2 Sturm Liouville Theory
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1 2 Sturm Liouville Theory So fr, we ve exmined the Fourier decomposition of functions defined on some intervl (often scled to be from π to π). We viewed this expnsion s n infinite dimensionl nlogue of expnding finite dimensionl vector into its components in n orthonorml bsis. But this is just the tip of the iceberg. Reclling other gmes we ply in liner lgebr, you might well be wondering whether we couldn t hve found some other bsis in which to expnd our functions. You might lso wonder whether there shouldn t be some role for mtrices in this story. If so, red on! 2.1 Self-djoint mtrices We ll begin by reviewing some fcts bout mtrices. Let V nd W be finite dimensionl vector spces (defined, sy, over the complex numbers) with dim V = n nd dim W = m. Suppose we hve liner mp M : V W. By linerity, we know wht M does to ny vector v V if we know wht it does to complete set {v 1, v 2,..., v n } of bsis vectors in V. Furthermore, given bsis {w 1, w 2,..., w m } of W we cn represent the mp M in terms of n m n mtrix M whose components re M i =(w,mv i ) for =1,..., m nd i =1,..., n, (2.1) where (, ) is the inner product in W. We ll be prticulrly interested in the cse m = n, when the mtrix M is squre nd the mp M tkes M : V W = V is n isomorphism of vector spces. For ny n n mtrix M we define it s eigenvlues {λ 1,λ 2,..., λ n } to be the roots of the chrcteristic eqution P (λ) det(m λi) = 0, where I is the identity mtrix. This chrcteristic eqution hs degree n nd the fundmentl theorem of lgebr ssures us tht we ll lwys be ble to find n roots (genericlly complex, nd not necessrily distinct). The eigenvector v i of M tht corresponds to the eigenvlue λ i is then defined by Mv i = λ i v i (t lest for non-degenerte eigenvlues). Given complex n n mtrix M, its Hermitin conjugte M is defined to be the complex conjugte of the trnspose mtrix, M (M T ), where the complex conjugtion cts on ech entry of M T. A mtrix is sid to be Hermitin or self-djoint if M = M. There s neter wy to define this: since for two vectors we hve (u, v) =u v, we see tht mtrix B is the djoint of mtrix A iff (Bu, v) =(u, Av) (2.2) becuse the vector (Bu) = u B. The dvntges of this definition re tht i) it does not require tht we pick ny prticulr components in which to write the mtrix nd ii) it pplies whenever we hve definition of n inner product (, ). Self-djoint mtrices hve number of very importnt properties. Firstly, since λ i (v i, v i )=(v i, Mv i )=(Mv i, v i )=λ i (v i, v i ) (2.3) 19
2 the eigenvlues of self-djoint mtrix re lwys rel. Secondly, we hve λ i (v j, v i ) = (v j, Mv i )=(Mv j, v i )=λ j (v j, v i ) (2.4) or in other words (λ i λ j )(v j, v i ) = 0 (2.5) so tht eigenvectors corresponding to distinct eigenvlues re orthogonl wrt the inner product (, ). After rescling the eigenvectors to hve unit norm, we cn express ny v V s liner combintion of the orthonorml set {v 1, v 2,..., v n } of eigenvectors of some self-djoint M. If M hs degenerte eigenvlues (i.e. two or more distinct vectors hve the sme eigenvlue) then the set of vectors shring n eigenvlue form vector subspce of V nd we simply choose n orthonorml bsis for ech of these subspces. In ny cse, the importnt point here is tht self-djoint mtrices provide nturl wy to pick bsis on our vector spce. A self-djoint mtrix M is non-singulr (det M 0 so tht M 1 exists) if nd only if ll its eigenvlues re non-zero. In this cse, we cn solve the liner eqution Mu = f for fixed vector f nd unknown u. Formlly, the solution is u = M 1 f, but prcticl wy to determine u proceeds s follows. Suppose {v 1, v 2,..., v n } is n orthonorml bsis of eigenvectors of M. Then we cn write f = f i v i nd u = u i v i where f i =(v i, f) etc. s before. We will know the vector u if we cn find ll its coefficients u i in the {v i } bsis. But by linerity Mu = u i Mv i = u i λ i v i = f = f i v i, (2.6) nd tking the inner product of this eqution with v j gives u i λ i (v j, v i )=u j λ j = f i (v j, v i )=f j (2.7) using the orthonormlity of the bsis. Provided λ j 0 we deduce u j = f j /λ j so tht u = f i λ i v i. (2.8) If M is singulr then either Mu = f hs no solution or else hs non-unique solution (which it is depends on the choice of f). 2.2 Differentil opertors In the previous chpter we lerned to think of functions s infinite dimensionl vectors. We d now like to think of the nlogue of mtrices. Sturm nd Liouville relised tht these 20
