## We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space: (i) $f$ is $n$ times continuously Fr\'{e}chet $\mathcal{S}^p$-differentiable at every bounded self-adjoint operator if and only if $f\in C^n(\mathbb{R})$; (ii) if $f',\ldots,f^{(n-1)}\in C_b(\mathbb{R})$ and $f^{(n)}\in C_0(\mathbb{R})$, then $f$ is $n$ times continuously Fr\'{e}chet $\mathcal{S}^p$-differentiable at every self-adjoint operator; (iii) if $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, then $f$ is $n-1$ times continuously Fr\'{e}chet $\mathcal{S}^p$-differentiable and $n$ times G\^{a}teaux $\mathcal{S}^p$-differentiable at every self-adjoint operator.

Higher order differentiability of operator functions in Schatten norms

## The paper is devoted to evolution equations of the form ∂ ∂t u(t) = −(A + B(t))u(t), t ∈ I = [0, T ], on separable Hilbert spaces where A is a non-negative self-adjoint operator and B(·) is family of non-negative self-adjoint operators such that dom(A α) ⊆ dom(B(t)) for some α ∈ [0, 1) and the map A −α B(·)A −α is Holder continuous with the Holder exponent β ∈ (0, 1).

Trotter product formula and linear evolution equations on Hilbert spaces On the occasion of the 100th birthday of Tosio Kato

## This problem is described by a quadratic operator polynomial with self-adjoint operators. 这个问题由具有自伴随算子的二次算子多项式来描述。

Asymptotics of the Eigenvalues of Self-adjoint Fourth Order Boundary Value Problems

## For any multi-graph $G$ with edge weights and vertex potential, and its universal covering tree ${\mathcal{T}}$, we completely characterize the point spectrum of operators $A_{{\mathcal{T}}}$ on ${\mathcal{T}}$ arising as pull-backs of local, self-adjoint operators $A_{G}$ on $G$. 对于任何具有边权重和顶点势的多图 $G$ 及其通用覆盖树 ${\mathcal{T}}$，我们完全刻画了算子 $A_{{\mathcal{T}}}$ 的点谱在 ${\mathcal{T}}$ 上，由于 $G$ 上的本地自伴算符 $A_{G}$ 的回调而产生。

Point Spectrum of Periodic Operators on Universal Covering Trees

## We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. 我们将摩尔关于有界自伴算子的谱定理的非标准证明推广到无界算子的情况。

A nonstandard proof of the spectral theorem for unbounded self-adjoint operators

## 41 , 613–639 ( 2002 ), creates a topos of presheaves over the poset V ( N ) $\mathcal {V}(\mathcal {N})$ of Abelian von Neumann subalgebras of the von Neumann algebra N $\mathcal {N}$ of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N $\mathcal {N}$ and measures on the spectral presheaf; and (d) a model of dynamics in terms of V ( N ) $\mathcal {V}(\mathcal {N})$. 41 , 613–639 (2002)，在冯诺依曼代数 N $\mathcal { N}$ 个与物理系统相关的有界算子，并建立了几个结果，包括： (b) 自伴算子的谱定理的一个版本； (c) N $\mathcal {N}$ 的状态与谱前层测量之间的联系； (d) 根据 V ( N ) $\mathcal {V}(\mathcal {N})$ 的动力学模型。

Topos Quantum Theory with Short Posets

## The objective of this paper is to reveal an operator version of the Jensen inequality and its reverse one for s-convex functions and self-adjoint operators on a Hilbert space. 本文的目的是揭示 Jensen 不等式的一个算子版本及其在希尔伯特空间上的 s-凸函数和自伴算子的逆版本。

An Operator Version of the Jensen Inequality for s-Convex Functions

## We study sufficient conditions that guarantee the boundedness of a C 0 -semigroup of operators on a Hilbert space whose generator can be decomposed as a product of self-adjoint operators. 我们研究了保证在希尔伯特空间上的 C 0 算子半群的有界性的充分条件，其生成器可以分解为自伴算子的乘积。

Bounded C0-semigroups and applications to linear stability of heteroclinic solutions in precipitation models

## Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i. 负定自伴算子的例子包括拉普拉斯算子的分数幂，即。

Stochastic generalized porous media equations driven by Lévy noise with increasing Lipschitz nonlinearities

