Search Results (111)

This work constructs an orthogonal function system on bounded intervals $[0,R]$ associated with the Atangana–Baleanu–Caputo (ABC) fractional derivative for $\alpha \in (1/2,1)$. Starting from a fractional Laguerre-type equation involving the ABC operator, solutions are obtained via a generalized Frobenius method, yielding series representations with characteristic exponent $\alpha -1$. Rather than postulating a weight function by analogy with classical or Caputo settings, the weight is derived directly from the requirement that the underlying fractional operator be formally self-adjoint on a suitable admissible domain. This operator-theoretic approach leads to the explicit Mittag–Leffler weight $w_{\alpha }\left(x\right)=x^{-(2\alpha -1)}{E}_{\alpha }(-x^{\alpha })$, which intrinsically reflects the nonlocal memory structure of the ABC kernel. A similarity transformation removes the universal singular factor and produces regularized eigenfunctions that are continuous on $[0,R]$ and orthogonal in the weighted $L^2$ space. The weight identity and formal self-adjointness are rigorously verified through a right-Volterra uniqueness argument. Numerical experiments confirm orthogonality up to machine precision, demonstrate spectral convergence for a model ABC differential equation, and illustrate consistency with classical Laguerre polynomials in the limit $\alpha \to 1^{-}$. The resulting framework provides a self-consistent orthogonal system suitable for spectral approximations of problems governed by the ABC operator on bounded domains. Full article

In this paper, we consider a partial differential equation with mixed derivatives of first order in time and second order in the spatial variable. Such equations are usually referred to as one-dimensional pseudoparabolic equations. We prove the existence and uniqueness of a classical solution to problems for a pseudoparabolic equation with a second-order differential operator involving pure involution, under certain requirements imposed on the initial data. The possibility of applying the Fourier method is based on the Riesz basis property of the eigenfunctions of the considered non-self-adjoint second-order differential operator with pure involution. Bessel-type inequalities are established for new systems of functions. The presence of a Bessel inequality for Fourier coefficients facilitates the proof of the uniform convergence of differentiated Fourier series. The solutions are obtained explicitly in the form of a Fourier series. Such representations can be used for performing numerical computations. Full article

In classical spectral theory, self-adjoint differential operators satisfy the relation $\sigma \left(L\right)=\sigma \left(L^{\ast }\right)$. However, this is not necessarily true for non-self-adjoint operators. In this study, we show that a similar spectral equality holds for non-self-adjoint with periodic PT-symmetric complex valued potential. These operators are often described in the literature as having self-adjoint-like spectral characteristics; however, here, we show that the reality of the spectrum depends on the isospectral relationship between the operator and its adjoint. Moreover, we prove that the adjoint operator $L^{\ast }$ inherits PT-symmetry and analyze its spectral properties under quasi-periodic boundary conditions, extending prior studies from the original operator to its adjoint. Full article

The first aim of this study is to point out new aspects of approximation theory applied to a few classes of holomorphic functions via Vitali’s theorem. The approximation is made with the aid of the complex moments of the functions involved, which are defined similarly to the moments of a real-valued continuous function. By applying uniform approximation of continuous functions on compact intervals via Korovkin’s theorem, the hard part concerning uniform approximation on compact subsets of the complex plane follows according to Vitali’s theorem. The theorem on the set of zeros of a holomorphic function is also applied. In the end, the existence and uniqueness of the solution for a multidimensional moment problem are characterized in terms of limits of sums of quadratic expressions. This is the application appearing at the end of the title. Consequences resulting from the first part of the paper are pointed out with the aid of functional calculus for self-adjoint operators. Full article

