a0 a1 star a2a3a4a5a6a7a8a9a10 4 star a11a12a13a14a15a16a17a18a19a15a16a20a21a22a23a24a25 a26a27a28a29 1 l 'lC {o( 11 1 l 'lC {o( y3 §? fi??n A?;. ' §‰)flK§0 ?)? ‰)fl K ?k {§'lC {'?? {§ ,k ‰ ?^^ §~X§ ? § ‰) ^ · 5 §ˇd‰)flK ) kU\5'31 o ￥§? Q?( N ?)L §§' ??){?u‰)flK ?'AO·§Q? ( 14.1!)?? {·?U ? H/A^u?) ' §‰)flK§3n § ?ue A flK 1. flK·? ‰k)§ ?{‘§3 o^ e§ flK ‰k)? 2. ‰)flK )·? ‰ –U , | …?—m§ ?{‘§3 o^ e§ …??·· ? 3. …??··? ‰ k 5' 3? ￥§? ln ￡ ?A flK§l 'lC {C‰ j￠ n ? :' ,§ ‘5§?p0 ·?'^ '3 nflK￥§? ^ ·U v ' x18.1 S¨ m 12 x18.1 S¨ m 3? K ‰′ n ￥ mV §§ (￥ )^x; y, ￠￠￠L?'? –rn ￥ m￥￥ Vg 2 n ￥ m' d§k‰′n ￥ S¨' ?u￠n ￥ m(=K ￠? )§3 ‰ |?fei, i = 1; 2; ￠￠￠ ; ng § m￥ ?? ￥ x –^§3? |? K( I) x1; x2; ￠￠￠ ; xnL?§ x= x1e1 +x2e2 +￠￠￠+xnen = nX i=1 xiei: ?u m￥ ￥ x y§ ~ S¨‰′ (x; y) = x1y1 +x2y2 +￠￠￠+xnyn = nX i=1 xiyi: ?· ￠?'w,k (x; y) = (y; x) (x; x) ? 0; ? § = x= 0 § k(x; x) = 0'3d?: § –‰′￥ x kxk kxk = (x; x)1=2: ?uEn ￥ m§XJE 3 aS¨‰′§N·w §? ￥ U · ￠?' –￥ E·￠?§ 3 – ‰′ cJe§rS¨‰′?U (x; y) = x?1y1 +x?2y2 +￠￠￠+x?nyn = nX i=1 x?iyi; ￥x?i·xi E 'w,§3E￥ m￥§ (x; y) = (y; x)?: ? S¨Vgw,·n ￥ I¨ { 2' ?H ? §AO·￥ S¨?w 6u? '? ,I lS¨ ? U‰′￥? § ' l ?nz S¨‰′(– ? S¨?n)' ‰′1 (‰′3￠?‰E? K )￥ m￥￥ x y S¨(x; y)·§ I …?§ v 1. (x; y) = (y; x)?? 2. (fix+fly; z) = fi?(x; z)+fl?(y; z)§ ￥fi fl ··?? K I ? 3. ?u? x§(x; x) ? 0? = x= 0 §(x; x) = 0' ~1 e x= 0 BB BB B@ x1 x2 ... xn 1 CC CC CA y= 0 BB BB B@ y1 y2 ... yn 1 CC CC CA x18.1 S¨ m 13 ·￠? ￥ §P ( ‰ )? §? Pii ￠?§K ‰′￥ x y S¨ (x; y) = ? x1; x2; ￠￠￠ ; xn · 0 BB BB B@ P11 0 ￠￠￠ 0 0 P22 ￠￠￠ 0 ... ... 0 0 ￠￠￠ Pnn 1 CC CC CA 0 BB BB B@ y1 y2 ... yn 1 CC CC CA: ~2 ￠C t ?kEX? ? “ 8 §3? “\{–9? “ E? ?{e ? E￥ m' b 0 ? t ? 1'ex(t) y(t)·d￥ m￥ ￥ (=? “)§K§ S¨ –‰′ (x; y) = Z 1 0 x?(t)y(t)‰(x)dt; ￥fi …?‰(x) ? 0 6· 0' § Aˇ /·‰(x) · 1§ (x; y) = Z 1 0 x?(t)y(t)dt: S¨?n￥ 11^ ?§ –w § ·￠ ‰E ￥ m§ ￥ §g S¨o··￠￠￠??§? 13^ ?￥ “ k??′′' 3d?: § r (x; x)1=2 = kxk ? ￥ x (=￥ x / 0)' l ?S¨?n￥ 11 12^ ?§ (x;fiy) = fi(x; y): ˇd kfixk = (fix; fix)1=2 = h fifi?(x; x) i1=2 = jfijkxk: ? "￥ –§ ? / 0 ￥ §‰? 