17 research outputs found
Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions
We perform the spectral analysis of a family of Jacobi operators
depending on a complex parameter . If the spectrum of
is discrete and formulas for eigenvalues and eigenvectors are
established in terms of elliptic integrals and Jacobian elliptic functions. If
, , the essential spectrum of covers
the entire complex plane. In addition, a formula for the Weyl -function as
well as the asymptotic expansions of solutions of the difference equation
corresponding to are obtained. Finally, the completeness of
eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied
previously by Carlitz, are proved.Comment: published version, 2 figures added; 21 pages, 3 figure
Asymptotic spectral properties of the Hilbert -matrix
We study asymptotic spectral properties of the generalized Hilbert -matrix
for large
order . First, for general , we deduce the asymptotic
distribution of eigenvalues of outside the origin. Second, for
, asymptotic formulas for small eigenvalues of are derived.
Third, in the classical case , we also prove asymptotic formulas for
large eigenvalues of . I particular, we obtain an
asymptotic expansion of improving Wilf's formula for the best
constant in truncated Hardy's inequality.Comment: 18 pages, dedicated to the memory of Harold Widom, accepted for
publication in SIAM J. Matrix Anal. App