3 could be thought of s liner differentil opertors L. This is just liner combintion of derivtives with coefficients tht cn lso be functions of x, i.e. L is liner differentil opertor of order p if L = A p (x) dp p + A p 1(x) dp 1 p A 1(x) d + A 0(x). When it cts on (sufficiently smooth) function y(x) it gives us bck some other function Ly(x) obtined by differentiting y(x) in the obvious wy. This is liner mp between spces of functions becuse for two (p-times differentible) functions y 1,2 (x) nd constnts c 1,2 we hve L(c 1 y 1 + c 2 y 2 )=c 1 Ly 1 + c 2 Ly 2. The nlogue of the mtrix eqution Mu = f is then the differentil eqution Ly(x) = f(x) where we ssume tht both the coefficient functions A p (x),..., A 0 (x) in L nd the function f(x) re known, nd tht we wish to find the unknown function y(x). For most of our pplictions in mthemticl physics, we ll be interested in second order 9 liner differentil opertors 10 L = P (x) d2 2 + R(x) d Q(x). (2.9) Recll tht for ny such opertor, the homogeneous eqution Ly(x) = 0 hs precisely two non-trivil linerly independent solutions, sy y = y 1 (x) nd y = y 2 (x) nd the generl solution y(x) =c 1 y 1 (x) +c 2 y 2 (x) with c i C is known s the complementry function. When deling with the inhomogeneous eqution Ly = f, we seek ny single solution y(x) = y p (x), nd the generl solution is then liner combintion y(x) =c p y p (x)+c 1 y 1 (x)+c 2 y 2 (x) of the prticulr nd complementry solutions. In mny physicl pplictions, the function f represents driving force for system tht obeys Ly(x) = 0 if left undisturbed. In the cses (I ssume) you ve seen up to now, ctully finding the prticulr solution required good del of either luck or inspired guesswork you noticed tht if you differentited such-nd-such function you d get something tht looked pretty close to the solution you re fter, nd perhps you could then refine this guess to find n exct solution. Sturm Liouville theory provides more systemtic pproch, nlogous to solving the mtrix eqution Mu = f bove. 2.3 Self-djoint differentil opertors The 2 nd -order differentil opertors considered by Sturm & Liouville tke the form Ly d ( p(x) dy ) q(x)y, (2.10) where p(x) is rel (nd once differentible) nd q(x) is rel nd continuous. This my look to be tremendous speciliztion of the generl form (2.9), with R(x) restricted to be 9 It s beutiful question to sk why only second order? Prticulrly in quntum theory. 10 The sign in front of Q(x) is just convention. 21
4 P (x), but ctully this isn t the cse. Provided P (x) 0, strting from (2.9) we divide through by P (x) to obtin d R(x) d P (x) Q(x) x P (x) =e 0 ( ) R(t)/P (t)dt d x e 0 R(t)/P (t)dt d Q(x) P (x) (2.11) Thus setting p(x) to be the integrting fctor p(x) = exp ( x 0 R(t)/P (t)dt) nd likewise setting q(x) = Q(x)p(x)/P (x), we see tht the forms (2.10) nd (2.9) re equivlent. However, for most purposes (2.10) will be more convenient. The beutiful feture of these Sturm Liouville opertors is tht they re self-djoint with respect the inner product (f, g) = f(x) g(x), (2.12) provided the functions on which they ct obey pproprite boundry conditions. To see this, we simply integrte by prts twice: [ d (Lf, g) = (p(x) df ) ] q(x)f (x) g(x) = [p df ] b g p(x) df dg q(x)f(x) g(x) = [p df ] dg b [ ( d g pf + f(x) p(x) dg ) ] q(x) g(x) [ ( df = p(x) g f dg )] b +(f,lg) (2.13) where in the first line we hve used fct tht p nd q re rel for Sturm Liouville opertor. So we see tht (Lf, g) =(f,lg) provided we restrict ourselves to functions which obey the boundry conditions [ ( df p(x) g f dg )] b =0. (2.14) Exmples of such boundry conditions re to require tht ll our functions stisfy b 1 f ()+b 2 f() =0 c 1 f (b)+c 2 f(b) =0, (2.15) where b 1,2 nd c 1,2 re constnts, not both zero. I emphsize tht we must choose the sme constnts for ll our functions. These boundry conditions ensure tht (2.14) vnishes t ech boundry seprtely. If the function p(x) obeys p() = p(b) then we cn likewise sk tht ll our functions re periodic, so tht f() =f(b) nd f () =f (b); this ensures tht the contributions t ech boundry cncel in (2.14). Finlly, it my sometimes be tht p() =p(b) = 0, though in this cse the endpoints of the intervl [, b] re singulr points of the differentil eqution. 22
