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Reconstruction of a radially symmetric potential from two spectral sequences. (English) Zbl 1011.35129
The authors want to recover the radial potential $q$ in the eigenvalue problem $$ -\Delta u(x)+q(|x|)u(x)=\lambda u(x),\quad \text{ if } |x|<1,\qquad u(x)=0,\quad \text{ if } |x|=1, $$ starting from solutions of the form $u(r,\theta,\varphi)=r^{-1}\psi(r)Y^m_l(\theta,\varphi)$, where $(r,\theta,\varphi)$ are spherical coordinates and $Y^m_l$ are spherical harmonics. \noindent Of course, the family of eigenvalue problems for $\psi$ $$\aligned & A_l(q)u(r)\equiv \psi''(r) + \big[ \lambda - q(r) - l(l+1)r^{-2}\big]\psi(r)=0,\ r\in (0,1),\\ & \psi(1)=0,\ \psi(r)=O(r)\text{ as } r\to 0+,\endaligned\tag 1$$ where $l\in {\bbfN}$, is highly overdetermined. Consequently, the authors consider the problem of recovering $q$ in (1) when a family of spectral data $\{\lambda_{l,n}\}_{(l,n)\in \Lambda}$ is given. Moreover, they introduce the nonlinear operator $$ F_\Lambda(q) = \{ \Psi_1(1,\lambda_{l,n}(q),q)\}_{(l,n)\in \Lambda}, \tag 2$$ $\Psi_1(r,\lambda,q)$ denoting the solution to (1) satisfying the normalization condition $$\lim_{r\to 0+} r^{-l-1}\Psi_1(r,\lambda,q)=1.$$ The main purpose of the paper consists of showing that the linearization of problem (2) around $q=0$ {\it uniquely} determines a small potential $q$. As a consequence, the authors must show that the kernel of the linearized operator, given by $$ D_qF_\Lambda(0)\zeta = \Big\{ c_{l,n}\int_0^1 r^2 j_l^2(\lambda_{l,n}(0)^{1/2}r)\zeta(r) dr\Big\}_{(l,n)\in \Lambda},\qquad c_{l,n}\neq 0,$$ coincides with $\{0\}$, where $j_l$ are the spherical Bessel functions with standard normalization. Moreover, from the asymptotic relationships for the $\lambda_{l,n}(q)$’s it follows $\int_0^1 \zeta(r) dr=0$. \noindent Assuming $\Lambda=\Lambda_{l_1,l_2}=\{l_1,l_2\}\times {\bbfN}$, the assertion will be implied by the (possible) completeness in $H=\big\{\zeta \in L^2(0,1):\int_0^1 \zeta(r) dr=0\big\}$ of the set $\Phi_{l_1,l_2}=\{\varphi_{l}(\lambda_{l,n}(0)^{1/2}r)\}_{(l,n)\in \Lambda_{l_1,l_2}}$, where $\varphi_{l}(r)=rj_l(r)$. However, the authors limit themselves to showing the completeness of the previous set when $\lambda_{l,n}(0)$ is replaced with $(n+l/2)^2\pi^2$, the leading term in its asymptotic expansion. Finally, the authors show that $\Phi_{l,l+1}$ is complete in $H$ if $l=0,1,2,3$. Numerical examples are also provided.

MSC:
35R30Inverse problems for PDE
31B20Boundary value and inverse problems (higher-dimensional potential theory)
35P05General topics in linear spectral theory of PDE
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References:
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