Self-adjoint Operators Research Articles

Self-adjoint Phase Difference Operator and Its Uncertainty Relation with Number Difference Operator

Self-adjoint Phase Difference Operator and Its Uncertainty Relation with Number Difference Operator

Approximation Properties of Chlodovsky-Type Two-Dimensional Bernstein Operators Based on (p, q)-Integers

In the present study, we introduce the two-dimensional Chlodovsky-type Bernstein operators based on the (p,q)-integer. By leveraging the inherent symmetry properties of (p,q)-integers, we examine the approximation properties of our new operator with the help of a Korovkin-type theorem. Further, we present the local approximation properties and establish the rates of convergence utilizing the modulus of continuity and the Lipschitz-type maximal function. Additionally, a Voronovskaja-type theorem is provided for these operators. We also investigate the weighted approximation properties and estimate the rate of convergence in the same space. Finally, illustrative graphics generated with Maple demonstrate the convergence rate of these operators to certain functions. The optimization of approximation speeds by these symmetric operators during system control provides significant improvements in stability and performance. Consequently, the control and modeling of dynamic systems become more efficient and effective through these symmetry-oriented innovative methods. These advancements in the fields of modeling fractional differential equations and control theory offer substantial benefits to both modeling and optimization processes, expanding the range of applications within these areas.

Crossing symmetry and the crossing map

We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita–Takesaki modular theory. In particular, crossing symmetric operators, namely those operators that are mapped to their adjoints by the crossing map, define endomorphisms of standard subspaces. Conversely, such endomorphisms can be integrated to crossing symmetric operators. We also investigate the relation between crossing symmetry and natural compatibility conditions with respect to unitary representations of certain symmetry groups, and furthermore introduce a generalized crossing map defined by a real object in an abstract [Formula: see text]-tensor category, not necessarily consisting of Hilbert spaces and linear maps. This latter crossing map turns out to be closely related to the (unshaded, finite-index) subfactor theoretical Fourier transform. Lastly, we provide families of solutions of the crossing symmetry equation, solving in addition the categorical Yang–Baxter equation, associated with an arbitrary Q-system.

Higher order $\mathcal{S}^{p}$-differentiability: The unitary case

Consider the set of unitary operators on a complex separable Hilbert space \mathcal{H} , denoted as \mathcal{U}(\mathcal{H}) . Consider 1<p<\infty . We establish that a function f defined on the unit circle \mathbb{T} is n times continuously Fréchet \mathcal{S}^{p} -differentiable at every point in \mathcal{U}(\mathcal{H}) if and only if f\in C^{n}(\mathbb{T}) . Take a function U \colon\R\rightarrow\mathcal{U}(\mathcal{H}) such that the function t\in\R\mapsto U(t)-U(0) takes values in \mathcal{S}^{p} and is n times continuously \mathcal{S}^{p} -differentiable on \R . Consequently, for f\in C^{n}(\mathbb{T}) , we prove that f is n times continuously Gâteaux \mathcal{S}^{p} -differentiable at U(t) . We provide explicit expressions for both types of derivatives of f in terms of multiple operator integrals. In the domain of unitary operators, these results closely follow the n -th order successes for self-adjoint operators achieved by the second author, Le Merdy, Skripka, and Sukochev. Furthermore, as for application, we derive a formula and \mathcal{S}^{p} -estimates for operator Taylor remainders for a broader class of functions. Our results extend those of Peller, Potapov, Skripka, Sukochev, and Tomskova.

A Study on Non-Homogeneous Multiplicative Boundary Value Problems

The paper deals with the non-homogeneous boundary value problems established in the multiplicative calculus. For this problem, the multiplicative meanings of concepts such as adjoint operator, self-adjoint operator, Lagrange identity, Green's formula are obtained. Then, a criterion is given for the existence of solutions to non-homogeneous multiplicative boundary value problems.

A Subclass of Bi-univalent Functions Defined by aSymmetric q-Derivative Operator and Gegenbauer Polynomials

This paper introduces a novel subclass of bi-univalent analytic functions by utilizing a symmetric q-derivative operator in conjunction with Gegenbauer polynomials. Within this newly defined subclass, we derive bounds for the first two Maclaurin coefficients and address the Fekete-Szeg o problem. By varying the parameters in our results between 0 and 1, we obtain a range of new insights and rediscover some previously established results. This approach not only broadens the scope of bi-univalent function theory but also deepens the understanding of coefficient bounds and extremal problems within this context.

