A point interaction for the discrete Schrödinger operator and generalized Chebyshev polynomials

We consider semi-infinite Jacobi matrices corresponding to a point interaction for the discrete Schrödinger operator. Our goal is to find explicit expressions for the spectral measure, the resolvent, and other spectral characteristics of such Jacobi matrices. It turns out that the spectral analysis of this toy problem leads to a new class of orthogonal polynomials generalizing the classical Chebyshev polynomials.

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