A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F16%3A00234545" target="_blank" >RIV/68407700:21340/16:00234545 - isvavai.cz</a>
Alternative codes found
RIV/68407700:21240/16:00234545
Result on the web
<a href="http://dx.doi.org/10.1080/03081087.2015.1064348" target="_blank" >http://dx.doi.org/10.1080/03081087.2015.1064348</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1080/03081087.2015.1064348" target="_blank" >10.1080/03081087.2015.1064348</a>

Result language
angličtina
Original language name
A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix
Original language description
A three-parameter family B = B(a, b, c) of weighted Hankel matrices is introduced where a, b, c are positive parameters fulfilling a < b + c, b < a + c, c <= a + b. The famous Hilbert matrix is included as a particular case. The direct sum B(a, b, c) + B(a + 1, b + 1, c) is shown to commute with a discrete analogue of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices, T (a, b, c), commuting with B(a, b, c). The orthogonal polynomials associated with T (a, b, c) turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping U diagonalizing T (a, b, c) can be constructed explicitly. At the same time, U diagonalizes B(a, b, c) and the spectrum of this matrix operator is shown to be purely absolutely continuous and ﬁlling the interval [0, M(a, b, c)] where M(a, b, c) is known explicitly. If the assumption c <= a + b is relaxed while the remaining inequalities on a, b, c are all supposed to be valid, the spectrum contains also a ﬁnite discrete part lying above the threshold M(a, b, c). Again, all eigenvalues and eigenvectors are described explicitly.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GA13-11058S" target="_blank" >GA13-11058S: Spectral analysis of operators and its applications in quantum mechanics</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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