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

Remark on square roots of self-adjoint weighted composition operators on H2

In this paper, we study square roots of self-adjoint weighted composition operators on H2. More precisely, we focus on square roots Wf,φ of a self-adjoint operator Wg,ψ=Wf,φ2 when φ is a linear fractional selfmap of D. We also investigate several properties of such Wf,φ. Finally, we show that the square roots Wf,φ may be other, nonself-adjoint weighted composition operators and have nontrivial invariant subspaces.

Submanifolds with constant scalar curvature in space forms

Let [Formula: see text] be an [Formula: see text]-dimensional oriented compact submanifold with constant normalized scalar curvature [Formula: see text] in the space form [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the mean curvature and the Ricci curvature of [Formula: see text] respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if [Formula: see text] is a hypersurface in a unit sphere, and [Formula: see text], then [Formula: see text] is totally umbilical. Furthermore, we investigate the submanifolds in [Formula: see text] with flat normal bundle satisfying [Formula: see text], and obtain a complete classification theorem.

Research on Asymmetrical Operation of Multilevel Converter-Type Solid-State Transformers Based on High-Frequency Link Interconnection

The large size of the sub-module (SM) capacitor is a typical problem in traditional modular multilevel converter-type solid-state transformers (MMC-SSTs). The MMC-SST based on high-frequency link interconnection is an effective solution for achieving lightweight capacitance. This structure can help to eliminate the symmetric SM fluctuating power, thereby reducing the SM capacitance. In a three-phase interconnected MMC-SST with low capacitance, potential risks may arise during transient processes, especially in cases of three-phase voltage asymmetry, such as large fluctuations in the SM voltage and unstable DC bus voltage. Aiming to solve this problem, this article re-analyzes the internal power characteristics of the MMC-SST under asymmetric operation and re-derives the SM capacitance constraint suitable for different degrees of three-phase voltage asymmetry. The new SM capacitance constraint enhances the asymmetric voltage ride-through capability of the MMC-SST. The new capacitance constraint is higher than that in symmetric operation, but it still has significant advantages in capacitance compared with the traditional MMC-SST.

First‐order asymptotic perturbation theory for extensions of symmetric operators

AbstractThis work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.

New Results on Differential Subordination and Superordination for Multivalent Functions Involving New Symmetric Operator

This article aims to significantly advance geometric function theory by providing a valuable contribution to analytic and multivalent functions. It focuses on differential subordination and superordination, which characterize the interactions between analytic functions. To achieve our goal, we employ a method that relies on the characteristics of differential subordination and superordination. As one of the latest advancements in this field, this technique is able to derive several results about differential subordination and superordination for multivalent functions defined by the new operator within the open unit disk . Additionally, by employing the technique, the differential sandwich outcome is achieved. Therefore, this work presents crucial exceptional instances that follow the results. The findings of this paper can be applied to a wide range of mathematical and engineering problems, including system identification, orthogonal polynomials, fluid dynamics, signal processing, antenna technology, and approximation theory. Furthermore, this work significantly advances the knowledge and understanding of the analytical functions of the unit and its interactive higher relations. The characteristics and consequences of differential subordination theory are symmetric to those of differential superordination theory. By combining them, sandwich-type theorems can be derived.

Determination of reduced density matrices in the doubly occupied configuration interaction space: A Hellmann-Feynman theorem approach.

In this work, the Hellmann-Feynman theorem is extended within the doubly occupied configuration interaction space to enable practical calculations of reduced density matrices and expected values. This approach is straightforward, employing finite energy differences, yet remains reliable and accurate even with approximate energies from successive approximation methods. The method's validity is rigorously tested against the Richardson-Gaudin-Kitaev and reduced Bardeen-Cooper-Schrieffer models using approximate excitation energies procured from the Hermitian operator method within the same space, effectively proving the approach's reliability with median error rates for reduced density matrix calculations around 0.1%. These results highlight the procedure's potential as a practical tool for computing reduced density matrices and expected values, particularly valuable as an ad hoc method in scenarios where only system energies are easily available.

