Self-adjoint Differential Operators Research Articles

A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators

A Simple Formula for the Generalized Spectrum of Second Order Self-Adjoint Differential Operators

On wave-like differential equations in general Hilbert space with application to Euler–Bernoulli bending vibrations of a beam

Wave-like differential equations occur in many engineering applications. Here the engineering setup is embedded into the framework of functional analysis of modern mathematical physics. After an overview, the L2–Hilbert space approach to free Euler–Bernoulli bending vibrations of a beam in one spatial dimension is investigated. We analyze in detail the corresponding positive, selfadjoint differential operators of 4-th order associated to the boundary conditions in statics. A comparison with free string wave swinging is outlined.

Direct and Inverse Problems for a Third Order Self-Adjoint Differential Operator with Periodic Boundary Conditions and Nonlocal Potential

Direct and Inverse Problems for a Third Order Self-Adjoint Differential Operator with Periodic Boundary Conditions and Nonlocal Potential

Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let H: L^{2}(mathbb{R})to L^{2}(mathbb{R}) have the form \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ H:=-\\frac{d^{2}}{dx^{2}}+Q, $$\\end{document} where Q is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, {1}_{(-infty ,rho ^{2}]}(H), has a complete asymptotic expansion in powers of ρ. This settles the 1-dimensional case of a conjecture made by the last two authors.

Special values of spectral zeta functions of graphs and Dirichlet L-functions

In this paper, we establish relations between special values of Dirichlet L-functions and those of spectral zeta functions or L-functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values are bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

BOUNDARY VALUE PROBLEM FOR THE EQUATION OF UNSTEADY THERMAL CONDUCTIVITY IN A NON-CYLINDRICAL REGION

The application of the method of decomposition by eigenfunctions of a self-adjoint differential operator to solving a non-stationary heat transfer problem with a phase transition in a non-automatic formulation under special initial conditions is presented for the example of the solidification process in a continuous medium. The one-dimensional problem is solved in spherical coordinates. Solving of the problem begins with its transformation to a problem in a domain with fixed boundaries, then a finite integral transformation with an unknown kernel is constructed to solve the transformed problem, the finding of which is associated with the formulation and solving of the corresponding spectral problem through degenerate hypergeometric functions. The eigenvalues and eigenfunctions are found, as well as the inversion formula for the introduced integral transformation, which makes it possible to write out an analytical solution to the problem. In the course of solving the problem, the parabolic law of motion of the interface of the two phases is established. Problems of this type arise in the mathematical modeling of heat transfer processes in construction, especially in permafrost areas, in oil and gas production during drilling and operation of wells, in metallurgy, etc.

On the spectrum of the differential operators of even order with periodic matrix coefficients

In this paper, we consider the band functions, Bloch functions and spectrum of the self-adjoint differential operator L with periodic matrix coefficients. Conditions are found for the coefficients under which the number of gaps in the spectrum of the operator L is finite.

Uncertainty Principles and Differential Operators on the Weighted Bergman Space

We classify self-adjoint first-order differential operators on weighted Bergman spaces on the unit disc and answer questions related to uncertainty principles for such operators. Our main tools are the discrete series representations of $$\textrm{SU}(1,1)$$ . This approach has the promise to generalize to other bounded symmetric domains.

A Remark on the Essential Self-adjointness for Klein–Gordon-Type Operators

We discuss a new simplified proof of the essential self-adjointness for formally self-adjoint differential operators of real principal type with the null non-trapping condition, previously proved by Vasy (J Spectr Theory 10: 439–461, 2020) and Nakamura-Taira (Ann Henri Lebesgue 4: 1035–1059, 2021). For simplicity, here we discuss the second-order cases, i.e., Klein–Gordon-type operators only.

