TY - JOUR
T1 - Spectral analysis of non–self–adjoint Jacobi operator associated with Jacobian elliptic functions
AU - Siegl, Petr
AU - Štampach, František
N1 - Funding Information:
Acknowledgements. The research of P. S. is supported by the Swiss National Science Foundation, SNF Ambizione grant No. PZ00P2 154786. F. Sˇ . gratefully acknowledges the kind hospitality of the Mathematisches Institut at Universität Bern and in particular of Professor Christiane Tretter; his research was also supported by grant No. GA13-11058S of the Czech Science Foundation. We also thank a referee for the very stimulating report, in particular containing plots of pseudospectra of J(α) that we re-computed in Mathematica and included as Appendix B.
Publisher Copyright:
© 2017, Element D.O.O. All rights reserved.
PY - 2017/12
Y1 - 2017/12
N2 - We perform the spectral analysis of a family of Jacobi operators J(α) depending on a complex parameter α. If |α| ≠ 1 the spectrum of J(α) is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If |α| = 1, α ≠ ±1, the essential spectrum of J(α) covers the entire complex plane. In addition, a formula for the Weyl m-function as well as the asymptotic expansions of solutions of the difference equation corresponding to J(α) are obtained. Finally, the completeness of eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied previously by Carlitz, are proved.
AB - We perform the spectral analysis of a family of Jacobi operators J(α) depending on a complex parameter α. If |α| ≠ 1 the spectrum of J(α) is discrete and formulas for eigenvalues and eigenvectors are established in terms of elliptic integrals and Jacobian elliptic functions. If |α| = 1, α ≠ ±1, the essential spectrum of J(α) covers the entire complex plane. In addition, a formula for the Weyl m-function as well as the asymptotic expansions of solutions of the difference equation corresponding to J(α) are obtained. Finally, the completeness of eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied previously by Carlitz, are proved.
KW - Jacobian elliptic functions
KW - Non-self-adjoint Jacobi operator
KW - Weyl m-function
UR - http://www.scopus.com/inward/record.url?scp=85035060975&partnerID=8YFLogxK
U2 - 10.7153/oam-2017-11-64
DO - 10.7153/oam-2017-11-64
M3 - Article
AN - SCOPUS:85035060975
SN - 1846-3886
VL - 11
SP - 901
EP - 928
JO - Operators and Matrices
JF - Operators and Matrices
IS - 4
ER -