Fundamental solution of a singular Bessel differential operator with a negative parameter

L. N. Lyakhov ; Yu. N. Bulatov ; S. A. Roshchupkin ; E. L. Sanina

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2023), pp. 52-65

Résumé

The singular differential Bessel operator $B_{-\gamma}$ with negative parameter $-\gamma0$ is considered. Solutions of the singular differential Bessel equation $B_{-\gamma} u+\lambda^2u=0$ are represented by linearly independent functions $\mathbb{J}_\mu$ and $\mathbb{J}_{-\mu},~{\mu}=\dfrac{\gamma+1}{2}$. Studied some properties of the functions $\mathbb{J}_\mu$, which are expressed in terms of the properties of the Bessel–Levitan j-function. Direct and inverse Bessel $\mathbb J_\mu$-transforms are introduced. Based on the $\mathbb T$-pseudo-shift operator introduced earlier, a a generalized $\mathbb T$-shift operator belonging to the Levitan class of generalized shifts, commuting with the Bessel operator $B_{-\gamma}$. A fundamental solution is found for the singular differential operator $B_{-\gamma}$ with a singularity at an arbitrary point on the semiaxis $[0,\infty).$

Détail

Keywords: spherical symmetry, singular Bessel differential operator, Bessel transforms, generalized Levitan shift, fundamental solution.