1,186 research outputs found
Discrete spectrum in a critical coupling case of Jacobi matrices with spectral phase transitions by uniform asymptotic analysis
For a two-parameter family of Jacobi matrices exhibiting first-order spectral
phase transitions, we prove discreteness of the spectrum in the positive real
axis when the parameters are in one of the transition boundaries. To this end
we develop a method for obtaining uniform asymptotics, with respect to the
spectral parameter, of the generalized eigenvectors. Our technique can be
applied to a wide range of Jacobi matrices.Comment: 27 pages, 2 figure
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
Inverse problems for Jacobi operators II: Mass perturbations of semi-infinite mass-spring systems
We consider an inverse spectral problem for infinite linear mass-spring
systems with different configurations obtained by changing the first mass. We
give results on the reconstruction of the system from the spectra of two
configurations. Necessary and sufficient conditions for two real sequences to
be the spectra of two modified systems are provided.Comment: 25 pages, 2 figures. Typos were corrected, two remarks were added,
material added to Sec.
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