The Classical Moment Problem as a Self-Adjoint Finite Difference Operator

The Classical Moment Problem as a Self-Adjoint Finite Difference Operator

This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Padé approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we o...

Self-adjoint difference operators and classical solutions to the Stieltjes-Wigert moment problem

Abstract. The Stieltjes–Wigert polynomials, which correspond to an indeterminate moment problem on the positive half-line, are eigenfunctions of a second order q-difference operator. We consider the orthogonality measures for which the difference operator is symmetric in the corresponding weighted L-spaces. Under some additional assumptions these measures are exactly the solutions to the q-Pear...

The Laplace operator is essentially self-adjoint

Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at ±∞. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of ...

Inverse eigenvalue problem via finite-difference three-diagonal Schrödinger operator

In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation with three-diagonal SturmLiouville operator on a finite interval. It is demonstrated that inverse problem procedure is nothing else than well known Gram-Schmidt orthonormalization in Euclidean space for special vectors numbered by space coordinate index. All the resu...