## We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. 我们研究了一类无限三对角矩阵的特征值扰动，该矩阵定义了具有离散谱的无界自伴随算子。

Perturbation series for Jacobi matrices and the quantum Rabi model

## This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. 这导致通过随机过程而不是自伴随算子来表示物理可观察量。

On the Stochastic Mechanics Foundation of Quantum Mechanics

## The integral representation is obtained for a family of bounded commutative self-adjoint operators which are connected by algebraic relationship. 通过代数关系连接的有界交换自伴算子族得到了积分表示。

Evenly positive definite function of Hilbert space and some algebraic relationship

## The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. 希尔伯特空间理论工具的使用、无界自伴算子的谱理论、Sturm-Liouville 理论、变分方法、算子的解析微扰理论和对称算子的可拓理论是我们研究的基础。

Nonlinear dispersive equations: classical and new frameworks

## To simplify exposition we consider here self-adjoint operators. 为了简化说明，我们在这里考虑自伴算子。

The Feshbach–Schur map and perturbation theory

## The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. 本文的目的是建立具有自伴随算子的时间分数伪抛物方程的直接和反问题的可解性结果。

Direct and inverse problems for time-fractional pseudo-parabolic equations

## Observables in quantum mechanics are represented by self-adjoint operators on Hilbert space. 量子力学中的可观测量由希尔伯特空间上的自伴算子表示。

Self-adjointness in quantum mechanics: a pedagogical path

## Non-self-adjoint operators have many applications, including quantum and heat equations. 非自伴算子有很多应用，包括量子和热方程。

Investigation of the Spectral Properties of a Non-Self-Adjoint Elliptic Differential Operator

## These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverio’s monograph, perturbed by an attractive δ-distribution supported on the spherical shell of radius r0. 这些结果将与 Albeverio 的专着中深入研究的两个显着的 3 维自伴算子的基态能量研究相关联，这些算子受到半径 r0 球壳上的有吸引力的 δ 分布的干扰。

The Schrödinger particle on the half-line with an attractive $$\delta$$-interaction: bound states and resonances

## Different self-adjoint operators can lead to new classes of M-indeterminate densities. 不同的自伴算子可以导致新的 M-不定密度类别。

Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

## It is proved that for any $\gamma \in \mathbb{C}$, $|\gamma| = 1$, the operator $\gamma I$ can be represented as the product of four unitary self-adjoint operators. 证明对于任何 $\gamma \in \mathbb{C}$, $|\gamma| = 1$，算子 $\gamma I$ 可以表示为四个酉自伴算子的乘积。

On products of some unitary operators

## We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator −d/dx on L[−a, a], a > 0, that is, the one dimensional infinite square well. 我们找到了自伴算子家族的超对称伙伴，它们是微分算子 -d/dx 在 L[-a, a], a > 0 上的自伴扩展，即一维无限平方阱。

Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

## Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. 使用分解算子，我们开发了一种算法，用于计算与自伴随算子相关的频谱测量的平滑近似值。

Computing spectral measures of self-adjoint operators

## Here, motivated by the spreading interest in non self-adjoint operators in quantum mechanics, we extend this situation to a set of four operators, c, d, r and s, satisfying dc = rs + γ 1 and cd = sr + δ 1 , and we show that they are also ladder operators. 在这里，由于对量子力学中非自伴算子的广泛兴趣，我们将这种情况扩展到一组四个算子，c、d、r 和 s，满足 dc = rs + γ 1 和 cd = sr + δ 1 ，并且我们证明他们也是梯形图运算符。

Coupled Susy, pseudo-bosons and a deformed su(1,1) Lie algebra

## We study generic fractal properties of bounded self-adjoint operators through lower and upper generalized fractal dimensions of spectral measures. 我们通过谱度量的下广义分形维数研究有界自伴算子的一般分形性质。

Some generic fractal properties of bounded self-adjoint operators

## The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. 他们证明的主要步骤是一种研究半经典状态下非自伴算子谱性质的新方法。

The density of states of 1D random band matrices via a supersymmetric transfer operator

## While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. 虽然已知分配律不适用于量子逻辑，并且等式公理在量子集合论中不成立，但他证明了量子集合论中的实数与自我一一对应-希尔伯特空间上的伴随算子，或等价于相应量子系统的物理量。

From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

## A common assumption in non-relativistic quantum mechanics is that self-adjoint operators mathematically represent properties of quantum systems. 非相对论量子力学中的一个常见假设是自伴随算子在数学上表示量子系统的属性。

How many properties of spin does a particle have?

