PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Function spaces and operator theory
Conditions of Validity, Ch Sturm-Liouville theory: the spectral theorem for a differential operator
Problem
On [0,L] take Ly=−y′′ (so p=1, q=0, w=1), and impose the phase-coupled conditions y(L)=eiθy(0),y′(L)=eiθy′(0), where θ is a fixed real phase with 0<θ<2π. These couple the two ends and are not the periodic conditions; the anti-periodic conditions of the text are the case θ=π.
((a)) Show that the boundary form −[f∗g′−f∗′g]0L vanishes for every pair f,g obeying the conditions. Keep f(0), f′(0), g(0), g′(0) general, and keep track of the complex conjugate.
((b)) Find the eigenvalues as functions of θ, and say for which θ any of them is repeated.