On the spectrum of some hybrid differential operators of second order (Q2768794)

The author considers the hybrid differential operator NEWLINE\[NEWLINEM_{(\mu)}=a_1^2\theta(r-R_0)\theta(R_1-r) d^2/dr^2+ a_2^2\theta(r-R_1)\theta(R_2-r)\Lambda_{\mu_2}+a_3^2\theta(r-R_2)\Lambda_{\mu_3}NEWLINE\]NEWLINE on the set \(I_{2}^{+}=\{r: r\in(R_0,R_1)\cup(R_1,R_2)\cup(R_2,\infty)\); \(R_0>0\}\). NEWLINENEWLINENEWLINEHere, \(a_{j}>0\), \(\theta(x)=\begin{cases} 0,&x<0,\\ 1,&x\geq 0,\end{cases}\) \(\Lambda_{\mu_{m}}=d^2/dr^2+\text{cth} r d/dr+1/4-\mu^2_{m}/\text{sh}^2 r\), \(\mu_{m}\geq 0\), \(m=2,3\). The author proves that the operator \(M_{(\mu)}\) is selfadjoint and has continuous spectrum.