral shift function for resolvent comparable perturbations

Given a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $(H+V-i)^{-1}-(H-i)^{-1}$ belongs to the Schatten-von Neumann ideal $\mathcal{S}^n$, $n\ge 2$, of operators on a separable Hilbert space, we establish higher order trace formulas for a broad set of functions $f$ containing several major classes of test functions and also establish existence of the respective locally integrable real-valued spectral shift functions determined uniquely up to a low degree polynomial summand. Our result generalizes the result of \cite{PSS13} for Schatten-von Neumman perturbations $V$ an…

Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra

We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of Hilbert-Schmidt type in a maximal angular region. In order to show this last property, we deduce a non-self-adjoint version of the classical Mehler's formula.

Removal of the resolvent-like energy dependence from interactions and invariant subspaces of a total Hamiltonian

The spectral problem $(A + V(z))ψ=zψ$ is considered where the main Hamiltonian $A$ is a self-adjoint operator of sufficiently arbitrary nature. The perturbation $V(z)=-B(A'-z)^{-1}B^{*}$ depends on the energy $z$ as resolvent of another self-adjoint operator $A'$. The latter is usually interpreted as Hamiltonian describing an internal structure of physical system. The operator $B$ is assumed to have a finite Hilbert-Schmidt norm. The conditions are formulated when one can replace the perturbation $V(z)$ with an energy-independent ``potential'' $W$ such that the Hamiltonian $H=A +W$ has the sam…

Asymptotic expansions for trace functionals

We obtain Taylor approximations for functionals $V\mapsto Tr(f(H_0+V))$ defined on the bounded self-adjoint operators, where $H_0$ is a self-adjoint operator with compact resolvent and $f$ is a sufficiently nice scalar function, relaxing assumptions on the operators made in [17], and derive estimates and representations for the remainders of these approximations.

算子伴随对的边值问题

Boundary value problems for adjoint pairs of operators

该研究将拟边界三元组及其Weyl函数概念推广至算子伴随对框架，借助抽象Titchmarsh-Weyl $M$-函数得到边值问题唯一可解的充分条件并给出解的Krein型预解表示，理论可应用于一大类椭圆微分算子。

Perturbations and operator trace functions

We study the spectral functional tr f(D+A) for a suitable function f, a self-adjoint operator D having compact resolvent, and a certain class of bounded self-adjoint operators A. Such functionals were introduce by Chamseddine and Connes in the context of noncommutative geometry. Motivated by the physical applications of these functionals, we derive a Taylor expansion of them in terms of Gâteaux derivatives. This involves divided differences of f evaluated on the spectrum of D, as well as the matrix coefficients of A in an eigenbasis of D. This generalizes earlier results to infinite dimensions…

$p$-Hyperbolic Zolotarev Functions in Boundary Value Problems for a $p\,$th order Differential Operator

For the self-adjoint operator of the $p$th derivative, a system of fundamental solutions is constructed. This system is analogues to the classical system of sines and cosines. The properties of such functions are studied. Classes of self-adjoint boundary conditions are described. For the operator of the third derivative, the resolvent is calculated and an orthonormal basis of eigenfunctions is given.

Asymptotic Completeness and S-Matrix for Singular Perturbations

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

Self-adjoint Dirac operators on domains in $\mathbb{R}^3$

In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Ω; \mathbb{C}^4)$, where $Ω\subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract bound…

On compressions of self-adjoint extensions of a symmetric linear relation

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with equal deficiency indices $n_\pm (A)\leq\infty$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an exit space extension of $A$; such an extension is called finite-codimensional if $\dim (\wt\gH\ominus\gH)< \infty$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of exit space extensions $\wt A=\wt A^*$. For a certain class of extensions $\wt A$ we parameterize the compressions $C (\wt A)$ by means of abstract boundary conditions. This enables us to characterize vari…

紧黎曼曲面上锥形拉普拉斯算子的Krein公式

Weyl Curves and Zeta Determinants of Conic Laplacians on Riemann Surfaces

通过临界渐近边界数据的有限维辛空间，建立紧黎曼曲面上锥形拉普拉斯算子自伴扩张的Krein公式，并导出迹恒等式用于比较不同Lagrangian边界条件下的正谱zeta行列式。

非自伴椭圆锥算子热迹的全渐近展开

Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators

本文研究了锥奇性流形上非自伴椭圆算子的热迹，在参数椭圆性条件下，通过参数依赖微积分构造拟基本解逼近预解式，获得了热迹的全渐近展开。

Differentiable perturbation of unbounded operators

If $A(t)$ is a $C^{1,\al}$-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized $C^1$ in $t$. If $A$ is $C^\infty$ then the eigenvalues can be parameterized twice differentiable.

Unbounded operators having self-adjoint or normal powers and some related results

We show that a densely defined closable operator $A$ such that the resolvent set of $A^2$ is not empty is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén-Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given, one of them being a proof that if $T$ is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\geq2$, then $T$ is normal. By a recent result by Pietrzycki-Stochel, we infer that a closed subnormal operator such that $T^n$ is normal,…

相对有界与相对迹类扰动下自伴算子的函数：相对算子 Lipschitz 函数

Functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. Relatively operator Lipschitz functions

本文引入相对算子 Lipschitz 与预解式 Lipschitz 函数类，利用双算子积分研究相对有界及相对迹类扰动下自伴算子函数，并证明迹公式成立的最大函数类及谱移位函数可积性的新方法。

Locally bounded perturbations and (odd) unbounded KK-theory

A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity is bounded. We prove that the perturbation of a regular self-adjoint operator by a locally bounded symmetric operator is again regular and self-adjoint. We use this result to show that the Kasparov class represented by an unbounded Kasparov module is stable under locally bounded perturbations. As an application, we show that we obtain a converse to the 'dou…