MIT OpenCourseWarehttp://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.132 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 22. Thursday April 30: Dirchlet problem continued I did not finish the proof last time:-Proof. Notice the form of the solution in case V ≥ 0 in (21.25). In general, we can choose a constant c such that V + c ≥ 0. Then the equation d2w d2w(22.1) − dx2 + V w = T wk ⇐⇒ − dx2 + (V + c)w = (T + c)w. Thus, if w satisfies this eigen-equation then it also satisfies (22.2) w = (T + c)A(Id +A(V + c)A)−1Aw ⇐⇒ Sw = (T + c)−1 w, S = A(Id +A(V + c)A)−1A. Now, we have shown that S is a compact self-adjoint operator on L2(0, 2π) so we know that it has a complete set of eigenfunctions, ek, with eigenvalues τk = 0� . From the discussion above we then know that each ek is actually continuous – since it is Aw� with w� ∈ L2(0, 2π) and hence also twice continuously differentiable. So indeed, these ek satisfy the eigenvalue problem (with Dirichlet boundary conditions) with eigenvalues (22.3) Tk = τk−1 + c → ∞ as k → ∞. The solvability part also follows much the same way.
View Full Document
Unlocking...