Spectral rigidity for radial Schrödinger operators
The authors study the problem of identifying the spectral data needed to determine the coefficients of the radial Schrödinger operators \(- \Delta+ p(\| x\|)\) in odd space dimensions \(n\) greater than 1 acting on functions on the unit ball with a boundary condition such as \(f(x)= 0\) when \(\| x\|= 1\) or \(\partial f/\partial r(x)+ bf(x)= 0\) when \(\| x\|= 1\). The eigenvalues of the operator can be obtained as the union of the eigenvalues of a sequence of singular ordinary differential operators \(-D^ 2+ p(x)+ m(m+1)/ x^ 2\) where \(m= m_ 0\), \(m+1\), \(m+2,\dots\) with \(m_ 0= (n- 3)/2\). A local diffeomorphism taking operators \(S(m,b,p)\) to those of the form \(S(m+1,b,p)\) is analyzed, the derivative of this diffeomorphism is computed, and these linear operators are related to convolution operators on the line. Finally it is proved that if \(m(0)+ m(1)= 1\text{ mod }2\) then the tangent spaces \(\text{TM}(m(0),b,Q)\) and \(\text{TM}(m(1),b,Q)\) are Fredholm for any \(Q\in L^ 2_ R[0,1]\). Fredholm operators and subspaces are discussed in the Appendix.
scientific article; zbMATH DE number 665704