5 2.4 Eigenfunctions nd weight functions Whtever boundry conditions we choose, provided they stisfy (2.14) we hve (Lf, g) = (f,lg) nd L is self-djoint. Just s in the finite dimensionl cse of self-djoint mtrices, these self-djoint differentil opertors utomticlly inherit mny useful properties. We strt by defining the notion of n eigenfunction of our differentil opertor. It s convenient to do this in slightly surprising wy. A weight function w(x) is rel-vlued, non-negtive function tht hs t most finitely mny zeros on the domin [, b]. A function y(x) is sid to be n eigenfunction of L with eigenvlue λ nd weight w(x) if Ly(x) =λw(x)y(x) (2.16) where we note the presence of the weight function on the right hnd side. In fct, given such n eigenfunction we cn lwys find corresponding eigenfunction ỹ(x) with weight function 1 by setting ỹ(x) = ( w(x) y(x) nd replcing Ly by 1 w L ỹ w ), so the weight function does not relly represent nything new, but it s conventionl (nd will turn out to be convenient) to keep it explicit. The weight function plys role in the inner product. We define the inner product with weight w to be (f, g) w f(x) g(x) w(x) (2.17) so tht the mesure includes fctor of w. Notice tht since w is rel (f, g) w =(f, wg) = (wf, g) (2.18) where the inner products on the rhs re the stndrd ones with mesure only. This inner product is gin non-degenerte in the sense tht (f, f) w = 0 implies f = 0 if f is continuous (t lest in neighbourhood of ny zeros of w(x)). This is becuse (f, f) w = f(x) 2 w(x) is the integrl of continuous positive function w(x) f(x) 2. By ssumption w hs only finitely mny zeros on [, b] while f is continuous, so the integrl gives zero if nd only if f(x) = 0 identiclly on [, b]. The first property of Sturm Liouville opertors is tht their eigenvlues re lwys rel. The proof is exctly the sme s in the finite dimensionl cse: if Lf = λwf then λ (f, f) w =(f, λwf) =(f,lf) =(Lf, f) =λ (f, f) w (2.19) using the self-djointness of L nd the fct tht the inner product (, ) is nti-liner in its first entry. Note tht if f hs eigenvlue λ, then becuse the eigenvlues, weight w nd coefficients p(x) nd q(x) re rel L(f )=(Lf) =(λwf) = λw(x) f(x) (2.20) 23
6 so tht f is lso n eigenfunction of L with the sme eigenvlue. Thus, tking Re f nd Im f if necessry, we cn lwys choose our eigenfunctions to be rel-vlued. Just s in the finite dimensionl cse, eigenfunctions f 1 nd f 2 with distinct eigenvlues, but the sme weight function, re orthogonl wrt the inner product with weight w, since: so tht if λ i λ j then λ i (f j,f i ) w =(f j, Lf i )=(Lf j,f i )=λ j (f j,f i ) w (2.21) (f j,f i ) w = f j (x) f i (x) w(x) =0. (2.22) Thus, exctly s in the finite dimensionl cse, given self-djoint opertor L we cn form n orthonorml set {Y 1 (x),y 2 (x),...} of its eigenfunctions by setting / Y n (x) =y n (x) y n 2 w (2.23) where y n (x) is the unnormlised eigenfunction. I emphsize gin the presence of the weight function in these orthogonlity nd normliztion conditions. Finlly, fter mking prticulr choice of boundry conditions, one cn lso show 11 tht the eigenvlues form countbly infinite sequence λ 1,λ 2,..., with λ n s n, nd tht the corresponding set of orthonorml eigenfunctions Y 1 (x),y 2 (x),... form complete bsis for functions on [, b] stisfying these boundry conditions. Tht is, ny function f(x) on [, b] tht obeys the chosen boundry conditions my be expnded s 12 f(x) = f n Y n (x), where f n = Y n (x)f(x) w(x) =(Y n,f) w. (2.24) The significnt feture here is tht the function f(x) is expnded s discrete sum, just s we sw for Fourier series. This is relly remrkble, becuse the definition of the Y n s tht they be normlised eigenfunctions of L involves no hint of discreteness. In fct, we ll see lter in the course tht the discreteness rises becuse the domin [, b] is compct, nd becuse of our boundry conditions (2.14). 2.5 Some exmples Let s tke look t some simple exmples of the generl theory bove. The simplest nontrivil cse is just to tke the domin [, b] to be [ L, L] nd impose the homogeneous boundry conditions tht ll our functions re periodic i.e. f( L) =f(l) nd f ( L) = f (L). If we lso choose p(x) = 1 nd q(x) = 0 then the Sturm Liouville opertor reduces to L = d2 2, (2.25) 11 But, sdly, not in this course. 12 As wrned, in this course we will no longer worry bout convergence of these infinite sums, lthough see section
