Reconstructing a differential operator from a part of its spectrum (Q1112224)

In a Hilbert space \(L_ 2(0,\pi)\) we consider an operator L generated by the differential expression \(\ell (y)=(-1)^ my^{(2m)}(x)+q(x)y(x)\) (m\(\geq 2)\) and the boundary conditions \(y^{(2k)}(0)=y^{(2k)}(\pi)=0\), \(k=0,1,...,m-1\), where \(q(x)\in L_ 2(0,\pi)\) is a complex-valued function (potential). Let \(\{\mu_ n\}_ 1^{\infty}\) be a sequence of eigenvalues of the operator L, where \(| \mu_ 1| <| \mu_ 2| <...\), and \(N_ 1\) is a subset of the set of natural numbers N. We propose to resolve the following problem: From given eigenvalues \(\{\mu_ n\}\) \((n\in N_ 1)\) of the operator L, define the potential q(x). For q(x) we assume that \(q(x)=q(\pi -x)\) and \(\int^{\pi}_{0}q(x)\cos 2kxdx=0\) \((k\in N\setminus N_ 1,k=0)\).