Self-adjoint Operators Research Articles

Criteria for the Uniqueness of a Solution to a Differential-Operator Equation with Non-Degenerate Conditions

In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and B. The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers, the criterion for the uniqueness of the boundary value problem formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required.

Characterization of self-adjoint domains for regular odd order C-symmetric differential operators

We enlarge the class of regular odd order differential operators and find the self-adjoint boundary condition of every operator. In this paper, we give the characterization of self-adjoint domains for regular odd order C-symmetric differential operators with two-point boundary conditions. The previously known characterization of self-adjoint operators for regular odd order symmetric differential operators is the special case of this one.

Maximal function characterizations of Hardy spaces associated with both non-negative self-adjoint operators satisfying Gaussian estimates and ball quasi-Banach function spaces

Maximal function characterizations of Hardy spaces associated with both non-negative self-adjoint operators satisfying Gaussian estimates and ball quasi-Banach function spaces

Simpson Type Tensorial Norm Inequalities for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities of the form $$\bigg|\bigg|\frac{1}{3}\left(2\operatorname{exp}\left(A\right)\otimes 1-\operatorname{exp}\left(\frac{A\otimes 1 +1\otimes B}{2}\right)+2\cdot 1\otimes \operatorname{exp}\left(B\right)\right)$$ $$-(1\otimes B-A\otimes 1)^{-1}\times\bigg(\operatorname{exp}(1\otimes B)-\operatorname{exp}(A\otimes 1)\bigg)\bigg|\bigg|$$ $$\leqslant \norm{1\otimes B-A\otimes 1}\frac{5}{12}\operatorname{exp}(M)$$ are obtained with variations due to the convexity properties of the mapping $f$.

The Dirac equation across the horizons of the 5D Myers–Perry geometry: separation of variables, radial asymptotic behaviour and Hamiltonian formalism

We analytically extend the 5D Myers–Perry metric through the event and Cauchy horizons by defining Eddington–Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster–Röken and Stone’s formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green’s functions of the radial ODE.

Functions of perturbed noncommuting unbounded selfadjoint operators

Let f f be a function on R 2 \mathbb {R}^2 in the inhomogeneous Besov space {\text \textit {\Russian {B}}}_{\infty ,1}^{1}(\mathbb {R}^2). For a pair ( A , B ) (A,B) of not necessarily bounded and not necessarily commuting self-adjoint operators, the function f ( A , B ) f(A,B) of A A and B B is introduced as a densely defined linear operator. It is shown that if 1 ≤ p ≤ 2 1\le p\le 2 , ( A 1 , B 1 ) (A_1,B_1) and ( A 2 , B 2 ) (A_2,B_2) are pairs of not necessarily bounded and not necessarily commuting selfadjoint operators such that both A 1 − A 2 A_1-A_2 and B 1 − B 2 B_1-B_2 belong to the Schatten–von Neumann class S p {\boldsymbol {S}}_p and f\in {\text \textit {\Russian {B}}} _{\infty ,1}^{1}(\mathbb {R}^2), then the following Lipschitz type estimate holds: \begin{equation*} \|f(A_1,B_1)-f(A_2,B_2)\|_{{\boldsymbol {S}}_p} \le \operatorname {const}\|f\|_{\text \textit {\Russian {B}}_{\infty ,1}^{1}}\max \big \{\|A_1-A_2\|_{{\boldsymbol {S}}_p},\|B_1-B_2\|_{{\boldsymbol {S}}_p}\big \}. \end{equation*}

A new application to cryptology using generalized Jacobsthal matrices, inner product and self-adjoint operator

In this paper, we first examine the properties of the matrices of the generalized [Formula: see text]-step Jacobsthal sequence, and we develop a public key cryptosystem using key exchange based on the relationship between inner product and orthogonality. Then we create encryption and decryption algorithms using this key exchange, self-adjoint operator, and generalized [Formula: see text]-step Jacobsthal sequence, and we provide a detailed example of how encryption and decryption algorithms work. Finally, a security strength and complexity analysis of these algorithms will be made.

Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces

In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type.

Hamacher Maclaurin symmetric mean aggregation operators and WASPAS method for multiple criteria group decision making under [formula omitted] - spherical fuzzy environment

T-spherical fuzzy sets can accurately represent fuzzy information and effectively simulate real-world decision making scenarios by adjusting the parameter t. The Maclaurin symmetric mean can combine multiple arguments by considering the relationships between them in any decision making process. The main goal of this paper is to develop Maclaurin symmetric mean aggregation operators based on the Hamacher operations of T-spherical fuzzy sets. The developed operators are thoroughly examined through their fundamental properties. The defined operators are adopted to develop a decision making methodology called WASPAS (Weighted Aggregated Sum Product ASsessment) for solving multiple criteria group decision making problems in a T-spherical fuzzy environment. A real-life example of project assessment is illustrated to demonstrate the practicality of the proposed decision making approach. Sensitivity analysis of the parameters is carried out to check their effect on the decision results. A comparison analysis with existing methods confirms the accessibility of the developed approach.

Relativistic Schrödinger equation and probability currents for free particles

In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.

