Self-adjoint Operators Research Articles

Consistent Treatment of Quantum Systems with a Time-Dependent Hilbert Space.

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.

Linear graviton as a quantum particle

Wave function of a single linear graviton and its interpretation are proposed. The evolution equation for this function is given. A Hermitian operator with mutually commuting components canonically conjugated to the momentum operator of the linear graviton is found. Second quantization of the linear graviton quantum mechanics as well as quantization of the classical free linear graviton field are investigated.

Symmetry and symmetric transformations in mathematical imaging

The article delves into the intricate relationship between symmetry and mathematical imaging, spanning various mathematical disciplines. Symmetry, a concept deeply ingrained in mathematics, manifests in art, nature, and physics, providing a powerful tool for understanding complex structures. The paper explores three types of symmetriesreflection, rotational, and translationalexemplified through concrete mathematical expressions. Evariste Galoiss Group Theory emerges as a pivotal tool, providing a formal framework to understand and classify symmetric operations, particularly in the roots of polynomial equations. Galois theory, a cornerstone of modern algebra, connects symmetries, permutations, and solvability of equations. Group theory finds practical applications in cryptography, physics, and coding theory. Sophus Lie extends group theory to continuous spaces with Lie Group Theory, offering a powerful framework for studying continuous symmetries. Lie groups find applications in robotics and control theory, streamlining the representation of transformations. Benoit Mandelbrots fractal geometry, introduced in the late 20th century, provides a mathematical framework for understanding complex, self-similar shapes. The applications of fractal geometry range from computer graphics to financial modeling. Symmetrys practical applications extend to data visualization and cryptography. The article concludes by emphasizing symmetrys foundational role in physics, chemistry, computer graphics, and beyond. A deeper understanding of symmetry not only enriches perspectives across scientific disciplines but also fosters interdisciplinary collaborations, unveiling hidden order and structure in the natural and designed world. The exploration of symmetry promises ongoing discoveries at the intersection of mathematics and diverse fields of study.

One-dimensional Schrödinger operator with decaying white noise potential

In this paper, we consider a random one-dimensional Schrödinger operator Hω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize Hω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that Hω is actually self-adjoint with probability one. When the white noise in Hω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of Hω. We shall further prove that the negative part of the spectrum of Hω is discrete, with no accumulation points other than 0.

Self-adjoint momentum operator for a particle confined in a multi-dimensional cavity

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.

On the decomposability for sums of complex symmetric operators

On the decomposability for sums of complex symmetric operators

ДИНАМІЧНА МОДЕЛЬ РЕЗОНАНСНОГО ПЕРЕТВОРЮВАЧА ДЛЯ ВПЛИВУ ЗІ СТОРОНИ ЖИВЛЕННЯ

In the paper, a discrete dynamic model of a full-bridge resonant converter with a symmetrical operating mode has been obtained, which describes the resonant converter as a transfer link with an supply voltage input side and a load current output side. The dynamic model is based on a linear mathematical model of the resonant converter built according to the superposition principle. The structure of the resonant converter with the processes outline function is given and analyzed. The structure of the discrete dynamic model of the resonant converter of the nth order is presented. It is proved that the transfer function of the discrete dynamic model for the outline function can be determined by the transfer function of the continuous system. The resulting dependencies describing the discrete transfer functions of the resonant converter are used to obtain the discrete dynamic model of the double-circuit transformer resonant converter in a synchronous rectifier. For this, the sequence of actions is defined: definition of the transfer function of the continuous dynamic model; obtaining the system of equations in vector-matrix form describing electromagnetic processes in the converter; definition of the system of discrete equations; definition of the transfer function of the circuit. The obtained theoretical results were verified by means of simulation modeling of the considered double-circuit resonant converter. The proposed dynamic model makes it possible to analyze the influence of instability and low-frequency fluctuations of the supply voltage of the resonant converter on the output values. The presented results can be used in the development and improvement of single- or multi-circuit bridge high-frequency resonant converters as the secondary power sources and converters used in ship demagnetization systems. References 25, figures 8, tables 2.

A Few Maclaurin Symmetric Mean Aggregation Operators for Spherical Fuzzy Numbers Based on Schweizer-Sklar Operations and Their Use in Artificial Intelligence

A Few Maclaurin Symmetric Mean Aggregation Operators for Spherical Fuzzy Numbers Based on Schweizer-Sklar Operations and Their Use in Artificial Intelligence

A new multi-attribute group decision-making method based on probabilistic multi-valued linguistic spherical fuzzy sets for the site selection of charging piles

Motivated by the concepts of low carbon and environmental protection, electric vehicles have received much attention and become more and more popular all around the world. The expanding demand for electric vehicles has driven the rapid development of the charging pile industry. One of the prominent issues in charging pile industry is to determine their sites, which is a complex decision-making problem. As a matter of factor, the process of charging piles sites selection can be regarded as multi-attribute group decision-making (MAGDM), which is the main topic of this paper. The recently proposed linguistic spherical fuzzy sets (LSFSs) composed of the linguistic membership degree, linguistic abstinence degree and linguistic non-membership degree are powerful tools to express the evaluation information of decision makers (DMs). Based on the concept of LSFSs, we introduce probabilistic multi-valued linguistic spherical fuzzy sets (PMVLSFSs), which can describe DMs’ fuzzy evaluation information in a more refined and accurate way. The operation rules of PMVLSFSs are also developed in this article. To effectively aggregate PMVLSFSs, the probabilistic multi-valued linguistic spherical fuzzy power generalized Maclaurin symmetric mean operator and the probabilistic multi-valued linguistic spherical fuzzy power weighted generalized Maclaurin symmetric mean are put forward. Based on the above aggregation operators, a new method for MAGDM problem with PMVLSFSs is established. Further, a practical case of suitable site selection of charging pile is used to verify the practicability of this method. Lastly, comparative analysis with other methods is performed to illustrate the advantages and stability of proposed method.

