Integral transform of Kontorovich-Lebedev type with spectral parameter on the three-component segment (Q2768799)

Using the method of delta-shaped sequences the author introduces the integral transform generated on the set \(I_{2}=\{r: r\in(0,R_1)\cup(R_1,R_2)\cup(R_2,R_3); 0<R_1<R_2<R_3<\infty\}\) by the hybrid differential Bessel operator \(B_{(\alpha)}=\sum_{j=1}^3a_{j}^2\Theta(r-R_{j-1})\Theta(R_{j}-r)B_{\alpha_{j}};\;R_0=0,\;(\alpha)=(\alpha_1,\alpha_2,\alpha_3), \alpha_{j}>0\), under the assumptions that the spectral parameter is presented in the boundary condition at \(r=R_3\) and in the conjunction conditions at \(r=R_{k}, k=1,2\). Here \(\Theta(x)=\begin{cases} 0,&x<0,\\ 1,&x\geq 0,\end{cases}\) \(B_{\alpha_{j}}=r^2{d^2\over dr^2}+(2\alpha_{j}+1)r{d\over dr}+\alpha_{j}^2-\lambda_{j}^2r^2\), \(2\alpha_{j}+1\geq 0\), \(\lambda_{j}\in (0,\infty)\). Theorems on the integral representation, on the spectrum of operator \(B_{(\alpha)}\) and on the fundamental identity are proved.