## As well as being particularly simple, it generalises previous no-signalling conditions in that it allows for degeneracies and can be applied to all bounded self-adjoint operators. 除了特别简单之外，它还概括了以前的无信号条件，因为它允许退化并且可以应用于所有有界自伴随算子。

Impossible measurements revisited

## To prove this, we use the $$L_1$$ -functional calculus for self-adjoint operators and a suitable similarity transformation. 为了证明这一点，我们使用 $$L_1$$ - 自伴算子的泛函演算和 合适的相似变换。

Spectral Properties of the Dirac Operator on the Real Line

## The variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. 考虑了加性有界扰动下线性自伴算子谱子空间的变化。

Unifying the treatment of indefinite and semidefinite perturbations in the subspace perturbation problem

## The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. 该问题的可解性研究基于紧自伴算子的谱理论。

Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers

## These fractional-order governing equations involve self-adjoint operators and admit unique solutions, in contrast to analogous studies following the local Cauchy’s hypothesis. 这些分数阶控制方程涉及自伴随算子并承认唯一解，这与遵循局部柯西假设的类似研究相反。

Thermodynamics of fractional-order nonlocal continua and its application to the thermoelastic response of beams

## We show that for any $(n+1)$-tuple ${\bf A}$ of bounded self-adjoint operators the multiple operator integral $T_{a^{[n]}}^{\bf A}$ maps $\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ to $\mathcal{S}_{1, \infty}$ boundedly with uniform bound in ${\bf A}$. 我们证明，对于任何有界自伴算子的 $(n+1)$-元组 ${\bf A}$，多重算子积分 $T_{a^{[n]}}^{\bf A}$ 映射$\mathcal{S}_{p_1} \times \ldots \times \mathcal{S}_{p_n}$ 到 $\mathcal{S}_{1, \infty}$ 有界，在 ${\bf一个}$。

Weak (1,1) estimates for multiple operator integrals and generalized absolute value functions

## Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. 最近的几篇论文将他们的注意力集中在证明非自伴算子的 Lieb-Thirring 不等式的正确模拟上，并在复平面上寻找其特征值分布的界限。

Bounds on eigenvalues of perturbed Lamé operators with complex potentials

## The aim of this short note is to study the kinematics of this noncommutative space using the tools developed for usual quantum mechanics, namely quantize the space associating to it an algebra of operators, obtain a concrete representation of them on some Hilbert space, whose vectors are pure states, diagonalize sets of completely commuting observables and use the known measurement theory, namely that the possible results of a measurement are given by the eigenvalues of the observables with probabilities given by the spectral decomposition of self-adjoint operators. 这篇简短笔记的目的是使用为通常的量子力学开发的工具来研究这个非对易空间的运动学，即将与其相关的空间量化为算子的代数，在某个希尔伯特空间上获得它们的具体表示，其向量为纯状态，对完全可交换的可观测量集进行对角化，并使用已知的测量理论，即测量的可能结果由可观测量的特征值给出，概率由自伴算子的谱分解给出。

