Energy eigenvalue level motion with two parameters

From the eigenvalue equation (H) over cap (lambda)\ psi (n)(lambda)> = E-n( lambda)\ psi (n)(lambda)> where (H) over cap (lambda) = (H) over cap (0) lambda(V) over cap one can derive an autonomous system of first order ordin ary differential equations for the eigenvalues E-n(lambda) and the matrix e lements V-mn(lambda) := < psi (m)(lambda)\(V) over cap \ psi (n)(lambda)> w here lambda is the independent variable. We derive the partial differential equations for the extended case (H) over cap (lambda1,lambda2) = (H) over cap (0) + lambda (1)(V) over cap (1) + lambda (2)(V) over cap (2), where la mbda (1) and lambda (2) are the independent variables. Some applications of this system of partial differential equations are discussed.