Renormalized Oscillation Theory for Symplectic Eigenvalue Problems with Nonlinear Dependence on the Spectral Parameter

Abstract: 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 arbitrary interval $(a,b]$ using 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 nonconstant rank with respect to the spectral parameter.