8 z ￥ § ? x kxk; x kxk · = 1: F‰′ S¨ ￥ m? S¨ m' F kS¨ ￠￥ m? Ap m(Euclidean space)? F kS¨ E￥ m? j m(unitary space)' 3 ? S¨‰′ § – \￥ Vg' x18.1 S¨ m 14 F = (x; y) = 0 § ￥ x; y ' F"￥ ? ￥ ' F? | 5?’￥ § ˇLIO ‰? ' ?| 5?’ ￥ fy1; y2; y3; ￠￠￠g§ 5| x1 = y1; x2 = y2 +fi21x1; x3 = y3 +fi31x1 +fi32x2; ... ?§ · (x1; x2) = (x1; y2 +fi21x1) = (x1; y2)+fi21(x1; x1) = 0; (x1; x3) = (x1; y3 +fi31x1 +fi32x2) = (x1;y3)+fi31(x1; x1) = 0; (x2; x3) = (x2; y3 +fi31x1 +fi32x2) = (x2;y3)+fi32(x2; x2) = 0; ... ?– fi21 = ?(x1; y2)(x 1; x1) ; fi31 = ?(x1; y3)(x 1; x1) ; fi32 = ?(x2; y3)(x 2; x2) : ?H (J· fijk = ?(xk; yj)(x k; xk) : ? ‰? Schmidt z' ‰′2 e?u?k i j§(xi; xj) = –ij§K?￥ |fx1;x2;￠￠￠g· 8 ' 8 ￥ ‰· 5?’ §?·ˇ XJ § 5| ?"￥ § fi1x1 +fi2x2 +fi3x3 +￠￠￠ = 0; K ‰k fij = 0; j = 1;2;3;￠￠￠ : ?–n ￥ m￥ ? |n 8 ￥ – ?d m ?§? 8 ?(‰? IO?)' J 8 ?§? 3n ‰￠^ § k4 ? 5' x18.1 S¨ m 15 3n ￥ mV￥§ k | 8 ￥ fx1; x2; ￠￠￠ ; xkg; k ? n: K?u?? ￥ x2 V § ? fii = (xi; x)'w,§A k x? kX i=1 fiixi 2 ·?x? kX i=1 fiixi; x? kX i=1 fiixi · ? 0: , ?§ ? x? kX i=1 fiixi; x? kX i=1 fiixi · = (x; x)? kX i=1 fi?i(xi; x)? kX i=1 fii(x; xi)+ kX i;j=1 fi?ifij(xi; xj) = (x; x)? kX i=1 fi?ifii ? kX i=1 fiifi?i + kX i;j=1 fi?ifij–ij = (x; x)? kX i=1 fi?ifii; u·§ ? “ (x; x) ? kX i=1 flfl(x i; x) flfl2; = kxk2 ? kX i=1 flfl(x i; x) flfl2; ? “? Bessel “' Bessel “ ? ·Schwarz “ ex; y·S¨ m￥ ￥ §K j(x; y)j?kxk￠kyk: XJy·"￥ §y = 0§ “￥ ??'XJy6= 0§K 3Bessel “￥ k 1§ x1 =y=kyk§u· k fl flfl fl x; ykyk ?flfl flfl 2 ?kxk2; Schwarz “=y' ‰′3 3k ￥ m￥§XJ | 8 ￥ (? 8 ￥ 8)§? ?3, 8 ￥ 8 ￥§K?T 8 ￥ 8· ' F3k ￥ m￥§ 8 ￥ 8￥￥ ?7, m ? ' x18.1 S¨ m 16 F￠SflK￥§ ? ·k ￥ m ?§ · ˇL? | 8 ￥ ( |Aˇ 5?’￥ |)5 m ?§ ?? ￥ m | ?' F3 S¨ mV￥§k | 8 ￥ fxi;i = 1; 2; ￠￠￠ ; kg; §·? §· ~y￠ flK' ~^ O{ke A 1. = x= 0 §(xi; x) = 0; i = 1; 2; ￠￠￠ ; k' 2. ?u?? x2 V § kx= kX i=1 (xi; x)xi' 3. Bessel “￥ ??§=?u?? x2 V § k kxk2 = kX i=1 j(xi; x)j2: 4. Parseval §§???§=?u?? x;y2 V § k (y; x) = kX i=1 (y; xi)(xi; x): § · 8 ￥ | ?'7 ^ §ˇ · d ' x18.2 …? m 17 x18.2 …? m …? m· aAˇ ￥ m m ·…?§ ( /‘§·‰′3 ‰? m( (‰ § 4?ma ? x ? b) E …?f(x)§? ¨' Z b a flflf(x)flfl2dx 3(/… ?f(x)? ¨0)' F‰′ f1 f2 \{f1 +f2 · …? \§ (f1 +f2)(x) = f1(x)+f2(x); F f E?fi ??fif· (fif)(x) = fif(x); ? ? ¨…? 8 §?u\{ ??