搜索
WuChong-shi Sturm-Liouville §25.1 25.1 L M !#$%&(’())*+,-./!#$%&0123!u v+45(v,Lu)=(Mv,u)6Zbav∗Ludx=Zba(Mv)∗udx,78M9L&:;=25.1,L=ddx+.9Zbav∗dudxdx=v∗u ba−Zbadv∗dxudx.?@+Au vBCDEFGHy(a)=y(b)I+ddx&J)*9−ddx= 25.1K&)*M L9L J)*+M NOM9L&J)*+7-.01!u v+P5Zbav∗Mudx=Zba(Mu)∗vdx#∗=Zbau∗Lvdx#∗=Zba(Lv)∗udx,?@+LP9M&J)*=25.2 L=d2dx2+QRSTZbav∗d2udx2dx=hv∗u0−(v∗)0uiba+Zbad2vdx2∗udx.?@+A!u vBCDUVUWXEFGHα1y(a)+β1y0(a)=0,α2y(b)+β2y0(b)=0(YK|α1|2+|β1|26=0,|α2|2+|β2|26=0)Z[\GHy(a)=y(b),y0(a)=y0(b)I+d2dx2&J)*]9^_‘=WuChong-shi§25.1abcdefghijk2l 25.2,)*L&J)*]9^_‘+6-./!#$%&0123!u v+45(v,Lu)=(Lv,u)6Zbav∗Ludx=Zba(Lv)∗udx,78L9m:;=25.3 n4opqr&GHs+)*iddx]9_J)*=Zbav∗idudxdx=−iZbadv∗dxudx=Zbaidvdx∗udx.)*&_Jt+u9 &!#$vw x&=yz+{|u9}~•! & $ +•! 5D & t(nN+-.V ’()*+]}~!&V + ( NO9 F $+7}~! )+M + u9 .Hilbert#$= + }~•!CD &EFGH+6u9 Hilbert#$K& #$%= EFGH& ¡¢£)*&_Jt=3)*+q-.⁄X!9_J&+¥-.ƒX!+] 9_J&=25.4 L=iddx+§¤EFGH'“«‹&›fiy(b)=αy(a),α (fl)z..9Zbav∗idudxdx=iv∗u ba−iZbadv∗dxudx=i(αα∗−1)u(a)v∗(a)+Zbaidvdx∗udx.?@ 5EFGHK&αCDαα∗=1I+)*iddx–9_J&= 25.3 L _J)*+7 †Ly(x)=λy(x)8 _J)*&‡· ¶•=‚„”5T»…‰ ¿EFGH+9M ^ `´ˆ _J)*L& K˜=WuChong-shi¯˘˙¨ Sturm-Liouville˚¸ efghijk3l_J)*&‡· ¶• 5s˝˛3ˇ}&—‡t • 1_J)*&‡· =( S)• 2_J)*&‡· = M Ly=λy,'fl (Ly)∗=λ∗y∗. .L9_J)*+?@Zba[y∗Ly−(Ly)∗y]dx=(λ−λ∗)Zbayy∗dx=0. M Zbayy∗dx6=0+?@λ=λ∗,6S ‡· λ =• 3_J)*&‡·! 5 t+6- r‡· &‡·! = λi λj9 qÆ&23‡· +- &‡·! yi yj+Lyi=λiyi,Lyj=λjyj. 1ª‡· λi,λj +.9Zba[y∗iLyj−(Lyi)∗yj]dx=(λj−λi)Zbay∗iyjdx.M λi6=λj+?@Zbay∗i(x)yj(x)dx=0.‚ ]ST˜‡·!& t= .‡·!9 ¿’( † ¿EFGHs& +?@¤‡·! @3 ŁzM Ø 9‡·!={|] @ŒAº ‚3zM + -.013‡· λi+B5Zbay∗i(x)yi(x)dx=1.‚ ª&]93 æ ı=Zbay∗i(x)yj(x)dx=δij.WuChong-shi§25.1abcdefghijk4l• 4_J)*&‡·!(&p ) “3oł!ø+6013 $[a,b]K5 V U CD _J)*Lqr&EFGH&!f(x)+œ ߇·!{yn(x)} -§ & f(x)=∞Xn=1cnyn(x),(#)YKcn=Zbaf(x)y∗n(x)dxZbayn(x)y∗n(x)dx. 9+NO‡·!ø9 &+7 fiK&( 1+ &›fi« =( S)r + &‡·!ø&ołtP @ “∞Xn=1yn(x)y∗n(x0)=δ(x−x0).• &t 3 4 @ ª+ }¤‡·!ŒA +7‡·!&p ] “˜3oł& ! =M + K5 oł& ! &¢£œ Œ =•‚„ I t+6- .3‡· 5 3( t &)‡·!+M§ =‚ ›¤ 25.3 ¢£=¥6 N +u @ Schmidt& ( 18.1 ) +M§Ø @ ª3oł& ! =•! + & GH @# -.01 [a,b]K &!+(#)fi œ limN→∞Zba f(x)−NXn=1cnyn(x) 2dx=0&1 sØ “$=%&’¡+ ._J)*‡· & t ‡·!&ołt&¢£+‡¡ A (()&( $ FZ* F Z9 5F $ ’( †5(+) ()&( $5F+ ’( † $ (+)‡· ¶•‚2 ›=¥ . ”5 ‰5 &ST+?@P],- (‚2X‡· ¶•=§ + ˜./& 0+ 5 & /KB ˜5F $&›fi=WuChong-shi¯˘˙¨ Sturm-Liouville˚¸ efghijk5l§25.2Sturm–Liouville123 4 ˛5K+{|¢£6˛3z’( †&‡· ¶•=78&’( †5X00+λX=0;ddx 1−x2dydx+hλ−m21−x2iy=0;1rddrrdRdr+hλ−m2r2iR=0.