7 which is esily seen to be self-djoint when cting on functions obeying these boundry conditions. Finlly, we choose the weight function to be w(x) = 1 identiclly. The eigenfunction eqution becomes Ly(x) = λy(x) (2.26) where we ve introduced minus sign for convenience (just by relbelling the eigenvlues). If λ< 0 then the only solution tht obeys the periodic boundry conditions is the trivil cse y(x) = 0. However, if λ 0 then bsis of solutions is given by ( y n (x) = exp i nπx ) L ( nπ ) 2 for λ n = with n Z. (2.27) L Thus we hve recovered the Fourier series of section 1.3 s specil cse! Note tht these eigenvlues re degenerte, with λ n = λ n ; s we sw before, whenever y(x) is complex vlued eigenfunction of SL opertor, then y (x) is lso n eigenfunction with the sme eigenvlue. If insted of sking for the functions to be periodic, we d sked specificlly tht f( L) =f(l) = 0, then we would find just the sinusoidl Fourier series which hs non-degenerte eigenvlues. For more interesting exmple, suppose we re interested in solving the differentil eqution H xh = λh(x) for x R, (2.28) subject to the condition tht H(x) behves like polynomil s x (so tht in prticulr e x2 /2 H(x) 0 s x ). The reson for this strnge looking condition will be reveled below. Eqution (2.28) is not yet in Sturm Liouville form, so we first compute the integrting fctor nd rewrite (2.28) s ( d e p(x) = x2 dh x 0 2t dt = x 2 (2.29) ) = 2λ e x2 H(x) (2.30) (multiply through by 2e x2 to recover the form (2.28)). This eqution is known s Hermite s eqution nd it plys n importnt role in combintorics, probbility nd in the quntum mechnics of hrmonic oscilltor. We cn now understnd the condition tht H(x) grows t most polynomilly t lrge x : in checking self djointness of the Sturm Liouville opertor on the unbounded domin R, we do not need to consider boundry terms, but we do need to ensure the integrls f d ( ) x2 dg df dg e = e x2 13 I m cheting here by working on n unbounded domin x R rther thn x [, b]. Much of the theory holds, but notice the rther strnge boundry condition we impose. This is just to ensure tht the integrls over the entire rel xis tht rise when we check self-dulity of L re bounded. 25
8 ctully remin finite! For f nd g regulr long R this will be so s long s the integrl is suppressed t lrge x. This is wht our decy condition is designed to ensure. I ll stte without proof tht eqution (2.30) hs non-trivil solutions tht re regulr for ll x R iff the eigenvlue λ is non-negtive integer n, nd you cn check tht these solutions re given by H n (x) =( 1) n e x2 n e x2. (2.31) Crrying out the differentition in (2.31) we find for exmple, H 0 (x) = 1, H 1 (x) = 2x, H 2 (x) = 4x 2 2 nd H 3 (x) = 8x 3 12x. In generl H n (x) is rel polynomil of degree n, known s Hermite polynomil. The Hermite polynomils re orthogonl with respect to the weight function w(x) =e x2 nd obey the normliztion condition dn (H m,h n ) e x 2 = H m (x) H n (x)e x2 = δ m,n 2 n πn!, (2.32) where we note the decy condition is gin crucil to ensure tht this inner product remins finite. 2.6 Inhomogeneous equtions nd Green s functions Finlly, we return to the infinite dimensionl nlogue of the inhomogeneous mtrix eqution Mu = f for self-djoint mtrix M. In the context of Sturm Liouville differentil opertors, we seek to solve the inhomogeneous differentil eqution Lφ(x) =w(x)f (x) (2.33) where gin we choose to include the weight function in the definition of the forcing term on the right hnd side. By the remrks bove, the functions φ(x) nd F (x) cn be expnded in complete set of eigenfunctions of L. So we suppose tht the set {Y 1 (x),y 2 (x),...} form complete set of such eigenfunctions with nd expnd LY n (x) =λ n w(x) Y n (x) nd (Y m,y n ) w = δ m,n (2.34) φ(x) = φ n Y n (x), F(x) = F n Y n (x). (2.35) As in the mtrix cse, it is ssumed tht the function F (x), nd hence the coefficients F n =(Y n,f) w re known, while the coefficients φ n must be found. But gin, this cn be done exctly in nlogy with the finite dimensionl cse. Since L is liner opertor we hve Lφ = φ n LY n = w φ n λ n Y n = wf = w (2.36) F n Y n 26
