Can someone help me out with a problem involving differential equations? The equation u"(x) = (V(x) - E) u(x), with V(x+P) = V(x) is the Schroedinger wave equation for an electron in a 1-dimensional crystal. V(x) is the periodic potential, and E is the energy of the electron. If u1(x) and u2(x) are linearly independent solutions, so are u1(x+P) and u2(x+P). Since the solution space is 2-dimensional, it follows that [u1(x+P)] = [A B] [u1(x)] [u2(x+P)] [C D] [u2(x)] for some constants A, B, C, D. Call the matrix M. The absence of a u'(x) term implies that the Wronskian is constant, from which it follows that det(M) = 1. A change of basis in the solution space induces a similarity transformation on M. The differential equation is real, so either u1 and u2 are real, or u2 can be chosen to be the complex conjugate of u1. Either way, tr(M) = A+D is real. So the eigenvalues of M are either (exp(+i t), exp(-i t)) or (r, 1/r), with r and t real. In the first case, an electron of energy E can exist in the crystal, while in the second case the energy is forbidden. Thus the band structure is characterized by the function tr(M(E)). The allowed energy bands are given by -2 <= tr(M(E)) <= 2, and energies E for which this inequality is false lie in the gap between bands. Now for my question. Suppose I'm only interested in the energy bands, and I don't care about the actual wave function u(x). Then, given some value of E, is there a way to calculate tr(M(E)) without having to solve the differential equation? Gene __________________________________ Do you Yahoo!? Yahoo! Mail Address AutoComplete - You start. We finish. http://promotions.yahoo.com/new_mail