STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH

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1 STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s tool minimizing the functionl is not the gol of this note. Consider the qudrtic functionl () K(y) = [P (x)y 2 + R(x)y 2 ]dx. The Euler eqution is the well-known Jcobi eqution (2) P y d dx (Ry ) = 0. Consider the isoperimetric problem: find the sttionry solution for K with the constrin (3) for y 2 dx =. The method of multiplier leds to the problem of finding the sttionry solution The corresponding Euler eqution is (Ry 2 + P y 2 λy 2 )dx. (4) P y d dx (Ry ) = λy. This is so clled the Sturm-Liouville eqution. We will only consider the simplest boundry conditions (5) y() = y(b) = 0. Assume tht R(x) nd P (x) re C functions, nd R(x) > 0 for x b.

2 2 XIAO-BIAO LIN Introducing the nottion (6) L(y) = P y d dx (Ry ). Jcobi eqution (2) nd S-L eqution (4) become L(y) = 0, L(y) = λy. Usully L(y) is clled the S-L opertor. It is liner for ny C 2 functions y, y 2 nd ny constnt α, L(y + y 2 ) = L(y ) + L(y 2 ), L(αy) = αl(y). Qudrtic nd biliner functionls Define (7) K(y, z) = (Ry z + P yz)dx. K(y, z) is biliner with respect to y = y(x) nd z = z(x). When y = z, we hve Observe tht K(y, y) = K(y). K( y + 2 z) = 2 K(y) K(y, z) + 2 2K(z), K( i y i ) = i= Using the boundry condition (5), Ry z dx = Thus, (7) cn be written s 2 i K(y i ) + 2 K(y i, y j ). j>i i= (8) K(y, z) = ( d dx Ry )zdx = where L(y) is from (6). Obviously we lso hve K(y, z) = L(y)zdx, L(z)ydx. ( d dx Rz )ydx.

3 STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH 3 Therefore (9) In prticulr, if y = z, L(y)zdx = L(z)ydx. (0) K(y) = L(y)ydx. Self-djoint opertors Assume tht liner opertor Ay(x) is defined on the function spce {y(x) : x b}. If for ny functions y(x), z(x) in the spce, we hve (Ay)zdx = z (Az)ydx, then Ay is clled self-djoint opertor. Eqution (9) shows tht the S-L opertor is self-djoint. Similrly, we cn show tht in the spce y C, L(y) is self-djoint under the conditions y () = 0, y (b) = 0. Also, the opertor L(y) is self-djoint under the periodic boundry condition with period ω = b : y(b) = y(), y (b) = y (). Orthogonlity: Two functions y (x) nd y 2 (x) re orthogonl on [, b] with respect to the weight ρ(x) if ρ(x)y (x)y 2 (x)dx = 0. The function y (x) is sid to be normlized with respect to the weight ρ(x) if ρ(x)y 2 (x)dx =. We ssume tht ρ(x) 0 nd is not identiclly zero. A sequence of functions y i is sid to be orthonorml with respect to ρ(x) if 0 i j, ρ(x)y i (x)y j (x)dx = i = j.

4 4 XIAO-BIAO LIN 2. Eigenvlues nd eigenfunctions For ny rel or complex λ, eqution (4) under condition (5) hs trivil solution y(x) 0. But for some λ, the system my hve nontrivil solution y 0. Those λ re clled eigenvlues of the opertor L, nd the corresponding nontrivil functions re clled eigenfunctions of L(y). The eigenfunction is sid to be normlized if y 2 (x)dx =. Bsic properties of eigenvlues nd eigenfunctions. If y(x) is n eigenfunction with y() = y(b) = 0, then y () 0, y (b) If y, y 2 re two eigenfunctions corresponding to the sme eigenvlue, then the liner combintion y(x) = c y (x) + c 2 y 2 (x), is either n eigenfunction or trivil solution. 3. If y nd y 2 re eigenfunctions corresponding to the sme eigenvlue λ, then y nd y 2 re linerly dependent. Moreover y 2 (x)y () y (x)y 2() There re only two normlized eigenfunctions for the sme eigenvlue λ. The two differ by multiple of. Theorem. If y nd y 2 re eigenfunctions corresponding to two distinct eigenvlues λ λ 2, then y nd y 2 re orthogonl to ech other y (x)y 2 (x)dx = 0. Theorem 2. All the eigenvlues for L re rel. Theorem 3. If λ is n eigenvlue for L nd y is normlized eigenfunction, then K[y] = λ.