Time discretization from noncommutativity

## $Similar properties are valid for evolution equations of the form $$u''+ cu' + (B+A(t))u = 0$$ where$A(t) $and$B$are self-adjoint operators on a real Hilbert space$H$with$B$coercive and$A(t)$bounded in$L(H)$with a sufficiently small bound of its norm in$L^{\infty}(R+, L(H))$. nan A sharp stability criterion for single well Duffing and Duffing-like equations ## Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space. Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces. ## It is shown that these problems generate self-adjoint operators. Regular third-order boundary value problems ## Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order ## The last section is devoted to symmetric operators and self-adjoint operators. Adjoint, Symmetric, and Self-adjoint Linear Operators ## In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. Sharp spectral multipliers without semigroup framework and application to random walks ## 41 , 613–639 ( 2002 ), creates a topos of presheaves over the poset V ( N )$\mathcal {V}(\mathcal {N})$of Abelian von Neumann subalgebras of the von Neumann algebra N$\mathcal {N}$of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global section of the spectral presheaf; (b) a version of the spectral theorem for self-adjoint operators; (c) a connection between states of N$\mathcal {N}$and measures on the spectral presheaf; and (d) a model of dynamics in terms of V ( N )$\mathcal {V}(\mathcal {N})$. Topos Quantum Theory with Short Posets ## They launched this theory in order to compute the derivative of the function t↦f(A(t)), where {A(t)}t is a family of bounded self-adjoint operators depending on the parameter t. Double Operator Integrals ## It is proved that in a quite general setting, generalized weighted quasi-arithmetic means on self-adjoint operators are not monotone in their variables which is an interesting property. Maps Preserving Norms of Generalized Weighted Quasi-arithmetic Means of Invertible Positive Operators ## We establish the following results on higher order$\mathcal{S}^p$-differentiability,$1<p<\infty$, of the operator function arising from a continuous scalar function$f$and self-adjoint operators defined on a fixed separable Hilbert space: (i)$f$is$n$times continuously Fr\'{e}chet$\mathcal{S}^p$-differentiable at every bounded self-adjoint operator if and only if$f\in C^n(\mathbb{R})$; (ii) if$f',\ldots,f^{(n-1)}\in C_b(\mathbb{R})$and$f^{(n)}\in C_0(\mathbb{R})$, then$f$is$n$times continuously Fr\'{e}chet$\mathcal{S}^p$-differentiable at every self-adjoint operator; (iii) if$f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, then$f$is$n-1$times continuously Fr\'{e}chet$\mathcal{S}^p$-differentiable and$n$times G\^{a}teaux$\mathcal{S}^p$-differentiable at every self-adjoint operator. Higher order differentiability of operator functions in Schatten norms ## Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces$H^s(\partial \Omega)$involving a family of positive self-adjoint operators. Functional characterizations of trace spaces in Lipschitz domains ## We construct self-adjoint operators in the direct sum of a complex Hilbert space H and a finite dimensional complex inner product space W. Self-adjoint operators and the general GKN-EM theorem ## Let $${\mathscr {H}}$$H be a Hilbert space, J be an open interval and $$B_J({\mathscr {H}})$$BJ(H) be the set of all self-adjoint operators on $${\mathscr {H}}$$H with spectra in J. A sufficient condition for that an operator map is of the form of functional calculus ## As an application, we give an atomic decomposition for weighted Hardy spaces associated to nonnegative self-adjoint operators on$X$. A q-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application. ## Motivated by applications of the discrete random Schrodinger operator, mathematical physicists and analysts, began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators $$H_\omega = H + V_\omega$$ on a separable Hilbert space$\mathcal{H}$, where the perturbation is given by $$V_\omega = \sum_n \omega_n (\cdot, \varphi_n)\varphi_n$$ with a sequence$\{\varphi_n\}\subset\mathcal{H}$and independent identically distributed random variables$\omega_n$. Rank-one perturbations and Anderson-type Hamiltonians ## Viewing them as self-adjoint operators on the space of radially symmetric functions in L2(RN ), we show that the following properties are generic with respect to the potential: (P1) the eigenvalues below the essential spectrum are nonresonant (that is, rationally independent) and so are the square roots of the moduli of these eigenvalues; (P2) the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set. Some generic properties of Schrödinger operators with radial potentials ## It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. Diffusion equations from master equations -- A discrete geometric approach -- ## The mathematical approach developed by Holevo is analyzed, which allows us to assign the corresponding observables to non-self-adjoint operators and to establish uncertainty relations for nonstandard canonical conjugate pairs. Aftermath of the Chernobyl Catastrophe from the Point of View of the Security Concept ## We prove new spectral enclosures for the non-real spectrum of a class of$2\times2$block operator matrices with self-adjoint operators$A$and$D$on the diagonal and operators$B$and$-B^*\$ as off-diagonal entries.

Spectral enclosures for a class of block operator matrices.

## Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem.

Regularization of linear ill-posed problems involving multiplication operators

## Then we present characterizations of normal projections and of compact and self-adjoint operators on c0.

On an Operator Theory on a Banach Space of Countable Type over a Hahn Field

## This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space.

On the linearity of order-isomorphisms

## The paper is devoted to evolution equations of the form ∂ ∂t u(t) = −(A + B(t))u(t), t ∈ I = [0, T ], on separable Hilbert spaces where A is a non-negative self-adjoint operator and B(·) is family of non-negative self-adjoint operators such that dom(A α) ⊆ dom(B(t)) for some α ∈ [0, 1) and the map A −α B(·)A −α is Holder continuous with the Holder exponent β ∈ (0, 1).

Trotter product formula and linear evolution equations on Hilbert spaces On the occasion of the 100th birthday of Tosio Kato

## The remaining sections give a brief account of the very beautiful classical theory of non-self-adjoint operators, taken from a section in Sjostrand (Lectures on Resonances) which is a brief account of parts of the classical book by Gohberg and Krein (Introduction to the Theory of Linear Non-Selfadjoint Operators.

Review of Classical Non-self-adjoint Spectral Theory

## We show that any linear quantization map into the space of self-adjoint operators in a Hilbert space violates the von Neumann rule about postcomposition with real functions.

On the von Neumann rule in quantization