· 4 §ˇd ( ? ￥ m' AO·§ˇ flflf 1(x)+f2(x) flfl2 +flflf 1(x)?f2(x) flfl2 = 2￡jf 1(x)j2 +jf2(x)j2 ?; ?–§ ? ¨…? E·? ¨ § flflf 1(x)+f2(x) flfl2 ? 2￡jf 1(x)j2 +jf2(x)j2 ?: ‰′4 f1(x) f2(x)·…? m￥ …?§§ S¨· (f1; f2) = Z b a f?1(x)f2(x)dx: dduu flflf 1(x) flfl2 +flflf 2(x) flfl2 ?2flflf 1(x) flfl￠flflf 2(x) flfl = ￡jf1(x)j?jf2(x)j?2 ? 0; ˇd fl flf?1(x)f2(x)flfl = flflf1(x)flfl￠flflf2(x)flfl? 1 2 ￡jf 1(x)j2 +jf2(x)j2 ?; ?–¨' Z b a flflf? 1(x)f2(x) flfldx 3'qˇ flfl fl Z b a f?1(x)f2(x)dx flfl fl? Z b a flflf? 1(x)f2(x) flfldx; ?–§ f1(x) f2(x)? ¨§@o§ S¨ ‰ 3' ￡ eS¨?n￥ n^ ?§c ^·w, v ' §?u m￥ ??… ?f(x)§ k(f; f) ? 0'3d?: § –‰′…?f(x) / 0 kfk = (f; f)1=2; ? …?f(x) ?' x18.2 …? m 18 FflK· 3? S¨‰′e§XJ(f; f) = 0§f(x)? 3 ?m ?? 0' fl￠·§f(x) –3k : 0§ ? 0 …? ? ?K ¨' §?– E –k(f; f) = 0' FO(/‘§XJ(f; f) = 0§Kf(x) –3 " :8 " '?– U ‘(f; f) = 0 ?Xf(x)A ?? 0' FXJ ^2′ "…? Vg§r? A ?? 0 …?? "…?§@o§?p‰′ S¨ ? S¨?n￥ 13^ ?' ? §‰′4 …?S¨ ‰′ (? S¨?n ?' …?S¨ ‰‰′ –? 2 (f1; f2) = Z b a f?1(x)f2(x)‰(x)dx; ￥‰(x) ? 0 6· 0'? §k’?“ I A ?U'AO·§’u…? ? ¨ ? AT?U ?¨' Z b a flflf(x)flfl2‰(x)dx 3' 3‰′ …? S¨ § –‰′…? 5 8 5§? ?…? 8 8 Vg' e…?f(x) g(x) v (f; g) · Z b a f?(x)g(x)dx = 0; K?§ ·(3?m[a; b] ) 'e…?f(x) §g S¨ (f; f) · Z b a f?(x)f(x)dx = 1; ‰= kfk = 1; K?f(x)·8 z ' e?u…?8 ffig§ k (fi; fj) · Z b a f?i (x)fj(x)dx = –ij; K?d…?8 · 8 ' ~3 …?8 'einx=p2…; n = 0;§1;§2;￠￠￠“3?m[?…; …] · 8 ' 8 …?8 5Vg'XJ?u(…? m￥ )??…?f(x)§o –L?? 8 …?8ffi; i = 1;2;￠￠￠g 5| f(x) = 1X i=1 cifi(x); (z) K? 8 …?8ffi; i = 1;2;￠￠￠g· ' x18.2 …? m 19 8 …?8 5Vgo· ??…?·? –UT…?8—m ?X ' F1 § ‘5§? …?8AT?k??? …?§?K(z)“ U???f(x) ? ?'? fl￠w ? §…? m·?? ￥ m' F1 §(z)“AT??m[a; b]S z :x ??§‰ ‘§?u?m[a; b]S z :x§?? 1P i=1 cifi(x) AT′?uf(x)'??′?5? ˉ:′?' F 2′"…? Vg ?A§ –r(z)“n) m 2′ "… ?§ ?{‘§r?? 1P i=1 cifi(x)n) ? ′?uf(x)§= limn!1 Z b a flfl flf(x)? nX i=1 cifi(x) flfl fl 2dx = 0: (#) F1n§d…?8ffi; i = 1;2;￠￠￠g 8 5§ ? ci = Z b a f?i (x)f(x)dx = (fi; f): (~) F1o§N·y? Z b a flfl flf(x)? nX i=1 cifi(x) flfl fl 2dx = (f; f)? nX i=1 c?i(fi; f)? nX i=1 ci(f; fi)+ nX i=1 flflc i flfl2 = (f; f)? nX i=1 flflc i flfl2; ˇd§ …?8ffi; i = 1;2;￠￠￠g· §@o§ (#)“§ k (f; f) = 1X i=1 flflc n flfl2 = 1X i=1 flfl(f i; f) flfl2: ? ·…?8ffi; i = 1;2;￠￠￠g 5’X§‰?Parseval §' …?8ffi; i = 1;2;￠￠￠g 5 , ?L /“' (~)“ \(z)“§ k f(x) = 1X i=1 Z b a f(x0)fi(x)f?i (x0)dx0 = Z b a f(x0) " 1X i=1 fi(x)f?i (x0) # dx0: x18.2 …? m 110 F? (J?u( ‰…? m￥ )??