### An inverse eigenvalue problem with rotational symmetry

T I Seidman
1988 Inverse Problems
We consider convergence of an approximation method for the recovery of a rotationally symmetric potential v from the sequence of eigenvalues. In order to permit the consideration of 'rough' potentials li, (having essentially H-'(O, 1) regularity), we first indicate the appropriate interpretation of -A + v (with boundary conditions) as a self-adjoint, densely defined operator on X:= L2(S2) and then show a suitable continuous dependence on for the relevant eigenvalues. The approach to the inverse
more » ... oach to the inverse problem is by the method of 'generalised interpolation' and, assuming uniqueness, it is shown that one has convergence to the correct potential li, (strongly, for an appropriate norm) for a sequence of computationally implementable approximations ( P c , N ) . T I Seidman We do not consider here the deep question of uniqueness: within which sets Y, is q uniquely determined by the given eigenvalue information? Rather, this is taken as an a priori assumption on the suitability of I # . + for EVP. On the other hand, as in [l], we are very much concerned with another aspect of the suitability of Y*: for how 'rough' a potential q can we construct a workable interpretation of ( -V -aV + q ) as a densely defined, self-adjoint operator A , on L2(Q) with compact resolvent so that discussion of the 'eigenvalues of . . . A,' makes sense? The approach, as in [l], is closely related to that of chapter 3 of [2] with modification to fit the setting under consideration. In [l] it was shown, for the onedimensional case, that A , is suitably defined for yl E 9* with 8:= HI( -1 , l ) and that the eigenvalues then depend continuouslya on q. A principal concern here will be to obtain comparable results for \$J E 8* with 8 much like H'(0, 1)-viewing yl = q ( r ) as given for Y E (0, 1), rather than on SZ-but now with 9 defined through a weighted H' norm, controlling the behaviour near Y = 0. We are able to get results quite comparable with the one-dimensional case treated in [ 11 precisely because the radial symmetry permits a treatment through separation of variables which reduces this to one-dimensional considerations. Once we have developed the setting in which EVP is a meaningful problem, our concern is to demonstrate convergence for an approximation method of 'generalised interpolation' type (see, e.g., [ 2 , 3 ] ) . We assume in EVP that we are given the sequence (AI, & , . . .) of the eigenvalues of A , corresponding to a unique potential E Y * and consider the approximation procedure: