New cases of Hankel operator diagonalization

Abstract

We obtain some new cases of diagonalization of Hankel operator on the semiaxis. The kernels of these operators contain hyperbolic functions. The integral transformations diagonalizing these operators are a composition of the classical Mehler-Fock transformations, sine and cosine Fourier transforms and some unitary operator. The latter is written out explicitly.

Keywords:

• Hankel equation
• diagonalization
• Fourier transforms
• Mehler–Fock transform

Mathematics Subject Classification (2010):

• MSC 45C99

Acknowledgments

The author is grateful to the reviewers for valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Apparently, the operator with the kernel $(x+y{)}^{-1}\exp \{-x-y\}$ was diagonalized simultaneously by Shanker [3], Magnus [4] and Lebedev [5] (see also Ref. [6]).

2 Here and in what follows, ‘eigenvalues’ and ‘eigenfunctions’ are understood as eigenvalues and eigenfunctions of the continuous spectrum.

3 Yafaev [1] normalizes an eigenfunction: $v(p,\lambda )=\sqrt{{\xi }_0\tanh \pi {\xi }_0}{P}_{-1/2+i{\xi }_0}(p),{\xi }_0={\xi }_0(\lambda ).$