Generalized integral transform of Legendre 1-st kind type with spectral parameter on the three-component polar axis (Q2768791)

Using the delta-shaped sequences method the author introduces an integral transform generated on the set \(I_{2}^{+}=\{r: r\in(0,R_1)\cup(R_1,R_2)\cup(R_2,\infty)\}\) by the hybrid differential operator \(M_{(\mu)}=a_1^2\theta(R_1-r)\Lambda_{(\mu)_1}+ a_2^2\theta(r-R_1)\theta(R_2-r)\Lambda_{(\mu)_2}+a_3^2\theta(r-R_2)\Lambda_{(\mu)_3}\). Here \(a_{j}>0\), \(\theta(x)=\begin{cases} 0,&x<0,\\ 1,&x\geq 0,\end{cases}\) \(\Lambda_{(\mu)_{j}}=d^2/dr^2+\text{cth} r d/dr+1/4+(1/2)(\mu^2_{1j}/(1-\text{ch} r)+\mu^2_{j2}/(1+\text{ch} r))\), \((\mu)=\{(\mu)_1,(\mu)_2,(\mu)_3\}\), \((\mu)_{j}=(\mu_{1j},\mu_{2j})\), \(\mu_{j1}\geq \mu_{j2}\geq 0\). A theorem on the integral representation is proved.