Stable difference schemes for hyperbolic–parabolic equations with unknown parameter

In the present paper, we study the first and second order of accuracy difference schemes for the approximate solution of the inverse problem for hyperbolic–parabolic equations with unknown time-independent source term. The unique solvability of constructed difference schemes and the stability estimates for their solutions are obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

Sturm-Liouville problems and global bounds by small control sets and applications to quantum graphs

We develop a Logvinenko–Sereda theory for one-dimensional vector-valued self-adjoint operators. We thus deliver upper bounds on L2-norms of eigenfunctions – and linear combinations thereof – in terms of their L2- and W1,2-norms on small control sets that are merely measurable and suitably distributed along each interval. An essential step consists in proving a Bernstein-type estimate for Laplacians with rather general vertex conditions. Our results carry over to a large class of Schrödinger operators with magnetic potentials; corresponding results are unknown in higher dimension. We illustrate our findings by discussing the implications in the theory of quantum graphs.

The basis invariant flavor puzzle

The flavor puzzle of the Standard Model quark sector is formulated in a non-perturbative way, using basis invariants that are independent of the choice of quark field basis. To achieve this, we first derive the algebraic ring of 10 CP even (primary) and 1 CP odd (secondary) basis invariants, using the Hilbert series and plethystic logarithm. An orthogonal basis in the ring of basis invariants is explicitly constructed, using hermitian projection operators derived via birdtrack diagrams. The thereby constructed invariants have well defined CP transformation behavior and give the most direct access to the flavor symmetric alignments of basis covariants. We firstly “measure” the orthogonal basis invariants from experimental data and characterize their location in the available parameter space. The experimentally observed orthogonal basis invariants take very close to maximal values and are highly correlated. Explaining the location of the invariants at close to maximal points, including the associated miniscule and highly correlated deviations, corresponds to solving the flavor puzzle in the invariant language. Once properly normalized, the orthogonal basis invariants are close to scale (RGE) invariant, hence, provide exquisite targets for fits of both, low- and high-scale (bottom-up and top-down) flavor models. Our result provides an entirely new angle on the flavor puzzle, and opens up ample opportunities for its ultimate exploration.

A Class of Operators Related to Skew m-Complex Symmetric Operators

A Class of Operators Related to Skew m-Complex Symmetric Operators

An inverse nodal problem for a fourth-order self-adjoint binomial operator

In this work we deal with inverse nodal problems of reconstructing a fourth-order self-adjoint binomial operator d4dx4+q with Dirichlet boundary conditions. We prove that a dense subset of nodal points uniquely determines the potential function q.

Eigenvalue properties of Sturm-Liouville problems with transmission conditions dependent on the eigenparameter

<abstract><p>This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Liouville problem are transformed into an eigenvalue problem of a certain self-adjoint operator. Next, the eigenfunctions of the problem and some properties of the eigenvalues are given via construction of the basic solution. The Green's function for the Sturm-Liouville problem is also given. Finally, the continuity of the eigenvalues and eigenfunctions of the problem is discussed. Especially, the differential expressions of the eigenvalues for some parameters have been obtained, including the parameters in the eigenparameter-dependent transmission conditions.</p></abstract>

Operator upper bounds for Davis-Choi-Jensen's difference in Hilbert spaces

In this paper we obtain several operator inequalities providing upper bounds for the Davis-Choi-Jensen's Difference Ph (f (A)) - f (Ph (A)) for any convex function f : I → R, any selfadjoint operator A in H with the spectrum Sp (A) ⊂ I and any linear, positive and normalized map Ph : B (H) → B (K), where H and K are Hilbert spaces. Some examples of convex and operator convex functions are also provided.

Sobolev capacities and trace inequalities related to $ \alpha- $Hermite operators with application to $ p $-Laplace type equations

Sobolev capacities and trace inequalities related to $ \alpha- $Hermite operators with application to $ p $-Laplace type equations

Evaluation of Propulsion Technologies for Sustainable Road Freight Distribution Using a Dual Probabilistic Linguistic Group Decision-Making Approach

For the first time, this study addresses the problem of evaluating propulsion technologies for sustainable road freight distribution from the multicriteria group decision-making perspective. It aims to help logistics companies involved in global supply chains significantly reduce GHG emissions from freight distribution activities. To evaluate sustainable propulsion technologies, this article introduces a novel group decision-making approach with DPL term sets, which takes into account the simultaneous occurrence of stochastic and nonstochastic uncertainty. Since the existing methods do not consider the dependency among multiple numbers of criteria, the evaluation results become insignificant. To overcome this, we employ the Archimedean operations on DPL elements to develop DPL weighted Archimedean Maclaurin symmetric mean operator. For measuring the weights of criteria, an optimization model using a cross-entropy measure of DPL elements is developed. A case study for one of the major logistics companies in Serbia is presented to exhibit the relevance of the suggested approach, as well as offer decision-making guidelines in real-world settings. The research findings show that hybrid electric trucks are the most advantageous propulsion technology solution to reduce GHG emissions from distribution activities. The superiority of the proposed approach is demonstrated through the comparative investigation.