Transcendence and the Expression of the Spectral Series of a Class of Higher Order Differential Operators

In this paper, a relationship between the spectral zeta series of a class of higher order self-adjoint differential operators on the unit circle and the integral of Green functions is established by Mercer’s Theorem. Furthermore, the explicit expression and the transcendental nature of the spectral series are obtained by the integral representation. Finally, several applications in physics about differential operators’ spectral theory, yellow some further works, and the transcendental nature of some zeta series are listed.

Perturbations of embedded eigenvalues for self-adjoint ODE systems

We consider a perturbation problem for embedded eigenvalues of a self-adjoint differential operator in $L^2(\mathbb R;\mathbb R^n)$. In particular, we study the set of all small perturbations in an appropriate Banach space for which the embedded eigenvalue remains embedded in the continuous spectrum. We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension. Our methods involve the use of exponential dichotomies, their roughness property and Lyapunov-Schmidt reduction.

On essential-selfadjointness of differential operators on closed manifolds

The goal of this note is to present some arguments leading to the conjecture that a formally self-adjoint differential operator on a closed manifold is essentially self-adjoint if and only if the Hamiltonian flow of its symbol is complete. This holds for differential operators of degree two on the circle, for differential operators of degree one on any closed manifold and for generic Lorentzian Laplacians on surfaces.

Sharp eigenvalue asymptotics of fourth-order differential operators

Two-term self-adjoint fourth-order differential operator with summable potential on the unit interval is considered. High energy eigenvalue asymptotics and the trace formula for this operator are obtained.

Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one

In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P\\colon L^2(\\mathbb{R})\\to L^2(\\mathbb{R})$ have the form $$ P:=-\\frac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\\mathbf{1}{(-\\infty,\\lambda^2]}(P)$ has a full asymptotic expansion in powers of $\\lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, the class of potentials includes certain potentials with \_dense pure point spectrum. The proof combines the gauge transform methods of Parnovski–Shterenberg and Sobolev with Melrose's scattering calculus.

A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables.

Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

Some new necessary and sufficient conditions for the existence of analytic resolving families of operators to the linear equation with a distributed Riemann–Liouville derivative in a Banach space are established. We study the unique solvability of a natural initial value problem with distributed fractional derivatives in the initial conditions to corresponding inhomogeneous equations. These abstract results are applied to a class of initial boundary value problems for equations with distributed derivatives in time and polynomials with respect to a self-adjoint elliptic differential operator in spatial variables.

On the Unique Solvability of Incomplete Cauchy Type Problems for a Class of Multi-Term Equations with the Riemann–Liouville Derivatives

Incomplete Cauchy-type problems are considered for linear multi-term equations solved with respect to the highest derivative in Banach spaces with fractional Riemann–Liouville derivatives and with linear closed operators at them. Some new existence and uniqueness theorems for solutions are presented explicitly and the analyticity of the solutions of the homogeneous equations are also shown. The asymmetry of the Cauchy-type problem under study is expressed in the presence of a so-called defect, which shows the number of lower-order initial conditions that should not be set when setting the problem. As applications, our abstract results are used in the study of a class of initial-boundary value problems for multi-term equations with Riemann–Liouville derivatives in time and with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

A characterization for totally real submanifolds using self-adjoint differential operator

<abstract><p>In this article, we study totally real submanifolds in Kaehler product manifold with constant scalar curvature using self-adjoint differential operator $ \Box $. Under this setup, we obtain a characterization result. Moreover, we discuss $ \delta- $invariant properties of such submanifolds and get an obstruction result as an application of the inequality derived. The results in the article are supported by non-trivial examples.</p></abstract>

Analyticity of resolvents of elliptic operators on quantum graphs with small edges

We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to ε, where ε is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator Hε with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in ε. We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that certain parts of the resolvent of Hε are analytic in ε. This allows us to represent the resolvent of Hε by a uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of ε. In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in ε-expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident. We also discuss possible generalizations of our model including both the cases of a more general geometry of the small parts of the graph and a non-analytic ε-dependence of the coefficients in the differential expression and vertex conditions.