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The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix

100%

Štampach F. , Šťovíček P.

Special Matrices

2014

tom 2

nr 1

A family T(ν), ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in ℓ2(ℤ+) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for |ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton q-Bessel function Jν(z; q) serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the q-Bessel function due to Koelink and Swarttouw.

Self-adjoint extensions of differential operators and exterior topological derivatives in shape optimization

75%

Nazarov S. A. , Sokołowski J.

Control and Cybernetics

2005

tom Vol. 34, no 3

903-925

Self-adjoint extensions are constructed for a family of boundary value problems in domains with a thin ligament and an asymptotic analysis of a Lq-continuous functional is performed. The results can be used in numerical methods of shape and topology optimization of integral functionals for elliptic equations. At some stage of optimization process the singular perturbation of geometrical domain by an addition of thin ligament can be replaced by its approximation denned for the appropriate self-adjoint extension of the elliptic operator. In this way the topology variation of current geometrical domain can be determined and used e.g., in the level-set type methods of shape optimization.