Self-adjoint Operators Research Articles

Tensorial and Hadamard product inequalities of Schwarz type for selfadjoint operators in Hilbert spaces

Tensorial and Hadamard product inequalities of Schwarz type for selfadjoint operators in Hilbert spaces

Direct and inverse problems for the Hilfer fractional differential equation

The article studies direct and inverse problems for subdiffusion equations involving a Hilfer fractional derivative. An arbitrary positive self-adjoint operator $A$ is taken as the elliptic part of the equation. In particular, as the operator $A$ we can take the Laplace operator with the Dirichlet condition. First, the existence and uniqueness of a solution to the direct problem is proven. Then, using the representation of the solution to the direct problem, the existence and uniqueness of the inverse problem of finding the right-hand side of the equation, which depends only on the spatial variable, is proved.

Optimization of Market Risk via Maclaurin Symmetric Mean Aggregation Operators: An Application of Interval-Valued Intuitionistic Fuzzy Sets in Multi-Attribute Group Decision-Making

Optimization of Market Risk via Maclaurin Symmetric Mean Aggregation Operators: An Application of Interval-Valued Intuitionistic Fuzzy Sets in Multi-Attribute Group Decision-Making

Certain New Applications of Symmetric q-Calculus for New Subclasses of Multivalent Functions Associated with the Cardioid Domain

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies.

Quantum particle localization observables on Cauchy surfaces of Minkowski spacetime and their causal properties

We introduce and study a general notion of spatial localization on spacelike smooth Cauchy surfaces of quantum systems in Minkowski spacetime. The notion is constructed in terms of a coherent family of normalized POVMs, one for each said Cauchy surface. We prove that a family of POVMs of this type automatically satisfies a causality condition which generalizes Castrigiano’s one and implies it when restricting to flat spacelike Cauchy surfaces. As a consequence, no conflict with Hegerfeldt’s theorem arises. We furthermore prove that such families of POVMs do exist for massive Klein–Gordon particles, since some of them are extensions of already known spatial localization observables. These are constructed out of positive definite kernels or are defined in terms of the stress–energy tensor operator. Some further features of these structures are investigated, in particular the relation with the triple of Newton–Wigner selfadjoint operators and a modified form of Heisenberg inequality in the rest 3-spaces of Minkowski reference frames.

L p positivity preservation and self-adjointness of Schrödinger operators on incomplete Riemannian manifolds

The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$ functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $p\neq 2$ , i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$ .

A family of single‐phase boost AC‐AC converters based on impedance network cells with symmetric bipolar operation

SummaryIn this paper, a family of single‐phase boost AC‐AC converters based on impedance network cells with symmetric bipolar operation is proposed. The proposed converters exhibit a range of voltage gains as a result of non‐inverting and inverting boost operations, which are influenced by the input voltage and various impedance source cells. The converters leverage inherent commutation to address the commutation problem without requiring snubber circuits and safe commutation strategy. Additionally, high‐speed switching is achieved by preventing power MOSFETs' body diode conduction and related poor reverse recovery issues. The design incorporates a simple and flexible switching strategy to produce step‐change frequency at the output, using only four switches to achieve both step‐changed frequency operation and output voltage regulation. Furthermore, by reducing the total stored energy of passive components, the proposed converters effectively decrease volume, weight, and cost. The operational modes of the converters are elucidated, and their key relationships are summarized in tabular format. Finally, experimental validation through a laboratory prototype confirms the performance accuracy of the proposed converters at frequencies 25, 50, and 100 Hz.

Local Spectral Multiplicity of Selfadjoint Couplings with General Interface Conditions

We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schrödinger operator on a star-graph with a finite number of finite or infinite edges and an interface condition at the common vertex. A wide class of “selfadjoint” interface conditions, subject to an assumption which is generically satisfied, is considered. We determine the spectral multiplicity function on the singular spectrum (continuous as well as point) in terms of the spectral data of decoupled operators.

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.

Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators

We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to [Formula: see text] potentials.