This paper introduces a rigorously defined fractional Heun operator constructed through a symmetric composition of left and right Riemann–Liouville fractional derivatives. By deriving a compatible fractional Pearson-type equation, a new weight function and Hilbert space setting are established, ensuring the operator’s self-adjointness under natural fractional boundary conditions. Within this framework, we prove the existence of a real, discrete spectrum and demonstrate that the corresponding eigenfunctions form a complete orthogonal system in $L_{{\omega }_{\alpha }}^2(a,b)$. The central theoretical result shows that the fractional eigenpairs $({\lambda }_n^{\left(\alpha \right)},u_n^{\left(\alpha \right)})$ converge continuously to their classical Heun counterparts $({\lambda }_n^{\left(1\right)},u_n^{\left(1\right)})$ as $\alpha \to 1^{-}$. This provides a rigorous analytic bridge between fractional and classical spectral theories. A numerical study based on the fractional Legendre case confirms the predicted self-adjointness and spectral convergence, illustrating the smooth deformation of the classical eigenfunctions into their fractional counterparts. The results establish the fractional Heun operator as a mathematically consistent generalization capable of generating new families of orthogonal fractional functions. Full article

The Hamiltonian of Reggeon field theory is defined by ${\displaystyle H_{\mu ,\lambda }=\mu A^{*}A + }$${\displaystyle i\lambda A^{*}(A+A^{*})A}$, where A and $A^{*}$ are the annihilation and creation operators satisfying $[A,A^{*}]=I$ and $\mu$, $\lambda$ are real parameters, and $i^2=-1$. This operator acts on Bargmann space $\mathbb{B}$ where $\mathbb{B}$ is a Hilbert space of holomorphic square integrable functions with respect to the Gaussian-weighted Lebesgue measure. In this work, we consider the operator ${\displaystyle H_{\lambda }=i\lambda A^{*}(A+A^{*})A}$ with maximum domain ${\displaystyle D\left(H_{\lambda }\right)=\{\phi \in \mathbb{B};H_{\lambda }\phi \in \mathbb{B}\}}$. If we limit the domain to polynomials and take the closure of the obtained operator, we denote it by ${\displaystyle H_{\lambda }^{min},}$ of which $H_{\lambda }$ is obviously an extension. Contrary to what happens for $\mu \ne 0$, it is well known that these two operators are different. The main purpose of the present work is to show that ${\displaystyle H_{\lambda }}$ admits a right-inverse ${\displaystyle K_{\lambda }}$, i.e., ${\displaystyle H_{\lambda }{K}_{\lambda }=I}$ on negative imaginary axis and that ${\displaystyle K_{\lambda }}$ is compact. Full article

We construct a family of self-adjoint operators on the prime numbers whose entries depend on pairwise arithmetic divergences, replacing geometric distance with number-theoretic dissimilarity. The resulting spectra encode how coherence propagates through the prime sequence and define an emergent arithmetic geometry. From these spectra we extract observables such as the heat trace, entropy, and eigenvalue growth, which reveal persistent spectral compression): eigenvalues grow sublinearly, entropy scales slowly, and the inferred dimension remains strictly below one. This rigidity appears across logarithmic, entropic, and fractal-type kernels, reflecting intrinsic arithmetic constraints. Analytically, we show that for the unnormalized Laplacian, the continuum limit of its squared Hamiltonian corresponds to the one-dimensional bi-Laplacian, whose heat trace follows a short-time scaling proportional to $t^{-1/4}$. Under the spectral dimension convention $d_s=-2d\log \Theta /d\log t$, this result produces $d_s=1/2$ directly from first principles, without fitting or external hypotheses. This value signifies maximal spectral compression and the absence of classical diffusion, indicating that arithmetic sparsity enforces a coherence-limited, non-Euclidean geometry linking spectral and number-theoretic structure. Full article

On a compact n-dimensional Riemannian manifold without boundary $(M,g)$, it is well-known that the $L^2$-normalized Laplace eigenfunctions with semiclassical parameter h satisfy the universal $L^{\infty }$ growth bound of $O(h^{{\displaystyle \frac{1-n}{2}}})\mathrm{as}h\to 0^{+}.$ In the context of a quantum completely integrable system on M, which consists of n commuting self-adjoint pseudodifferential operators $P_1\left(h\right),\dots ,P_n\left(h\right)$, where $P_1\left(h\right)=-h^2{\mathrm{\Delta }}_g+V\left(x\right)$, Galkowski-Toth showed polynomial improvements over the standard $O(h^{{\displaystyle \frac{1-n}{2}}})$ bounds for typical points. Specifically, in the two-dimensional case, such an improved upper bound is $O(h^{-1/4})$. In this study, we aim to further enhance this bound to $O\left(\right|lnh{|}^{1/2})$ at the points where a strictly monotonic condition is satisfied. Full article

Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects. Full article

In the present paper we study a class of Toeplitz operators called concentration operators that are self-adjoint and compact in the linear canonical Dunkl setting. We show that a finite vector space spanned by the first eigenfunctions of such operators is of a maximal phase-space concentration and has the best phase-space concentrated scalogram inside the region of interest. Then, using these eigenfunctions, we can effectively approximate functions that are essentially localized in specific regions, and corresponding error estimates are given. These research results cover in particular the classical and the Hankel settings, and have potential application values in fields such as signal processing and quantum physics, providing a new theoretical basis for relevant research. Full article

A composite-coated inclusion is embedded in a matrix, where the conductivity (permittivity) of the phases is assumed to be complex-valued. The purpose of this paper is to demonstrate that a non-zero flux can arise under specific conditions related to the conductivities of the components in the absence of external sources. These conditions are unattainable with conventional positive conductivities but can be satisfied when the conductivities are negative or complex—a scenario achievable in the context of metamaterials. The problem is formulated as a spectral boundary value problem for the Laplace equation, featuring a linear conjugation condition defined on a smooth curve L. This curve divides the plane ${\mathbb{R}}^2$ into two regions, $D^{+}$ and $D^{-}\ni \infty$. The spectral parameter appears in the boundary condition, drawing parallels with the Steklov eigenvalue problem. The case of a circular annulus is analyzed using the method of functional equations. The complete set of eigenvalues is derived by applying the classical theory of self-adjoint operators in Hilbert space. Full article

The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work. Full article

Assume that A, $A_1$, and $A_2$ are three selfadjoint linear relations (multi-valued linear operators) in a certain complex Hilbert space. In this study, conditions are presented for the multi-valued operator equality $A=A_1{A}_2$ to hold when the inclusion $A\subset A_1{A}_2$ is assumed to be satisfied. The present study is strongly motivated by the invalidity of a classical result from A. Devinatz, A. E. Nussbaum, and J. von Neumann in the general case of selfadjoint linear relations. Two types of conditions for the aforementioned equality to hold are presented. Firstly, a condition is given in terms of the resolvent sets of the involved objects, which does not depend on the product structure of the right-hand side, $A_1{A}_2$. Secondly, a condition is also presented where the structure of the right-hand side is taken into account. This one is based on the notion of the $\mathfrak{L}$-stability of a linear operator under linear subspaces. It should be mentioned that the classical Devinatz–Nussbaum–von Neumann theorem is obtained as a particular case of one of the main results. Full article

This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space $\left({\ell }_{\mathtt{q}(·)}{log}^{\beta }\left({\mathtt{L}}^{\mathtt{p}(·)}\right)\right)$, which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when $\beta =0$ and to classical Lebesgue spaces when $\mathtt{q}=\infty ,\beta =0$. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis. Full article

We recently proved that in our model of quantum gravity, the solutions to the quantized version of the full Einstein equations or to the Wheeler–DeWitt equation could be expressed as products of spatial and temporal eigenfunctions, or eigendistributions, of self-adjoint operators acting in corresponding separable Hilbert spaces. Moreover, near the big bang singularity, we derived sharp asymptotic estimates for the temporal eigenfunctions. In this paper, we show that, by using these estimates, there exists a complete sequence of unitarily equivalent eigenfunctions which can be extended past the singularity by even or odd mirroring as sufficiently smooth functions such that the extended functions are solutions of the appropriately extended equations valid in $\mathbb{R}$ in the classical sense. We also use this phenomenon to explain the missing antimatter. Full article