On the conditions for a Hamiltonian matrix to have an eigen-value density with some prescribed characteristics

The following problem is considered: “Find the necessary and sufficient conditions to be fulfilled by the components of a hamiltonian operator to have an eigenvalue density with certain prescribed characteristics”. The solution when the hamiltonian operator is a Jacobi matrix and the prescribed characteristic is the unimodality, is shown. A sufficient condition for a real Jacobi matrix to have a prescribed density is also given.