5 STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH 5 Theorem 4. If y(x) is n eigenfunction nd if y is orthogonl to y, then K(y, y ) = 0. We now consider the minimiztion of K(y) under the condition y 2 (x)dx =, y() = y(b) = 0. It cn be proved tht there exists C function y = y which solves the minimiztion problem. At some λ = λ, this function y stisfies the Euler eqution (4). Therefore, for the S-L eqution there exist t lest one eigenvlue λ with corresponding y. From Theorem 3, K(y ) = λ. By definition, under conditions (3), (5), λ is the minimum of K with the function y. By the sme theorem, ny other eigenvlue is lso vlue of K(y) with the corresponding normlized eigenfunction y. However, they re not the minimum of K. Thus, λ is the smllest eigenvlue. The fct is summrized in the following Theorem 5. The smllest eigenvlue λ conditions (3) nd (5). is the conditionl minimum of K under Theorem 6. If λ is the smllest eigenvlue with eigenfunction y, then for ny C function y(x) stisfying (3) nd (5), K(y) λ y 2 (x)dx. The equl sign holds iff y(x) = ±ky. 3. Vritionl method on eigenvlues Theorem 5 provides method of getting the smllest eigenvlue λ. Other eigenvlues cn lso be obtined by conditionl minimiztion process. Theorem 7. Eigenvlues of L cn be rrnged s incresing infinite sequence λ < λ 2 < < λ n <...,

6 6 XIAO-BIAO LIN with corresponding eigenfunctions y, y 2,.... conditionl minimum of K(y) under the conditions () y 2 dx =, y() = y(b) = 0. For ech n, the eigenvlue λ n is the yy i dx = 0, i =, 2,..., n, Theorem 8. For ny C function y tht is orthogonl to the first n eigenfunctions nd stisfies (5), we hve (2) K(y) λ n y 2 dx. The equl sign hppens iff y = ky n. Theorem 9. If the coefficients P (x) nd R(x) increse by positive δp (x) nd δr(x), then the nth eigenvlue λ n, n =, 2, 3,... increses. Theorem 0. (Cournt) The nth eigenvlue, denoted λ n (b) is monotone decresing function of the right boundry b. more over λ n (b), s b. Theorem. (Oscilltion theorem) The eigenfunction y n (x) corresponding to the nth eigenvlue λ n (b) hs n zeros in (, b). 4. Completeness of the eigenfunctions Let {y n } be the orthonorml set of eigenfunctions corresponding to eigenvlues Then λ < λ 2 < < λ n <.... ( i y i ) 2 dx = i= 2 i. For ny y C[, b], c i = yy idx is the Fourier coefficient of y with respect to y i, i= iy i (x) is the Fourier series for y(x) nd n i= iy i (x) is the prtil sum. The reminder is defined s R n (x) := y(x) i= c i y i (x). i=

7 STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH 7 y 2 dx = c 2 i + i= R 2 ndx. (Prsevl s inequlity) c 2 i i= y 2 dx. If the equl sign holds for ny continuous function y(x), then we sy {y n (x)} is complete. Theorem 2. Eigenfunctions of the S-L eqution form complete orthonorml bsis. Proof. Consider K(y) = (P y2 +Ry 2 )dx nd its first n eigenfunctions: y, y 2,..., y n. Let y(x) = n c iy i + R n. K(y) = K( c i y i + R n ) = K(R n ) + c 2 i K(y i ) + 2 Bsed on K(y i ) = λ i nd the orthogonlity, c i K(y i, R n ) + 2 c i c j K(y i, y j ). i j K(y i, R n ) = 0, K(y i, y j ) = 0, if i j. K(y) = λ i c 2 i + K(R n ). Since R n is orthogonl to y,..., y n, from Theorem 8, K(R n ) λ n+ Rndx. 2 λ i > 0 strting from some index i > i 0, thus, K(R n ) = K(y) λ i c 2 i, decreses if n i 0, hence bounded with respect to n. Rndx 2 (K(y) λ i c 2 i ). λ n+ Since λ n+, we hve lim n R 2 ndx = 0.