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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

-Δu(x)+q(|x|)u(x)=λu(x),if|x|<1,u(x)=0,if|x|=1,

starting from solutions of the form u(r,θ,φ)=r -1 ψ(r)Y l m (θ,φ), where (r,θ,φ) are spherical coordinates and Y l m are spherical harmonics.

Of course, the family of eigenvalue problems for ψ

A l (q)u(r)ψ '' (r)+λ - q (r) - l (l+1) r -2 ψ(r)=0,r(0,1),ψ(1)=0,ψ(r)=O(r)asr0+,(1)

where l, is highly overdetermined. Consequently, the authors consider the problem of recovering q in (1) when a family of spectral data {λ l,n } (l,n)Λ is given. Moreover, they introduce the nonlinear operator

F Λ (q)={Ψ 1 (1,λ l,n (q),q)} (l,n)Λ ,(2)

Ψ 1 (r,λ,q) denoting the solution to (1) satisfying the normalization condition

lim r0+ r -l-1 Ψ 1 (r,λ,q)=1·

The main purpose of the paper consists of showing that the linearization of problem (2) around q=0 uniquely determines a small potential q. As a consequence, the authors must show that the kernel of the linearized operator, given by

D q F Λ (0)ζ=c l,n 0 1 r 2 j l 2 (λ l,n (0) 1/2 r) ζ (r) d r (l,n)Λ ,c l,n 0,

coincides with {0}, where j l are the spherical Bessel functions with standard normalization. Moreover, from the asymptotic relationships for the λ l,n (q)’s it follows 0 1 ζ(r)dr=0.

Assuming Λ=Λ l 1 ,l 2 ={l 1 ,l 2 }×, the assertion will be implied by the (possible) completeness in H=ζ L 2 (0,1) : 0 1 ζ (r) d r = 0 of the set Φ l 1 ,l 2 ={φ l (λ l,n (0) 1/2 r)} (l,n)Λ l 1 ,l 2 , where φ l (r)=rj l (r). However, the authors limit themselves to showing the completeness of the previous set when λ l,n (0) is replaced with (n+l/2) 2 π 2 , the leading term in its asymptotic expansion.

Finally, the authors show that Φ 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