Discrete Variable Transformations on Symplectic Systems and Even Order Difference Operators

Change of independent variable t D 1=x motivates variable step size discretizations of even order differential operators. We develop variable change methods for discrete symplectic (i.e., J-orthogonal) systems. This enables us to perform simultaneous change of independent and dependent variables on discrete linear Hamiltonian systems and on newly defined even order variable step size formally self adjoint difference operators. These variable changes yield a new system which is related to the original system by an operator identity. We generalize results of Bohner and Doˇsl´y on transformations of formally self-adjoint scalar difference operators. They only considered a change of dependent variable whereas these methods allow yxn‘ D xn‘ztn‘ where tn D f xn‘: These variable change results bring the subject of transformation theory for even order difference operators closer to the known transformation theory in the continuous case