…?f(x) ??§ 1X i=1 fi(x)f?i (x0) = –(x?x0): F3d?: §q – (f; g) = 1X i=1 (f; fi)(fi; g): r…?8ffi; i = 1;2;￠￠￠g ^ §b §· 8 § ‰ §E,`a ^? …?8 5| 1P i=1 aifi(x)5%Cf(x)'y3 flK· X J| X?ai ( n? ’)§ – Z%C§? f(x)? nX i=1 aifi(x) 2 ·Z b a flfl flf(x)? nX i=1 aifi(x) flfl fl 2dx 4 ” Parseval § y?§ –? Z b a flfl flf(x)? nX i=1 aifi(x) flfl fl 2dx = (f; f)? nX i=1 a?i(fi; f)? nX i=1 ai(f; fi)+ nX i=1 flfla i flfl2 = (f; f)? nX i=1 a?ici ? nX i=1 aic?i + nX i=1 a?iai = (f; f)+ nX i=1 flfla i ?ci flfl2 ? nX i=1 c?ici; ˇd§ ai = ci · (fi; f) § ‰ 4 § (f; f)? nX i=1 flflc i flfl2 ? 0; § X ?n O\§ 5 ' ? X §ok (f; f) ? 1X i=1 flflc i flfl2: ? —·…? m￥ Bessel “' ?Au…?8· /' …? m 5Vg'XJd mS …?|? CauchyS 4 E –3 T mS§K?T m ' ? ¨…? ? m· ' ˇ~§r S¨ m? Hilbert m'? Vg§3 n?￥k2 A ^'e? ? §￠S ·3Hilbert m S?1 ' x18.3 g ? flK 111 x18.3 g ? flK ‰′5 L M ‰′3 ‰…? mS ( ') ?§e?uT…? mS ?? …?u v§ k (v; Lu) = (Mv; u) = Z b a v?Ludx = Z b a (Mv)?udx; K?M·L ?' ~4 eL= ddx§u· Z b a v?dudxdx = v?u flfl fl b a ? Z b a dv? dx udx: ?–§ u v v>.^ y(a) = y(b) § ddx ?·? ddx' ‰′5￥ ?M L·p ?§ˇ XJM·L ?§K?u??… ?u v§ k Z b a v?Mudx = ?Z b a (Mu)?vdx ?? = ?Z b a u?Lvdx ?? = Z b a (Lv)?udx; ?–§L ·M ?' ~5 L= d 2 dx2 §N·y? Z b a v?d 2u dx2dx = h v?u0 ?(v?)0u ib a + Z b a ?d2v dx2 ·? udx: ?–§ …?u v v ! !na>.^ fi1y(a)+fl1y0(a) = 0; fi2y(b)+fl2y0(b) = 0 ( ￥jfi1j2 +jfl1j2 6= 0; jfi2j2 +jfl2j2 6= 0)‰–ˇ^ y(a) = y(b); y0(a) = y0(b) § d 2 dx2 ? ·§g ' ‰′6 e ?L ? ·§g §=?uT…? mS ?? …?u v§ k (v; Lu) = (Lv; u) = Z b a v?Ludx = Z b a (Lv)?udx; x18.3 g ? flK 112 K?L·g ?' ~6 3 ~4 ^ e§ ?i ddx ·g ?' Z b a v? idudx ? dx = ?i Z b a dv? dx udx = Z b a idvdx ?? udx: ? g 5§o· ‰‰ …? m?X3 'ˇ~§? o· ? ? …?‰′3 ‰‰ ?m § ? …? kv oY5(~X§?u ' ?§ ?…? ?o Y§ 'aoY?XJ·?.?m§K ?…?? ¨)§ ˇd§￠S o· uHilbert m'? § ? ? …? v ‰‰ >.^ §=o· 3Hilbert m￥ ‰f mS' U?l>.^ 5? ? g 5' ?§ ?u, a…?·g § ?u, a…?