104 14. EIGENFUNCTION EXPANSIONS Exercises for Chapter 14 Exercise 14.1. Show that if K is a compact <strong>in</strong>terval, then C(K) is a Banach space with the norm sup x∈K|u(x)|. Also show that if I is an arbitrary <strong>in</strong>terval, then C(I) is a Fréchet space (a l<strong>in</strong>ear Hausdorff space with the topology given by a countable family of sem<strong>in</strong>orms, which is also complete), under the topology of locally uniform convergence. Exercise 14.2. With the assumptions of Corollary 14.5 the Fourier series for u ∈ D(T ) actually converges absolutely and uniformly to u. This may be proved just as for the case of a Sturm-Liouville equation, which was considered <strong>in</strong> Exercise 11.2. Do it!
CHAPTER 15 S<strong>in</strong>gular problems We now have a satisfactory eigenfunction expansion theory for regular boundary value problems, so we turn next to s<strong>in</strong>gular problems. We then need to take a much closer look at Green’s function. To do this, we fix an arbitrary po<strong>in</strong>t c ∈ I; if I conta<strong>in</strong>s one of its endpo<strong>in</strong>ts, this is the preferred choice for c. Next, let F (x, λ) be a fundamental matrix for JF ′ + QF = λW F with λ-<strong>in</strong>dependent, symplectic <strong>in</strong>itial data <strong>in</strong> c. We will need the follow<strong>in</strong>g theorem. Theorem 15.1. A solution u(x, λ) of Ju ′ + Qu = λW u with <strong>in</strong>itial data <strong>in</strong>dependent of λ is an entire function of λ, locally uniformly with respect to x. This means that u(x, λ) is analytic as a function of λ <strong>in</strong> the whole complex plane, and that the difference quotients 1 (u(x, λ+h)−u(x, λ)) h converge locally uniformly <strong>in</strong> x as h → 0. The proof is given <strong>in</strong> Appendix C. We can now give the follow<strong>in</strong>g detailed description of Green’s function. Theorem 15.2. Green’s function has the follow<strong>in</strong>g properties: (1) For λ ∈ ρ(T ) we have Rλu(x) = 〈u, G ∗ (x, ·, λ)〉W . (2) As functions of y the columns of G ∗ (x, y, λ) satisfy the equation Ju ′ + Qu = λW u for y = x. (3) As functions of y, the columns of G ∗ (x, y, λ) satisfy the boundary conditions that determ<strong>in</strong>e T as a restriction of T1, for any x <strong>in</strong>terior to I. (4) G ∗ (x, y, λ) = G(y, x, λ), for all x, y ∈ I and λ ∈ ρ(T ). (5) G(x, y, λ) − G(x, y, µ)) = (λ − µ)〈G ∗ (y, ·, µ), G ∗ (x, ·, λ)〉W = (λ − µ)RλG ∗ (y, ·, µ)(x), for all x, y ∈ I and λ, µ ∈ ρ(T ). Furthermore, there exists an n×n matrix-valued function M(λ), def<strong>in</strong>ed <strong>in</strong> ρ(T ) and satisfy<strong>in</strong>g M ∗ (λ) = M(λ), such that (15.1) G(x, y, λ) = F (x, λ)(M(λ) ± 1 2 J −1 )F ∗ (y, λ), where the sign of 1J should be positive for x > y, negative for x < y. 2 Proof. We already know (1). Now let K be a compact sub<strong>in</strong>terval of I, (u, v) ∈ T0 with support <strong>in</strong> K, and suppose x /∈ K. We have u ∈ D(T0) ⊂ D(T ) and (u, v) = (u, λu + (v − λu)) so that u = Rλ(v − λu). 105
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