Density of states engineering: normalized energy density of states band structure using the tridiagonal representation approach

We expand the quantum mechanical wavefunction in a complete set of orthonormal basis such that the matrix representation of the Hamiltonian is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. The solution of this recursion is a set of orthogonal polynomials in the energy whose weight function is the energy density of states of the system. The latter is constructed using Green's function, which is written as a continued fraction in terms of the Hamiltonian matrix elements. We study the distribution of zeros of the orthogonal polynomials on the real energy line based exclusively on their three-term recursion relations. We show that the zeros are grouped into sets belonging to separated bands on the orthogonality interval. The number of these bands is equal to the periodicity (multiplicity) of the asymptotic values of the recursion coefficients and the location of their boundaries depends only on these asymptotic values. Bound states (if they exist) are located at isolated zeros found in the gaps between the density bands that are stable against variation in the degree of the polynomial for very large degrees. We give examples of systems with a single, double, triple, and quadruple energy density bands.