Department Mathematics Stanford University Math 175 Homework 8
D UE AT L ECTURE W EDNESDAY M AY 27
This is the last homework which will be graded; There will be a hw9, but that will not be graded 1. (a) H is a Hilbert space and {enP 1;2;::: , {fp }pD1;2;::: are complete orthonormal sequences in H . }nD P 1 Prove that if anp D .en ; fp / then pD1 anp amp D inm and 1 1 anp anq D ipq , where iij is the nD Kronecker delta. P (b) If T 2 L.H; K/ where H; K are Hilbert spaces, and if {en }; {fp } are as in (a) prove that n kT .en /k2 P converges if and only if p kT .fp /k2 converges and in this case the two sums agree. P Hint: Start by assuming p kT .fp /k2 converges, and, to avoid convergence difficulties, first consider PN the finite rank operator TN .x/ D pD1 .x; fp /T .fp / instead of T . 2. (a) Suppose that for u 2 C 2 .OEa; b/ (real valued twice continuously differentiable functions on OEa; b) we define Lu D .pu 0/ 0 C qu with p 2 C 1 .OEa; b/; q 2 C 0 .OEa; b/ given and p > 0 everywhere in OEa; b. 2 If ; ; ; i are given real numbers with .; / 0 and . ; i/ 0 and if CBC .OEa; b/ denotes the set of 2 .OEa; b/ with u.a/ C u 0.a/ D 0 and u.b/ C iu 0.b/ D 0, prove the "Green's identity" function u 2 C
2 .Lv; w/ D .v; Lw/; v; w 2 CBC .OEa; b/; Rb where .f; g/ is the inner product .f; g/ D a f .t /g.t / dt on the (real) space L2 .OEa; b/.
Hint: Use integration by parts; a key point is to show that the boundary terms are zero.
2 (b) With the notation of (a), show that if v; w 2 CBC .OEa; b/ with Lv D v and Lw D w with ; 2 R and , then .v; w/ D 0. 2 (c) If we instead consider complex-valued functions u D u1 C i u2 (u1 ; u2 2 CBC .OEa; b/) and use the Rb usual complex inner product .f; g/ D a f g , then nevertheless all possible eigenvalues are real. (i.e. 2 Lu D u where u D u1 C i u2 0 and u1 ; u2 2 CBC .OEa; b/ ) 2 R).
3. With as notation in 2(a) show that there are at most countably many 2 R such that there exists 2 u 2 CBC .OEa; b/ n {0} with Lu D u. (i.e. there are at most countably many eigenvalues of the 2 Sturm-Liouville problem Lu D u, u 2 CBC .OEa; b/).
N 2 Hint: If {c }2 ( any indexing set) is a set of non-zero real numbers such that j D1 cj 1 for every choice of N 1 and every choice of distinct 1 ; : : : ; N 2 , then is a countable set (i.e. there is a sequence 1 ; 2 ; : : : in which every element of appears exactly once).
P
4. All functions here are real valued. Suppose p 2 C 1 .OEa; b/ with p nowhere zero on OEa; b, q 2 C 0 .OEa; b/ and for f 2 C 2 .OEa; b/ let L.f / D .pf 0/ 0 Cqf . Prove: If u; v are both C 2 .OEa; b/ solutions of the equation L.f / D 0 and neither u nor v is identically zero on OEa; b then either u is a real constant multiple of v or else .u.t/; u 0.t//; .v.t /; v 0.t // are l.i. vectors in R2 for each t 2 OEa; b. Hint for (ii): The general ODE uniqueness theorem guarantees that f 0 on OEa; b if L.f / D 0 on OEa; b and if there is a point t0 2 OEa; b with .f .t0 /; f 0.t0 // D .0; 0/. ...