Three-spectra inverse problem for the perturbed Bessel operators (Q6584931)

The authors study the inverse problem for recovering the potential function \(q(.)\) and the angular-momentum quantum number \(l\) of the perturbed Bessel equation\N\[\N-y''(x)+\frac{l(l+1)}{x^{2}}y(x)+q(x)y(x)=\lambda y(x), \ \ \ x\in (0,1],\N\]\Nby using three spectra \(\sigma(L)=\left\{\lambda_{n}|n\in\mathbb{N}\right\}\), \(\sigma(L^{-})=\left\{\lambda_{n}^{-}|n\in\mathbb{N}\right\}\)\Nand \(\sigma(L^{+})=\left\{\lambda_{n}^{+}|n\in\mathbb{N}\right\}\), where \(L,\) \(L^{-}\) and \(L^{+}\) are generated by\N\(-D^{2}+\frac{l(l+1)}{x^{2}}+q(x)\) in \(L^{2}(0,1)\), \(L^{2}(0,a)\) and \(L^{2}(a,1)\) with the following boundary conditions\N\[\N\lim_{x\rightarrow 0}\frac{y(x)}{x^{l+1}}<\infty, \ \ \ y'(1)+Hy(1)=0,\N\]\N\[\N\lim_{x\rightarrow 0}\frac{y(x)}{x^{l+1}}<\infty, \ \ \ y'(a)+hy(a)=0,\N\]\Nand\N\[\Ny'(a)+hy(a)=0, \ \ \ y'(1)+Hy(1)=0,\N\]\Nrespectively. Here the fixed \(a\in (0,1)\), \(\lambda\) is the spectral parameter, and \(l\in\left[-\frac{1}{2},+\infty\right)\) is the so-called angular-momentum quantum number. The boundary condition parameters \(H\) and \(h\) belong to \(\mathbb{R}\cup\left\{\infty\right\}.\)\NAssume that the potential function \(q(x)\) is real-valued and \(q\in L_{l}^{1}(0,1),\) where\N\[\NL_{l}^{1}(0,1)=\begin{cases}\NL^{1}(0,1), &l>-\frac{1}{2}, \\\N\{f\in L^{1}(0,1)\mid \int_{0}^{1} |(1-\ln x)f(x)| dx<\infty\}, &l=-\frac{1}{2}.\N\end{cases}\N\]\NTo obtain the uniqueness of inverse problem, if eigenvalues of the operators defined on the sub-intervals \((0,a]\) and \([a,1]\) have some common eigenvalues, then the norming constants corresponding to these common eigenvalues are added as the prior data.