Finite hybrid integral transforms of (Hankel 1st kind)-(Kontorovich-Lebedev 2nd kind)-(Kontorovich-Lebedev) type (Q2768765)

The author introduces the finite 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 Bessel operator \(B_{(\nu,\alpha)}=a_1^2\theta(r)\theta(R_1-r)B_{\nu_1,\alpha_1}+ a_2^2\theta(r-R_1)\theta(R_2-r)B_{\alpha^2}+a_3^2\theta(r-R_2)B_{\alpha^3}\), where \(a_{j}>0, \theta(x)=\begin{cases} 0,&x<0,\\ 1,&x\geq 0,\end{cases}\) NEWLINE\[CARRIAGE_RETURNNEWLINEB_{\nu,\alpha}={d^2\over dr^2}+{2\alpha+1\over r}{d\over dr}-{\nu^2-\alpha^2\over r^2},\;\nu\geq\alpha\geq-{1\over 2};CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINEB_{\alpha_{j}}=r^2{d^2\over dr^2}+(2\alpha_{j}+1)r{d\over dr}+\alpha_{j}^2-\lambda_{j}^2r^2,\;2\alpha_{j}+1\geq 0,\;\lambda_{j}\in (0,\infty);\;j=2,3.CARRIAGE_RETURNNEWLINE\]NEWLINE Theorems on the discrete spectrum, on the discrete function and on the fundamental identity are proved.