§ U ·g ' ~7 L= i ddx§ >.^ ? /“ y(b) = fiy(a); fi (E)~?: u· Z b a v?idudxdx =iv?u flfl fl b a ?i Z b a dv? dx udx =i(fifi? ?1)u(a)v?(a)+ Z b a idvdx ?? udx: ?– k>.^ ￥ fi vfifi? = 1 § ?i ddx ·g ' ‰′7 L g ?§K § Ly(x) = ?y(x) ? g ? flK' ?pvk?( g>.^ §·ˇ §fi? ?3g ?L ‰‰′￥ ' g ? flK ke A ? ? 5 F5 1 g ? 7, 3'( y) F5 2 g ? 7 ￠?' y ˇ Ly = ?y; x18.3 g ? flK 113 E (Ly)? = ??y?: duL·g ?§?– Z b a ￡y?Ly ?(Ly)?y?dx = (????)Z b a yy?dx = 0: qˇ Z b a yy?dx 6= 0§?– ? = ??; =y ? ￠?' F5 3 g ? …? k 5§=?A …? ‰ ' y ?i ?j· §?A …? yi yj§ Lyi = ?iyi; Lyj = ?jyj: 5? ?i;?j ￠?§u· Z b a ￡y? iLyj ?(Lyi) ?yj?dx = (?j ??i) Z b a y?i yjdx: ˇ ?i 6= ?j§?– Z b a y?i (x)yj(x)dx = 0: ? y? …? 5' du …?· g ' §3 g>.^ e )§?– …??– "~? ˇfE,· …?'? –? J? ~?ˇf§? ?u?? ?i§ k Z b a y?i (x)yi(x)dx = 1: ? · 8 …?|' Z b a y?i (x)yj(x)dx = –ij: F5 4 g ? …?( N) ? …?|§=?? 3?m[a;b]￥ koY ?! v g ?L >.^ …?f(x)§ U … ?fyn(x)g—m ? ′? ?? f(x) = 1X n=1 cnyn(x); (#) x18.3 g ? flK 114 ￥ cn = Z b a f(x)y?n(x)dx Z b a yn(x)y?n(x)dx : AO·§XJ …?|·8 z §K “￥ '1 1§—m /“ \{ '( y) § 8 …?| 5 –L?? 1X n=1 yn(x)y?n(x0) = –(x?x0): Fd ? 5 3 4 –w § …?? 8 z§K …? N ? 8 …?8'ˇd§ !￥k’ 8 …?8 ? ? ^' F?p6 K ? U5§=?Au Uk ( 5?’ ) …?§ˇ U? *d '?? / 318.5!? ' =?Xd§o – ^Schmidt z ‰( 18.1!)? z§ˇ E, – 8 …?8' Ffl￠ § ? —m^ – ?u??3[a;b]￥? ¨ …?§(#)“3? ′? lim N!1 Z b a flfl flf(x)? NX n=1 cnyn(x) flfl fl 2dx = 0 ?′eE,??' ‘5§ ?’ug ? 35 …? 5 ? § 5 A ?' (?m?.‰ ?.?‰·3k.?m ' §k :) (?mk.§ ' §3?m ? :) flK? ? /' du ?vk k’ y?§?– Q?'? a flK' § Qa B§3k’ La￥ ^ k.?m /“' x18.4 Sturm–Liouville. § flK 115 x18.4 Sturm–Liouville. § flK 3c?A ￥§? ? LA ~ ' § flK' 9 ' §k X00 +?X = 0; d dx ?? 1?x2 · dy dx ? + h ?? m 2 1?x2 i y = 0; 1 r d dr rdRdr ? + h ?? m 2 r2 i R = 0: § –8B e? /“ d dx ? p(x)dydx ? +[?‰(x)?q(x)]y = 0: (#) ??a. §? Sturm–Liouville.({?S–L.) §' F rS–L. §￥ …?p(x); q(x) ‰(x) ·￠…?§ v7 oY 5 ?' F ‰(x)§? ?…?' F ?…?‰(x) =~? § – 1' F ~? ?…?§ –5 u ?? IX ?^(? –lLaplace ? NL “￥Jˇ ?…? l,?l ‘§§ N I ·TC … ?' –? 5 u m A ￡a !5)§ U5 uflK? 9 n5 !5(~X§ ' !)'ˇd§ ? ?’% nflK § b ‰(x) ? 0§ §A 0' ;n§ – ? ? L·? ddx ? p(x) ddx ? +q(x) (>) P '? §S–L. § –U ? Ly(x) = ?‰(x)y(x): (##) S–L. §N\ ? >.