Q-Rung orthopair fuzzy 2-tuple linguistic WASPAS algorithm for patients’ prioritization based on prioritized Maclaurin symmetric mean aggregation operators

Due to the fuzziness of the medical field, q-rung orthopair fuzzy 2-tuple linguistic (q-RF2L) set is the privileged way to aid medical professionals in conveying their assessments in the patient prioritization problem. The theme of the present study is to put forward a novel approach centered around the merging of prioritized averaging (PA) and the Maclaurin symmetric mean (MSM) operator within q-RF2L context. According to the prioritization of the professionals and the correlation among the defined criteria, we apply both PA and MSM to assess priority degrees and relationships, respectively. Keeping the pluses of the PA and MSM operators in mind, we introduce two aggregation operators (AOs), namely q-RF2L prioritized Maclaurin symmetric mean and q-RF2L prioritized dual Maclaurin symmetric mean operators. Meanwhile, some essential features and remarks of the proposed AOs are discussed at length. Based on the formulated AOs, we extend the weighted aggregated sum product assessment methodology to cope with q-RF2L decision-making problems. Ultimately, to illustrate the practicality and effectiveness of the stated methodology, a real-world example of patients’ prioritization problem is addressed, and an in-depth analysis with prevailing methods is performed.

On the closedness of the range of (fractional) powers of certain classes of possibly unbounded operators

In this article, we establish certain identities about the closedness of the range of several classes of possibly unbounded operators defined on a Hilbert space, such as self-adjoint, positive, normal, hyponormal, and quasinormal operators as well as their powers, and fractional powers when the latter applies. In particular, we show that an unbounded self-adjoint positive operator has closed range if and only if one (and all) of its fractional powers has closed range. A substantial part of the paper is devoted to additivity properties of operator ranges, again in the context of non-necessarily bounded operators. The paper is accompanied by many counterexamples that demonstrate the boundaries of possible assertions.

Singular solutions for space-time fractional equations in a bounded domain

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann–Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.

Flux and symmetry effects on quantum tunneling

AbstractMotivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for the tunneling approximation recently established in Fefferman et al. (SIAM J Math Anal 54: 1105–1130, 2022), Helffer & Kachmar (Pure Appl Anal, 2024), thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

Strong and Weak Convergence Theorems for the Split Feasibility Problem of (β,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature.

A distribution network traveling wave localization method based on CEEMD-DEO3S

A traveling wave head calibration way according to Complementary Ensemble Empirical Mode Decomposition (CEEMD) and Three-point Symmetric Differential Energy Operator (DEO3S) is proposed to address the issues of noise interference and difficulty in precise positioning in local area network wave positioning. The way filters out low-frequency bandwidth in the fault data by performing CEEMD decomposition on the signal and then uses the DEO3S algorithm to define the time of arrival of the fault wave head. The emulation results indicate the proposed way can precisely extract wave heads even in the presence of a large amount of noise interference, and has good robustness and accuracy. Compared with traditional traveling wave positioning methods, it has obvious advantages and further improves the accuracy and reliability of traveling wave positioning.

Stability of Krein-von Neumann self-adjoint operator extension under unbounded perturbations

We have considered a fourth order difference operator defined on the Hilbert space of square summable sequences on N. We investigated the stability of existence of Krein-von Neumann self-adjoint extension of difference operators under bounded and unbounded coefficients. Using asymptotic summation based on discretised Levinson's theorem and appropriate smoothness and decay conditions, we have shown that unlike the case of deficiency indices and discrete spectrum, the existence of positive self-adjoint operator extensions is stable under unbounded perturbations. These results now exhaustively characterise the spectral and structural properties of difference operators with unbounded coefficients.

Some properties of a complex symmetric operator

A square matrix with complex entries which is equal to its transpose is called a complex symmetric matrix. Such matrices are found in many areas of pure and applied mathematics. A conjugation on H is defined as a conjugate linear isometric involution C on a Hilbert Space H. A bounded linear operator T : H → H has a complex symmetric matrix with regard to an orthonormal basis if, for every conjugation C on H, T = CT *C. Such an operator is called a Complex symmetric operator. Several characteristics of Complex Symmetric matrices and associated operators are discussed in this article.

Amplifying the capacity of quantum super dense coding by non-Hermitian operations under phase decoherence source

We study the dynamic behavior of quantum super dense coding between two uncoupled qubits, which are immersed in a common Ohmic environment. We examine the validity of local non-Hermitian operations in the amplification of quantum super dense coding. In particular, using analytical and numerical investigations, we show explicitly that the capacity of quantum super dense coding can be greatly amplified with the assistance of a local non-Hermitian operation, i.e. a parity-time symmetric operation. This sheds new light on the protection of quantum super density.

On the existence of an eigenvalue of the generalized Friedrichs model

We consider a family of bounded self-adjoint matrix operators (generalized Friedrichs models) acting on the direct sum of one-particle and two-particle subspaces of the Fock space. Conditions for the existence of eigenvalues of the matrix operators are obtained.