On the Riemann–Liouville Operators with Variable Limits

Siberian Mathematical Journal -     

Numerical Solution of Riemann–Hilbert Problems: Random Matrix Theory and Orthogonal Polynomials

Recently, a general approach to solving Riemann–Hilbert problems numerically has been developed. We review this numerical framework and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with the approach of Bornemann to compute Fredholm determinants, we are able to calculate spectral densities and gap statistics for a broad class of finite-dimensional unitary invariant ensembles. We show that the accuracy of the numerical algorithm for approximating orthogonal polynomials is uniform as the degree grows, extending the existing theory to handle g-functions. As another example, we compute the Hastings–McLeod solution of the homogeneous Painlevé II equation.     

Existence and Uniqueness of Solutions of Some Cauchy Problems for the Emden–Fowler Equation

Differential Equations - We consider the Cauchy problem for the Emden–Fowler equation $$y^{prime {}prime }-x^ay^{sigma }=0 $$ with parameters $$ain mathbb {R} $$ and $$sigma <0...     

Some Remarks on One-dimensional Functions and Their Riemann–Liouville Fractional Calculus

A one-dimensional continuous function of unbounded variation on [0,1] has been constructed.The length of its graph is infnite,while part of this function displays fractal features.The Box dimension of its Riemann–Liouville fractional integral has been calculated.     

Approximation of Weyl–Heisenberg Frame Operators

We present approximations of a large class of Weyl–Heisenberg frame operators on \(L^{2}({\mathbb {R}})\) by Gabor Bessel sequence operators generated by compactly supported smooth functions.     

Weyl’s Theorem for Functions of Operators and Approximation

Let H{\mathcal{H}} be a complex separable infinite dimensional Hilbert space. In this paper, we characterize those operators T on H{\mathcal{H}} satisfying that Weyl’s theorem holds for f(T) for each function f analytic on some neighborhood of σ(T). Also, it is proved that, given an operator T on H{\mathcal{H}} and ε > 0, there exists a compact operator K with ||K|| < e{\|K\| < \varepsilon} such that Weyl’s theorem holds for T + K.     

Numerical Solution of Riemann–Hilbert Problems: Painlevé II

We describe a new, spectrally accurate method for solving matrix-valued Riemann–Hilbert problems numerically. The effectiveness of this approach is demonstrated by computing solutions to the homogeneous Painlevé II equation. This can be used to relate initial conditions with asymptotic behavior.     

Left–Right Fredholm and Weyl Spectra of the Sum of Two Bounded Operators and Applications

In this paper, we consider the sum of two bounded linear operators defined on a Banach space and we present some new and quite general conditions to investigate their essential spectra.     

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.     

Local and Nonlocal Boundary Value Problems for Degenerating and Nondegenerating Pseudoparabolic Equations with a Riemann–Liouville Fractional Derivative

We study local and nonlocal boundary value problems for degenerating and nondegenerating third-order pseudoparabolic equations of the general form with variable coefficients and with a Riemann–Liouville fractional derivative. For their solutions, we obtain a priori estimates that imply the uniqueness of the solution and its stability with respect to the right-hand side and the initial data.     

Inverse Scattering Problems for Sturm–Liouville Operators with Spectral Parameter Dependent on Boundary Conditions

In this paper, we consider the inverse scattering problem for the Sturm–Liouville operator on the half-line [0,∞) with Herglotz function of spectral parameter in the boundary condition. The scattering data of the problem is defined, and its properties are investigated. The main equation is obtained for the solution of the inverse problem and it is shown that the potential is uniquely recovered in terms of the scattering data.     

Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives

In this paper, we derive the existence and uniqueness of mild solutions for inhomogeneous fractional evolution equations in Banach spaces by means of the method of fractional resolvent. Furthermore, we give the necessary and sufficient conditions for the existence of strong solutions. An example of the fractional diffusion equation is also presented to illustrate our theory.     

Riemann–Hilbert Problems,Toeplitz Operators and {\mathfrak{Q}}-Classes

We generalize the notion of \({\mathfrak{Q}}\) -classes \({C_{{Q_1} {Q_2}}}\) , which was introduced in the context of Wiener–Hopf factorization, by considering very general 2 × 2 matrix functions Q ₁, Q ₂. This allows us to use a mainly algebraic approach to obtain several equivalent representations for each class, to study the intersections of \({\mathfrak{Q}}\) -classes and to explore their close connection with certain non-linear scalar equations. The results are applied to various factorization problems and to the study of Toeplitz operators with symbol in a \({\mathfrak{Q}}\) -class. We conclude with a group theoretic interpretation of some of the main results.     

Multiple Solutions for Systems of Sturm–Liouville Boundary Value Problems

In this paper, the authors discuss the existence of multiple solutions to a class of second-order Sturm–Liouville boundary value systems. Their proofs are based on variational methods and critical point theory.     

On the Approximation of Riemann–Liouville Integral by Fractional Nabla h-Sum and Applications

First of all, in this paper, we prove the convergence of the nabla h-sum to the Riemann–Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Second, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case.     

Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals

Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν  >   ? 1, in which the inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$ . We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann–Liouville differintegrals of functions comprising the Bessel functions J _ν used in the fractional calculus, where orders of differintegration are determined by ν ? κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.