Interval-valued picture fuzzy decision-making framework with partitioned maclaurin symmetric mean aggregation operators

Interval-valued picture fuzzy (IVPF) set is an extension of the picture fuzzy set theory used to represent uncertainty and vagueness in the processes of decision-making. This study focuses on exploring the interrelationships among multiple IVPFSs and criteria partitions. We investigate the IVPF partitioned Maclaurin symmetric mean operator and the weighted IVPF partitioned Maclaurin symmetric mean operator and discuss their respective properties. Subsequently, we identify certain special cases of these operators based on IVPF sets. Furthermore, we deploy a multi-criteria decision-making procedure utilizing the suggested IVPF partition operators. Through a numerical example, we demonstrate the practicality and validity of the presented approach. Finally, a thorough comparison with existing approaches is conducted to elucidate the superiority of the proposed method.

Classical echoes of quantum boundary conditions

Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.

Some new results on the property (R) under compact perturbations

The property (R) for a bounded linear operator T on a Hilbert space means that the points λ in the approximate point spectrum of T for which T − λI is upper semi-Browder are exactly the isolated points of the spectrum of T which are eigenvalues of finite multiplicity. For an operator T, if T + K satisfies the property (R) for all compact operators K, then T is said to have the stability of the property (R). In this paper, we give new criteria for the fulfillment of the stability of the property (R) for an operator. Moreover, the property (R) for complex symmetric operators is discussed.

Secure and Efficient Authentication using Linkage for permissionless Bitcoin network

The cryptocurrency’s permissionless and large-scale broadcasting requirements prohibit the traditional authentication implementation on the blockchain’s underlying peer-to-peer (P2P) networking. Thus, blockchain networking implementations remain vulnerable to networking integrity threats such as spoofing or hijacking. We design Secure and Efficient Authentication using Linkage (SEAL) to build connection security for permissionless Bitcoin networking. SEAL uses the linkage between the packets for a symmetric operation, in contrast to the traditional authentication approach relying on identity-credential-based trust. To make it appropriate for cryptocurrency networking, SEAL utilizes the packet header, protects the end-to-end connection, and separates the online process and the offline process so that the real-time overhead is minimal for greater efficiency and practicality. We implement SEAL on a functioning Bitcoin node and demonstrate that SEAL operates efficiently with minimal overhead. Specifically, it reduces the hash rate by only 1.3% compared to an unsecured node. Additionally, we use a network simulator to emulate the Bitcoin Mainnet and analyze SEAL’s impact on block propagation delay. SEAL yields 2.04 times smaller delay and 1.25 times smaller delay in block propagation than HMAC and ChaCha20-Poly1305, respectively. The key advantage of SEAL is that it requires fewer hash computations and simpler mixing operations, resulting in significantly lower computational overhead compared to traditional authentication schemes based on message authentication codes (MACs).

Effect of parity–time symmetry operation on entanglement distribution in two parallel Heisenberg spin chains

Abstract The effect of the parity–time ( PT ) symmetry operation on the entanglement transfer in two parallel Heisenberg spin chains is investigated. We give two schemes, i.e. pre- and post- PT symmetric operations, to demonstrate the performance of the PT symmetric operation on the entanglement dynamics of two qubits. In our investigation, we find that the effect of pre- PT symmetric operation schemes on the protection of entanglement of two qubits in the case of double spin chains is superior to that of the single spin chain case. Conversely, the effect of the post- PT symmetric operation scheme on the protection of entanglement of two qubits in the single spin chain case is better than that in the double spin chain case.

New subclass of meromorphic harmonic functions defined by symmetric q-calculus and domain of Janowski functions

In this article, by applying the convolution principle and symmetric q-calculus, we develop a new generalized symmetric q-difference operator of convolution type, which is applicable in the domain E⁎={τ:τ∈C and 0<|τ|<∞}. Utilizing this operator, we construct, analyze, and evaluate two new sets of meromorphically harmonic functions in the Janowski domain. Furthermore, we investigate the convolution properties and necessary conditions for a function F to belong to the class MSHP,R(q,q−1), examining the sufficiency conditions for F to satisfy these properties. Moreover, we examine key geometric properties of the function F in the class MSH‾P,R(q,q−1), including the distortion bound, convex combinations, the extreme point theorem, and weighted mean estimates.

A stable difference scheme for the solution of a source identification problem for telegraph-parabolic equations

In the present paper, we construct a first order of accuracy difference scheme for the approximate solution of the inverse problem for telegraph-parabolic equations with an unknown spacewise dependent source term. The unique solvability of constructed difference scheme and the stability estimates for its solution were obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.