Functionals of eigenvalues on the manifold of potentials (Q6588156)

The paper deals with the family of periodic self-adjoint boundary value problems \N\[\N-y''+p(x)y=\lambda y, \] \[ y(0)-y(2\pi)=y'(0)-y'(2\pi)=0, \N\]\Nwhere the parameter of the family is the potential \(p\) from the Hilbert space \(L_2(0,2\pi)\) of \(2\pi\)-periodic functions that are square integrable on the period and have zero mean: \N\[\N\int_0^{2\pi}p(x)dx=0. \N\]\NThe analytical and topological properties of the functional of eigenvalues are completely described. Relying on the Hilbert property of the family of potentials, the author proves the Morse property of the functional and calculates its index. It is found, possibly, for the first time that the functional of eigenvalues exhibits specific behavior with increasing spectral gap.