Introducing of the hybrid integral transformation of the second kind Kontorovich-Lebedev -- second kind Legendre-Legendre type (Q2730551)

Using the method of delta-like sequences the authors construct the integral transformation generated on the set \(I_2^{+}=\{r:\;r\in (R_0,R_1)\cup(R_1,R_2)\cup(R_2,\infty); R_0>0\}\) by hybrid differential operator NEWLINE\[CARRIAGE_RETURNNEWLINEM_{\alpha,\mu}=a_1^2\theta(r-R_0) \theta(R_1-r)B_{\alpha}+a_2^2\theta(r-R_1)\theta(R_2-r) \Lambda_{\mu_1}+a_3^2\theta(r-R_2)\Lambda_{\mu_2}, \mu=(\mu_1,\mu_2).CARRIAGE_RETURNNEWLINE\]NEWLINE The domain of definition of operator \(M_{\alpha,\mu}\) is the set of vector-valued functions \(f(r)=(f_1(r)\), \(f_2(r),f_3(r))\), \(f_{j}\in C^2(R_{j-1},R_{j}), j=1,2,3\), \(R_3=\infty\), which satisfy the conjunction conditions NEWLINE\[CARRIAGE_RETURNNEWLINE\left.\left[\left(\alpha_{j1}^{k}{d\over dr}+\beta_{j1}^{k}\right)f_{k}- \left(\alpha_{j2}^{k}{d\over dr}+\beta_{j2}^{k}\right)f_{k+1} \right ]\right|_{r=R_{k}}=0,\;j,k=1,2CARRIAGE_RETURNNEWLINE\]NEWLINE and ~the boundary conditions NEWLINE\[CARRIAGE_RETURNNEWLINE\left(\alpha_{11}^{0}{d\over dr}+\beta_{11}^{0}\right)f_{1}(r) \left|_{r=R_0}=0,\quad |\alpha_{11}^{0}|+ |\beta_{11}^{0}|\neq 0,\quad (e^{r/2}|f_3(r)|)\right|_{r=\infty}<\infty. CARRIAGE_RETURNNEWLINE\]NEWLINE Here \(a_{j}^2>0,\;c_{1k}\cdot c_{2k}>0\), \(c_{jk}=\alpha_{2j}^{k}\beta_{1j}^{k}- \alpha_{1j}^{k}\beta_{2j}^{k}\); \(B_{\alpha}=r^2{d^2\over dr^2}+ (2\alpha+1)r{d\over dr}+\alpha^2-\lambda^2r^2\), \(\lambda\in(0,\infty)\), \(2\alpha+1\geq 0\), \(\Lambda_{\mu}={d^2\over dr^2}+{\text cth} r{d\over dr}+ {1\over 4}-{\mu^2\over {\text sh}^2 r}\), \(2\mu+1>0\); \(\theta(x)=\begin{cases} 1, & x>0;\\ 0, & x<0\end{cases}\).