^ § ?S–L. § flK'?? '?u , ?§ vS–L §9 A >.^ ") · …?' l ' §5w§du‰(x) y§S–L. §(#)‰(##)?w u § L0u(x) = ?u(x): (z) ·§ˇLC C u(x) = p ‰(x)y(x); x18.4 Sturm–Liouville. § flK 116 – §(#)z (z)§ ￥ L0 = ? ddx ? `(x) ddx ? +?(x); `(x) = p(x)‰(x); ?(x) = ? 1p‰(x) ddx h p(x) ddx 1p‰(x) i + q(x)‰(x): §(z) , ·S–L. § § L· ?Aˇ S–L. § § ?…? 1 S–L. §' ‰n1 ?u??…?u1(x) u2(x)§ k u?1L0u2 ??L0u1￠?u2 = ? ddx h `(x) ? u?1du2dx ?u2du ?1 dx ·i ; ￥ L0 = ddx ? `(x) ddx ? ??(x): 2 ?L' L·? ddx ? p(x) ddx ? +q(x) ˇ 3C u1(x) = p‰(x)y1(x); u2(x) = p‰(x)y2(x) e§k u?1L0u2 ??L0u1￠?u2 = y?1Ly2 ?(Ly1)?y2: ?–§?u??…?y1(x) y2(x)§ y?1Ly2 ?(Ly1)?y2 = ? ddx h p(x) ? y?1 dy2dx ?y2dy ?1 dx ·i : ‰n2 3>.^ `(x) ? u?1du2dx ?u2du ?1 dx ·flflfl fl b a = 0 e§ ?L0·g ' ‰n1 ‰n2( 5§?= 3>.^ p(x) ? y?1 dy2dx ?y2dy ?1 dx ·flflfl fl b a = 0 (~) e§ ?L ·g ' 3 o ?e§>.^ (~)U ??” x18.4 Sturm–Liouville. § flK 117 F1 ? ?·3 :x = a x = b§ k p(x) ? y?1 dy2dx ?y2dy ?1 dx · = 0: (M) 1. XJy1 y23 : v1 ! !na>.^ §K(M)“??' ~X§3x = a:§ fiyi(a)?fly0i(a) = 0; i = 1;2; fi fl ( )￠?§ E § – fiy?i (a)?fly?i0(a) = 0; i = 1;2: dufi fl U 0§ k flfl flfly?1(a) y?01 (a) y2(a) y02(a) flfl flfl = y?1(a)y02(a)?y2(a)y?01 (a) = 0: 2. XJp(x) 3 :(~X, x = a) ? 0, ? x = a :· § :. b‰p(x); q(x) ‰(x) v ‰ ? ,? x = a:· § K :, 1 )k.,1 ) ?.. 3N\ k.^ K?.) , k p(x) ? y?1 dy2dx ?y2dy ?1 dx ·flflfl fl x=a = 0: F, ? ?· p(x) ? y?1 dy2dx ?y2dy ?1 dx ·flflfl fl x=a = p(x) ? y?1 dy2dx ?y2dy ?1 dx ·flflfl fl x=b ; 0§? (M)“ ??'XJ p(a) = p(b); q(a) = q(b); ‰(a) = ‰(b); ? yi(a) = yi(b); y0i(a) = y0i(b); i = 1;2; w, – v? ?'? ·? L –ˇ^ /' ~X p(a) = 0; p0(a) 6= 0; ‰(x) (x?a)q(x) 3x = a:) ‰ p(a) = 0; p0(a) = 0; p00(a) 6= 0; ‰(x) q(x) 3x = a:) ; ?3? ? L ￠SflK￥·U v ' x18.5 Sturm–Liouville. § flK {?y 118 x18.5 Sturm–Liouville. § flK {?y ?A k ( 5?’ ) …? y §? {?‰ z' duS–L. §· 5~ ' §§?–§?A ? Uk ( 5?’ ) …?' 3 o^ e§S–L. § flK·{? ”3 o^ e· {? ” ‰n3 XJS–L. § flK …?·E § ￠ J 5?’§ Kd flK· ?{? ' y ‰n? § …?y(x)·E § ￠ J 'O f(x) g(x)§ y(x) = f(x)+ig(x): KS–L. § – ? L(f +ig) = ?‰(f +ig): du ?L·￠ ?§ ?…?‰(x)·￠…?§ ? ￠?§ “'O’ ￠ J § Lf = ?‰f; Lg = ?‰g: ?‘?f(x) g(x) ·?Au ? …?