Renormalized oscillation theory for symplectic eigenvalue problems with nonlinear dependence on the spectral parameter

In this paper, we establish new renormalized oscillation theorems for discrete symplectic eigenvalue problems with Dirichlet boundary conditions. These theorems present the number of finite eigenvalues of the problem in the arbitrary interval using the number of focal points of a transformed conjoined basis associated with Wronskian of two principal solutions of the symplectic system evaluated at the endpoints a and b. We suppose that the symplectic coefficient matrix of the system depends nonlinearly on the spectral parameter and that it satisfies certain natural monotonicity assumptions. In our treatment, we admit possible oscillations in the coefficients of the symplectic system by incorporating their non-constant rank with respect to the spectral parameter.