Unbounded Coefficients Research Articles

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

Sublinear Elliptic Equations with Unbounded Coefficients in Lipschitz Domains

This paper is concerned with the homogeneous Dirichlet problem for a sublinear elliptic equation with unbounded coefficients in a Lipschitz domain. Bilateral a priori estimates for positive solutions and a priori upper estimates for their gradients are presented as a byproduct of the boundary Harnack principle. These estimates allow us to show the uniqueness of a positive solution of the homogeneous Dirichlet problem under no information about normal derivatives unlike in smooth domains.

A Note on Some Properties of Unbounded Bilinear Forms Associated with Skew-Symmetric $L^q(\Omega)$-Matrices

We study the bilinear forms on the space of measurable $p$-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We give an example showing that if a skew-symmetric matrix contains a locally unbounded $L^q$-elements, then the corresponding quadratic forms can be alternating. These questions are closely related to the existence issues of the Nuemann boundary value problem for $p$-Laplace elliptic equations with non-symmetric and locally unbounded anisotropic diffusion matrices.

Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients

Abstract In this paper, we initiate the study of asymptotic and oscillatory properties of solutions to second-order functional differential equations with noncanonical operators and unbounded neutral coefficients, using a recent method of iteratively improved monotonicity properties of nonoscillatory solutions. Our results rely on ideas that essentially improve standard techniques for the investigation of differential equations with unbounded neutral terms with delay or advanced argument. The core of the method is presented in a form that suggests further generalizations for higher-order differential equations with unbounded neutral coefficients.

Uniqueness of second-order elliptic operators with unbounded and degenerate coefficients in L 1-spaces

Let . Let and . Consider the formal second-order differential operator in . We show that the closure of is quasi-m-accretive under certain conditions on the coefficients.

A weighted $$L_q(L_p)$$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

A weighted $$L_q(L_p)$$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

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.

Oscillation criteria for third-order semi-canonical differential equations with unbounded neutral coefficients

In this paper, we investigate the oscillatory behavior of solutions to a class of third-order differential equations of the form Lz(t) + f (t)yβ (σ(t)) = 0, where Lz(t) = (p(t)(q(t)zt(t))t)t is a semi-canonical operator and z(t) = y(t) + g(t)y(τ (t)). The main idea is to convert the semi-canonical operator into canonical form and then obtain some new sufficient conditions for the oscillation of all solutions. The obtained results essentially improve and complement to the known results. Examples are provided to illustrate the main results. Mathematics Subject Classification (2010): 34C10, 34K11, 34K40. Received 19 September 2021; Accepted 20 January 2022

Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number μ

We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number μ, approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a τ preconditioning when the variable coefficient wave number μ is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.

Dirichlet problems with anisotropic principal part involving unbounded coefficients

This article establishes the existence of solutions in a weak sense for a quasilinear Dirichlet problem exhibiting anisotropic differential operator with unbounded coefficients in the principal part and full dependence on the gradient in the lower order terms. A major part of this work focuses on the existence of a uniform bound for the solution set in the anisotropic setting. The unbounded coefficients are handled through an appropriate truncation and a priori estimates. For more information see https://ejde.math.txstate.edu/Volumes/2024/11/abstr.html

The maximal regularity of the third‐order differential equation and its applications

In this work, we study the solvability and maximal regularity questions for the singular third‐order differential equation with unbounded coefficients and some applications. Unlike previously studied cases, the leading and intermediate coefficients of this equation can grow independently. We obtain sufficient coefficient conditions for correctness of this equation and compactness for inverse of the corresponding differential operator. We also prove the maximal regularity estimate for a generalized solution. Using these results, we obtain upper and lower estimates for the number of Kolmogorov ‐diameters of the set associated with the linear Korteweg–de Vries equation's solutions. We give an example and show that from the above inequalities follow two‐sided estimates of the Kolmogorov ‐diameters.

Fourth-order operators with unbounded coefficients

Fourth-order operators with unbounded coefficients

Absolute stability of time-delayed Lurie direct control systems with unbounded coefficients

Absolute stability of time-delayed Lurie direct control systems with unbounded coefficients

OPERATOR-THEORETICAL FORMULATLONS OF BOUNDARY VALUE PROBLEM FOR STURM–LIOUVILLE OPERATOR EQUATlON

Abstract. Our aim in that paper is to set the operator theoretical formulation of boundary value problem for Sturm-Liovuille equation with unbounded operator coefficient and boundary condition involving Herglotz-Nevanlinna function of spectral parameter in boundary condition. We define minimal operator and give description of maximal operator.

Oscillatory behavior of a fifth differential equation with unbounded neutral coefficients

"The authors study the oscillatory behavior of solutions to a class of fifth-order differential equations with unbounded neutral coefficients. The results are obtained by a comparison with first order delay differential equations whose oscillatory characters are known. Two examples illustrating the results are provided, one of which is applied to Euler type equations. Keywords: Oscillation, fifth-order, neutral differential equation."

The accuracy estimates of the Cayley transform method for the abstract Cauchy problem

We obtain the accuracy estimates of the Cayley transform method for solving the initial value problem for a homogeneous first-order differential equation with an unbounded operator coefficient in a Hilbert space. In the case of finite (in some sense) smoothness of the initial vector, our method has a power-law rate of convergence and, moreover, the rate automatically depends on this regularity (i.e. the Cayley transform method is a method without saturation of accuracy). If the initial vector is infinitely smooth, then our method is exponentially convergent. In addition, we substantiate that the estimates are unimprovable in the order of N (the discretization parameter N characterizes the number of summands in the partial sum of the approximate solution).

Generation of semigroups associated to strongly coupled elliptic operators in [formula omitted

A class of vector-valued elliptic operators with unbounded coefficients, coupled up to the second-order is investigated in the Lebesgue space Lp(Rd;Rm) with p∈(1,∞), providing sufficient conditions for the generation of an analytic C0-semigroup T(t). Under further assumptions, a characterization of the domain of the infinitesimal generator is given.

Spectral properties of a class of operator functions with applications to the Moore-Gibson-Thompson equation with memory

In this study, we present spectral enclosures and accumulation of eigenvalues of a class of operator functions with several unbounded operator coefficients. Our findings have direct relevance to the third-order Moore-Gibson-Thompson equation with memory and additional damping. The new results include sufficient conditions for the accumulation of branches of eigenvalues to the essential spectrum and new spectral enclosures for operator functions with several unbounded operator coefficients. To illustrate the analytical results, we apply the abstract findings to concrete equations of the Moore-Gibson-Thompson type. Additionally, we employ numerical computations to further elucidate the analytical results.

Estimates for the Approximation and Eigenvalues of the Resolvent of a Class of Singular Operators of Parabolic Type

In this paper, we study a differential operator of parabolic type with a variable and unbounded coefficient, defined on an infinite strip. Sufficient conditions for the existence and compactness of the resolvent are established, and an estimate for the maximum regularity of solutions of the equation Lu=f∈L2(Ω) is obtained. Two-sided estimates for the distribution function of approximation numbers are obtained. As is known, estimates of approximation numbers show the rate of best approximation of the resolvent of an operator by finite-dimensional operators. The paper proves the assertion about the existence of positive eigenvalues among the eigenvalues of the given operator and finds two-sided estimates for them.

L 2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients

The aim of this article is to obtain convergence in mean in the uniform topology of piecewise linear approximations of stochastic differential equations (SDEs) with C 1 drift and C 2 diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although L q convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider L 2 convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.