§§ 5?’53‰n fi ^ ￥fi? ?( ‰' 7Ly?f(x) g(x) v flK >.^ '? 5? >.^ · 5 g §? U y X? ·￠?§u·3>.^ ￥ 'O’ ￠ J = ' ‰n4 y1(x) y2(x) ·S–L. § flK Ly(x) = ?‰(x)y(x): ￠ 5?’ …?§? 3x = a x = b: v>.^ p(x) ? y?1 dy2dx ?y2dy ?1 dx ·flflfl fl x=a = p(x) ? y?1 dy2dx ?y2dy ?1 dx ·flflfl fl x=b = 0; (#) Ky1(x) y2(x) U?Au ?' y ^ y{' y1(x) y2(x)?Au ?§ Ly1 = ?‰y1; Ly2 = ?‰y2; ˇd y1Ly2 ?y2Ly1 = 0; 5?y1(x) y2(x) ·￠…?§y?1(x) = y1(x)§y?2(x) = y2(x)§?– !‰n1 § k d dx ? p(x) y1dy2dx ?y2dy1dx ?? = 0: u· p(x) y1dy2dx ?y2dy1dx ? =~?C: x18.5 Sturm–Liouville. § flK {?y 119 ‰n fi ^ (#)§ Ak p(x) y1dy2dx ?y2dy1dx ? · 0: ˇ p(x) 6· 0§ k y1dy2dx ?y2dy1dx · 0; = W￡y1(x); y2(x)?· flfl flfly1(x) y2(x) y?1(x) y?2(x) flfl flfl· 0: ?‘?y1(x) y2(x) 5 ’§ fi ^ g?'?–y1(x) y2(x) U?Au ' ? ‰n w ? §3 ! !na( g)>^ ‰( )k.^ e§S–L. § flK U·{? ' ?? L A?a. >.^ § k3–ˇ^ e§ …?3?m z :? v(#)§ k U u){?yy ' x18.6 lSturm–Liouville. § flKw'lC { 120 x18.6 lSturm–Liouville. § flKw'lC { E–u ?flK ~' ?u ‰u gd ?§‰)flK· @2u @t2 ?a 2@2u @x2 = 0; 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t > 0; uflflt=0 = `(x), @u@t flfl fl t=0 = ?(x), 0 < x < l: 18.3! 18.4! ? §XJ 3 S–L § flK LX = ?‰X; X(0) = 0; X(l) = 0; @o§du§ >.^ ‰)flK >.^ /“ §ˇd§ – ‰)flK )u(x;t)U …? NfXn(x); n = 1; 2; 3; ￠￠￠g ( B §b …? fi8 z)—m§ u(x;t) = 1X n=1 Tn(t)Xn(x): ?p§ …?| 5 ?‰5 ^' y 1P n=1 Tn(t)Xn(x)U ′ ?( ·? ′?) )u(x;t)§?p ? 7LH9 …?' – ?nd/V eZZ …?' ?K§?+3/“ q EU? ??/)0§ § U′? )u(x;t)' )“ \ §§k 1X m=1 T00m(t)Xm(x)?a2 1X m=1 Tm(t)X00m(x) = 0: ^X?n(x)? “ §, 3?m[0; l] ¨'§ T00n(t)?a2 1X m=1 (Xn; X00m)Tm(t) = 0; m = 1;2;3;￠￠￠ : 2 —'^ U? | …?—m§ Tn(0) = (Xn; `); T0n(0) = (Xn; ?): XJU ? Tn(t)§ ￡ )“￥§ , ? ‰)flK )u(x;t)' ?p ?) ·’u …?fTn(t);n = 1;2;3;￠￠￠g ~ ' §|' ‘ 5§? ·’ (J ' x18.6 lSturm–Liouville. § flKw'lC { 121 ? g>.^ 3'lC {￥ ?‰5 ^ § g § / HA ) 'XJr‰)flK￥ §U @2u @t2 ?a 2@2u @x2 = f(x;t); @o§y3w5§?)L§?vk § ? 3u § g f(x;t) U …?—m§u·§ g ~ ' §|C? g §| T00n(t)?a2 1X m=1 (Xn; X00m)Tm(t) = (Xn; f); m = 1;2;3;￠￠￠ : f(x;t) x …?§ fXn(x)gA ?u …? m' § Jn)§XJ‰)flK >.^ · g § ?k7L >.^ g z' F y3 §? X?' g>.^ 3'lC {￥ ?‰5 ^' F?u …?§ ?§ v ‰)flK >.