9 nd tking the inner product with Y m gives φ m λ m = F m. Thus, provided none of the eigenvlues of L re zero we hve found the prticulr solution φ p (x) = F n λ n Y n (x). (2.37) As bove, the generl solution is now sum of this prticulr solution nd the complementry function φ c (x) stisfying Lφ c = 0. It s worth looking little more closely t the structure of the solution (2.37). Substituting in the definition of the forcing term coefficients F n nd exchnging the sum nd integrl we hve φ p (x) = = (Y n,f) w λ n Y n (x) = G(x, t)f(t)dt [ 1 λ n Y n (x) ] Yn (t) F (t) w(t)dt (2.38) where f(t) = w(t)f (t) is the right hnd side of the originl inhomogeneous eqution (2.33) nd we ve defined the Green s function G(x; t) Y n (x)y n (t) λ n. (2.39) The Green s function is function of two vribles (x, t) [, b] [, b]. The importnt point bout the Green s function is tht it depends on the differentil opertor L both through its eigenfunctions nd (more subtly) through the boundry conditions we chose to ensure L is self-djoint, but it does not depend on the forcing function f. Thus if we know the Green s function we cn use (2.38) to construct prticulr solution of Ly = f for n rbitrry forcing term. In this wy, the Green s function provides forml inverse to the differentil opertor L in the sense tht if Ly(x) =f(x) then y(x) = G(x, t) f(t)dt gin in nlogy with the finite dimensionl cse where Mu = f implies u = M 1 f for non-singulr mtrix. The notion of Green s function nd the ssocited integrl opertor s n inverse of L is very importnt. We ll meet it gin lter in mny more generl contexts. One of them is depicted here 14 : 2.7 Prsevl s identity II Recll tht Pythgors theorem sys tht the length squred of vector is the sum of the (mod-)squred of its components in ny orthonorml bsis. In the cse of the Fourier bsis, we obtined n infinite dimensionl version of this in eqution (1.50). We now estblish 14 I stole this picture from Hnnh Wilson Illustrtion, s you ll quickly discover if, like me, you google Feynmn digrms. 27
10 version of Prsevl s identity in this more generl context of the weighted inner product (, ) w. Let {Y 1 (x),y 2 (x),...} be complete set of functions tht re orthonorml with respect to some weight function w(x), so tht (Y m,y n ) w = δ m,n. Then expnding f(x) = f n Y n (x) with f n =(Y n,f) w s in (2.24) we hve (f, f) w = [ ][ ] fm Ym(x) f n Y n (x) w(x) m=1 = n,m f m f n (Y m,y n ) w = f n 2. (2.40) This is Prsevl s identity for the cse of the inner product with weight w. 2.8 Lest squres pproximtion In the rel world, our computers hve finite power nd memory, nd we typiclly don t hve the resources to hndle very lrge number of eigenfunctions. So in prcticl pplictions, it s importnt to know how ccurtely we cn represent function by expnding it in just limited, incomplete set of eigenfunctions. Suppose we consider the finite sum g(x) c i Y i (x) (2.41) tht just includes some finite number n of the eigenfunctions, for some constnts c i. We sk how we should choose these constnts if we wnt g to represent given function f(x) 28
11 s closely s possible. One notion of wht we men by closely is to sk tht the distnce between g nd f should be minimized in the (, ) w norm, or in other words tht (f g, f g) w = f g 2 w(x) should be mde s smll s possible by vrying the c i s. Using the definition (2.41) of g(x) nd the expnsion (2.24) of f we hve c k (f g, f g) w = [(f, f) (g, f) (f, g)+(g, g)] c k [ = c k c i f i + f i c i c i 2 ] (2.42) nd likewise c k = f k + c k (f g, f g) w = f k + c k, (2.43) where the evlution of (g, g) uses Prsevl s identity. These derivtives vnish iff c k = f k, nd since 2 c j c k (f g, f g) w = 0 = 2 c, while j c k 2 c j c (f g, f g) w = δ j,k 0 k the extremum is indeed minimum. Therefore, if we wish to pproximte function f(x) by representing it s liner combintion of just few eigenfunctions of some Sturm Liouville opertor, the best we cn do is to choose the coefficients c k =(Y k,f) w exctly s in its true expnsion. This is lso n importnt first step in checking tht the expnsion (2.24) of f(x) does indeed converge on the originl function s the number of included terms tends to infinity. 29
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