^| @ 9 s &‹›fiddxp(x)dydx+[λρ(x)−q(x)]y=0.(#)‚ X:& †8 Sturm–Liouville:( 8S–L:) †=• ;S–L: †K&!p(x),q(x) ρ(x) = B9 !+§ BCD }& t}~=•ρ(x)+8 ˇ!=•Aˇ!ρ(x)=zI+ @' 1=• 4 z&ˇ!+ @¡?. @ ABw& (‚I @CLaplace)*& DfiKEFªˇ!&GH CI‡ ’+^JK˜ABLM N9/OP&!= @8 ¡?.#$&˛QR/& œSt)+P ¡?.¶•?78&TUt & œSt(nN+VM(W& œS)=M +]{|? X&TU¶•§Y+ ;Z ρ(x)≥0+§ + A 4 0= ˜ …&[\+ @]^)*L≡−ddxp(x)ddx+q(x)()&_‘=‚ +S–L: †] @a…“Ly(x)=λρ(x)y(x).(##)S–L: †b ŒA&EFGH+] “S–L: †&‡· ¶•=λ8 ‡· =-.⁄3‡· λ+CDS–L †8q &EFGH& Ł ]9‡·!=C’( †¡ + .ρ(x)&‰c+S–L: †(#)Z(##)Td r. †L0u(x)=λu(x).(z)WuChong-shi§25.2Sturm–Liouville˚¸ efghijk6l¥9+y6OPOeu(x)=pρ(x)y(x),] @¤ †(#) (z)+YKL0=−ddxφ(x)ddx+ψ(x),φ(x)=p(x)ρ(x),ψ(x)=−1pρ(x)ddxhp(x)ddx1pρ(x)i+q(x)ρ(x). †(z)A P 9S–L: †+ 69 f&S–L: †+ˇ! 1&S–L: †= g25.1-.01!u1(x) u2(x)+45u∗1L0u2− L0u1∗u2=−ddxhφ(x)u∗1du2dx−u2du∗1dxi,YKL0=ddxφ(x)ddx−ψ(x).hij;L=L≡−ddxp(x)ddx+q(x)M Oeu1(x)=pρ(x)y1(x),u2(x)=pρ(x)y2(x)s+5u∗1L0u2− L0u1∗u2=y∗1Ly2−(Ly1)∗y2.?@+-.01!y1(x) y2(x)+y∗1Ly2−(Ly1)∗y2=−ddxhp(x)y∗1dy2dx−y2dy∗1dxi. g25.2 EFGHφ(x)u∗1du2dx−u2du∗1dx ba=0s+)*L09_J&=¤ U1&k£ U2lmx¡+$6 ª EFGHp(x)y∗1dy2dx−y2dy∗1dx ba=0(~)s+)*LP9_J&=WuChong-shi¯˘˙¨ Sturm-Liouville˚¸ efghijk7l no ps+EFGH(~) “$q•r p9 s+x=a x=b+œ5p(x)y∗1dy2dx−y2dy∗1dx=0.(M)1.NOy1 y2 2s+œCDrUVUWXEFGH+7(M)fi“$=nN+ x=a++αyi(a)−βy0i(a)=0,i=1,2,α βœ ( ) +'fl + @ ‰αy∗i(a)−βy∗i0(a)=0,i=1,2. .α β rI 0+t5 y∗1(a)y∗01(a)y2(a)y02(a) =y∗1(a)y02(a)−y2(a)y∗01(a)=0.2.NOp(x) s+(nN+x=a)u 0,‚Ix=a+9 †&(+=Z p(x),q(x) ρ(x)CD &}~+ x=a+9 †& 7(++§ r 5F+rV F= b 5FGHv F w+]5p(x)y∗1dy2dx−y2dy∗1dx x=a=0.nNp(a)=0,p0(a)6=0,ρ(x) (x−a)q(x)œ x=a+ xZp(a)=0,p0(a)=0,p00(a)6=0,ρ(x) q(x)œ x=a+ x,‚ {|¢£6& ¶•K9 CD&=•ƒ p9p(x)y∗1dy2dx−y2dy∗1dx x=a=p(x)y∗1dy2dx−y2dy∗1dx x=b,¥ 0+‚I(M)fiP“$=NOp(a)=p(b),q(a)=q(b),ρ(a)=ρ(b), yi(a)=yi(b),y0i(a)=y0i(b),i=1,2,d ] @CD‚3}~=‚ 9¢£6&[\GH& ›=WuChong-shi§25.3Sturm–Liouville˚¸ fghijeyz{|k8l§25.3Sturm–Liouville123 }~ - 3‡· 5 3( t &)‡·!&c +8 Z = .S–L: †9V tz’( †+?@+- 3‡· 523( t &)‡·!= noGHs+S–L: †&‡· ¶•9 &q noGHs9 &q g25.3NOS–L: †‡· ¶•&3‡·!9fl&+ Y t +7 ‡· ¶•9Vˇ &= I U? +‡·!y(x)9fl&+Y ( f(x) g(x)+y(x)=f(x)+ig(x).7S–L: † @…“L(f+ig)=λρ(f+ig). .)*L9 )*+ˇ!ρ(x)9 !+ ‡· λ +t¤ fi( +] ªLf=λρf,Lg=λρg.‚’Tf(x) g(x)B9- .r3‡· λ&‡·!+^|& t t U& GHK ` ˜T»& = STf(x) g(x)PCD ‡· ¶•&EFGH=‚I } 1ªEFGHP9 t ¿&+ ‰c&wP9 +.9 EFGHKP( 6 = g25.4 y1(x) y2(x)B9S–L: †‡· ¶•Ly(x)=λρ(x)y(x).&23 & t &‡·!