On self-adjoint operators generated by the fourth-order Laguerre type differential expression

In this paper we address the problem of finding all self-adjoint operators generated by a Lagrangian symmetrizable differential expression in a Hilbert space with a mixed Sobolev inner product strictly discrete on the derivatives. We start giving survey on the initial results in the literature: the Glazman-Krein-Naimark (GKN) theory, the later developed GKN-EM theory as an extension of the first, and the works of Littlejohn and Wellman on the subject.We apply those to the fourth-order Laguerre type differential expression known for having a sequence of orthogonal polynomials as eigenfunctions and a symmetrizable differential expression. As a result, we give a complete description of the operator space defined from the GKN-EM theory, and also a characterization of all possible self-adjoint operators in the Sobolev space where the operator is defined. We validate these results with known examples previously studied in the literature.

Polynomial extension of some symmetric partition regular structures

The study of symmetric structures is a new trend in combinatorial number theory. Recently in [9], Di Nasso proved symmetrized versions of some classical theorems like Hindman's theorem, Van der Waerden's theorem and Deuber's theorem. This opens the question of which other classical theorems can be symmetrized, as well as if other symmetric operations allow for such generalizations. Here we give positive answers to both questions, by showing that symmetrization of the polynomial extension of Van der Waerden's and Deuber's Theorem is possible.

An extended COPRAS method based on complex q-rung orthopair fuzzy 2-tuple linguistic Maclaurin symmetric mean aggregation operators

An extended COPRAS method based on complex q-rung orthopair fuzzy 2-tuple linguistic Maclaurin symmetric mean aggregation operators

Multi-criteria Radio Frequency Identification Approach for Manufacturing Company Selection Based on Partitioned Maclaurin Symmetric Mean Operators Under Complex Intuitionistic Fuzzy Set Environment

Multi-criteria Radio Frequency Identification Approach for Manufacturing Company Selection Based on Partitioned Maclaurin Symmetric Mean Operators Under Complex Intuitionistic Fuzzy Set Environment

Analysis of a nonlinear model for the thermoelastic beam

This paper addresses the Cauchy problem associated with the nonlinear thermoelastic beam featuring thermal dissipation. We examine the problem in an open domain, which may be either bounded or unbounded. The existence of a solution is derived through the diagonalization theorem for self-adjoint operators using Hilbertian integral theory.

Narrow operators on [formula omitted]-complete lattice-normed spaces

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C-complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C-compact orthogonally additive operator T:X→Y from an atomless C-complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator TN:E(X)→E(X) to be C-compact. We also show that the sum of two orthogonally additive operators G and T, where G is narrow and T is laterally-to-norm continuous and C-compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space CE.

Differential and fuzzy differential sandwich theorems involving quantum calculus operators

The principle of subordination is useful in comparing two holomorphic functions when the range of one holomorphic function is a subset of the other and they comply at a single point. The subordination, when spoken in fuzzy set theory, becomes fuzzy subordination as the comparison between two holomorphic functions is made using the fuzzy membership function. In this article, differential and fuzzy differential Subordination, superordination, and sandwich theorems have been discussed for the classes defined by using q-derivative and symmetric q-derivative operators.

Browder operators on quaternionic Hilbert space

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces Related to Young's Result

Let $H$ be a Hilbert space. In this paper we show among others that, if the selfadjoint operators $A$ and $B$ satisfy the condition $0$ $<$ $m\leq A,$ $B\leq M,$ for some constants $m,$ $M,$ then \begin{align*} 0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes 1+1\otimes B^{2}}{2}-A\otimes B\right) \\ & \leq \left( 1-\nu \right) A\otimes 1+\nu 1\otimes B-A^{1-\nu }\otimes B^{\nu } \\ & \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes 1+1\otimes B^{2}}{2}-A\otimes B\right) \end{align*} for all $\nu \in \left[ 0,1\right] .$ We also have the inequalities for Hadamard product \begin{align*} 0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}% \circ 1-A\circ B\right) \\ & \leq \left[ \left( 1-\nu \right) A+\nu B\right] \circ 1-A^{1-\nu }\circ B^{\nu } \\ & \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}% \circ 1-A\circ B\right) \end{align*} for all $\nu \in \left[ 0,1\right] .$

On Perturbation of Operators and Rayleigh-Schrödinger Coefficients

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Stability of the gapless pure point spectrum of self-adjoint operators

We consider a self-adjoint operator T on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of T and on the bounded perturbation V ensuring the global stability of the spectral nature of T + ɛV, ε∈R.