^ §?u§? v ' § · ?7L·S–L. §§ ?u § N/“?vk ' F …? v ' § §fXn(x); n = 1;2;3;￠￠￠g /“ §ˇ ’ uTn(t) ~ ' §| /“ §? Tn(t) ' F‰)flK ) 3 5§ y ? ) ' FI 5?§? ·? ~ ' §| ·N·?) ' F3￠S?)L§￥§ I T / J …?|fXn(x); n = 1;2;3;￠￠￠g§? Tn(t) ?)flK? U/{ ' F { / · ? (Xn; X00m) = 0; n 6= m; ?{‘§ (Xn; X00m) = ??m–nm: ˇd§Tn(t) v~ ' § T00n(t)+a2?nTn(t) = 0 ‰ T00n(t)+a2?nTn(t) = (Xn; f): ·~ ' §|' F ? …?|fXn(x); n = 1;2;3;￠￠￠g· 8 § (Xn; Xm) = –mn; x18.6 lSturm–Liouville. § flKw'lC { 122 a ? du (Xn; X00m) = ??m(Xn; Xm) = (Xn; X00m +?mXm) = 0 ?? X …?A v~ ' § X00n(x)+?nXn(x) = 0; ? ·? ^'lC { IO ‰ ' §' ?–§'lC { ?? J? J …?| Z Y' …? 5·3n y ‰ – ‰)flK )UT …?|— m(?·k^ §‰)flK …? v g>.^ )§ ^/ A gflK …?0K y – B/? —mX?(￠S · …?)§ y ??){3￠^ 1155' 3 \n) 'lC { ￠ §?) ' §‰)flK … g d'? =Ny3? ? ?u ?a. ‰)flK( § g‰ g§>.^ g‰ g) ?)k \ @￡§ Ly3 ?u, ‰)flK ?)g ·' ~X§?u ￥S ?‰flK§ r2u = f; x2 +y2 +z2 < 1; uflflx2+y2+z2=1 = 0;? ^￥ IX?),U L {,A u(r; ;`)U/ A gflK …?0Yml ( ;`) —m§ u(r; ;`) = 1X l=0 lX m=?l Rlm(r)Yml ( ;`); , § § 1 r2 d dr ? r2dRlmdr ? ? l(l +1)r2 Rlm(r) = ZZ Ym?l ( ;`)f(r; ;`)sin d d` ( ￥ ¨'H9 4…?N ) >.^ Rlm(0)k.; Rlm(1) = 0 ? Rlm(r)' ?p ~ ' §· CX? g §§·?U N·?) ?u g N/“' x18.6 lSturm–Liouville. § flKw'lC { 123 U c? ' §XJU? | …?§ § vd‰)flK g>.^ §@ o§ – u(r; ;`)U? | …?—m' N‘5§ –k?) flK ?r2w = ?w; x2 +y2 +z2 < 1; wflflx2+y2+z2=1 = 0; ?nl = k2nl; n = 1;2;3;￠￠￠ ; l = 0;1;2;￠￠￠ …? wnlm(r; ;`) = jl(knlr)Yml ( ;`); ￥knl·l ￥Bessel…?jl(x) 1n ":' 5?§?p m = 0;§1;￠￠￠ ;§l?’§ ?{‘§d flK· ?m{? §{? 2l +1' , § u(r; ;`)Uwnlm(r; ;`)—m§ u(r; ;`) = 1X n=1 1X l=0 lX m=?l cnlm jl(knlr)Yml ( ;`); \ ' §§ ?k2nlcnlm Z 1 0 j2l(knlr)r2dr = Z 1 0 jl(knlr)r2dr ZZ Ym?l ( ;`)f(r; ;`)sin d d`; ￥Bessel…? ‰′9k’(J§ –? Z 1 0 j2l(knlr)r2dr = …2k nl Z 1 0 J2l+1=2(knlr)rdr = …4k nl ￡J0 l+1=2(knl) ?2 = 12￡j0l(knl)?2; ?– cnlm = ? 2 k2nl￡j0l(knl)?2 ￡ Z 1 0 jl(knlr)r2dr ZZ Ym?l ( ;`)f(r; ;`)sin d d`: F3?) …? §fi?^ >.^ § )3= ￥ IX y –ˇ^ k.^ '? § y )u(r; ;`) v? >.^ ' F??){ ‘:· ? ? …? § I 2 ?)~ ' §' x18.6 lSturm–Liouville. § flKw'lC { 124 F?·–? ???—m d ' F (J u Rlm(r) U￥Bessel…?jl(knlr)—m' F?? { k 5 ?^u ? m ?‰flK§? ? A flKk )§$ ?0 · ' – {§ , – B/ 2 ?A /G n ? §‰ ?? ‰? '