Notes on the Eigenfunction Method for solving differential equations
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1 Notes on the Eigenfunction Metho for solving ifferentil equtions Reminer: Wereconsieringtheinfinite-imensionlHilbertspceL 2 ([, b] of ll squre-integrble functions over the intervl [, b] (ie, b f(x 2 <1 with the inner prouct efine s n the norm inuce by this inner prouct hf gi : f (xg(x " kfk : p hf fi f(x 2 # 1/2 One cn introuce set of linerly inepenent bsis functions {y n } 1, where y n 2 L 2 ([, b] for n 0, 1,, such tht ny function f 2 L 2 ([, b] cn be represente s liner combintion of these functions f(x c n y n (x Such set is not unique n, in generl, the functions y n re not orthonorml, but they o form complete bsis over the intervl [, b] Furthermore, one cn prouce n orthonorml bsis for the Hilbert spce L 2 ([, b] from {y n } 1 using the Grm-Schmit proceure (ctully, this proceure cn be use to generte n orthonorml bsis for ny inner prouct spce 8 >< >: 0 y 0, ˆ0 0 k 0k 1 y 1 ˆ0h ˆ0 y 1 i, ˆ1 1 k 1k n y n P n 1 k0 ˆkh ˆk y n i, ˆk k k k k Using the set of orthonorml functions { ˆn} 1 one cn represent ny function f 2 L 2 ([, b] s f(x c n ˆn(x with the coefficients c n given by c n h ˆn fi This is true only if the bsis is orthonorml ˆ n (xf(x
2 1 Hermitin Conjugte of n Opertor Definition The joint of n opertor L,enotebyL,isefineby f (x[l g(x] [L f(x] g(x + bounry terms, (1 where the bounry terms (bt re evlute t the en-points of the intervl [, b] Using the efinition of n inner prouct one my re-write it s hf L gi hl f gi + bt For given liner ifferentil opertor, the metho for formlly fining the joint is to integrte by prts enough times to move the erivtives from g to f Exmple Compute the Hermitin conjugte of the following opertors: 1 L 1 hf gi pple ( f (x g(x u (f (xg(x b pple v 0 u f (x u 0 f (x f (xg(x f(x g(x +(f (xg(x b bt v 0 g(x v g(x 2 L 2 ı hf ı gi pple ( f (x ı g(x u v 0 u f (x u 0 f (x ı(f (xg(x b ı f (xg(x pple ı f(x g(x + ı(f (xg(x b bt v 0 ı g(x v ıg(x ı ı Remrk If L L then it is si tht L is self-joint Definition 1 If in Eq (1 the bt vnishes, then n opertor L is si to be Hermitin over the intervl
3 [, b] n f (x[l g(x] [L f(x] g(x (2 (equivlently, hf L gi hl f gi 2 Properties of Hermitin opertors From now on we re going to ssume tht the opertor L is Hermitin Relity of the eigenvlues Let y p n y q be two eigenfunctions of n opertor L corresponing to eigenvlues p n q,respectively L y p (x p y p (x L y q (x q y q (x + h y q i + hy p i hl y p y q i h p y p y q i hy p L y q i hy p qy q i hl y p y q i phy p y q i hl y p y q i q hy p y q i + L L hl y p y q i q hy p y q i Thus one hs phy p y q i qhy p y q i 0 p q hy p y q i 0 We get the sme result using the integrl form of the inner prouct Multiplying the eqution L y p (x p y p (x by y q n eqution L y q (x q y q (x by y p,nthenintegrtingovertheintervl[, b] les to The complex conjugte of Eq (3 becomes y q (x[l y p (x] p y p(x[l y q (x] q y q (x y p (x, (3 y p(x y q (x (4 y q (x[l y p (x] p y q (x y p(x Using now the efinition of Hermitin opertor (Eq 2 one gets [L y q (x] y p(x p y q (x y p(x (5
4 Compring Eq (4 n Eq (5 les to p y q (x y p(x q y p(x y q (x p q yp(x y q (x 0 Now, if p q, then it follows tht p p,ie, p is rel, since b y p(x y p (x hy p y p i n 0, 1,,functionsy n 60 An wht if p 6 q? 0 (for ll Orthogonlity n normliztion of the eigenfunctions We re going to consier two cses: 1 when y p n y q correspon to ifferent eigenvlues, p n q respectively, n 2 when y p n y q correspon to the sme eigenvlue 0 1 If y p n y q correspon to ifferent eigenvlues, then or, in terms of integrls, p q hy p y q i 0 hy p y q i 0 p q yp(x y q (x 0 y p(x y q (x 0 n we get two equivlent sttements of the orthogonlity of y p n y q 2 If 0 is egenerte eigenvlue, ie, L y`(x 0 y` for ` 0, 1,,k 1 then it my hppen tht y` re not orthogonl orthonormliztion proceure one cn obtin k orthogonl vectors However, by performing the Grm-Schmit To normlize the set of bsis functions {y n } 1 it is enough to ivie ech function by its norm ŷ n (x y n(x ky n (xk, obtining in this wy orthonorml set of bsis functions { yˆ n } 1 tht re stisfying the conition hŷ p ŷ q i 8 < ŷp(xŷ 0, if p 6 q q (x : 1, if p q Completeness of the eigenfunctions The infinite set of normlize eigenfunctions of Hermitin opertor form complete bsis set over [, b], ie,nyfunctiony 2 L 2 ([, b] obeying the bounry conitions my be expresse s liner combintion of these functions s y(x c n ŷ n (x,
5 where, s we know, c n hŷ n yi Thus y(x ŷ n (x On the other hn ŷ n(z y(zz ŷ n(x y(x y(z y(x y(z (x zz, ŷ n (xŷn(z where (x z stns for the Dirc elt function Compring these two equlities les to the completeness (or closure propertyoftheeigenfunctions ŷ n (xŷn(z (x z For continuous spectrum one hs to replce summtion by integrtion 3 Why is it importnt? ŷ n (xŷ n(z (x z We re concerne with the solution to the inhomogeneous eqution of the form L y(x f(x, (6 where f is some known function n the bounry conitions re given The ie for solving this problem is to use the linerity n Hemiticity of n opertor L Howoweoit? Step-1: Fin ll eigenfunctions of n opertor L,ie,solve the eqution L y n (x n y n (x Trnsform eigenfunctions {y n } 1 so tht they form n orthonorml bsis Step-2: Re-write both y n f s superpositions of eigenfunctions of L y(x f(x c n y n (x, f n y n (x where c n n f n stn for some constnt
6 Step-3: Mke use of the linerity of the opertor L! L y(x L c n y n (x c n L y n (x c n n y n (x f n y n (x f(x Step-4: Tke vntge of the orthonormlity of functions {y n } 1 -tkeninnerproucthy m i of both sies in the bove equlity Since hy m y n i mn,thus c n n hy m y n i f n hy m y n i c m m f m c m f m Step-5: Put your coefficients in to get generl solution of the problem (6 y(x 4 The Sturm-Liouville equtions Let us consier the following eqution p(x 2 y(x 2 + r(x y(x f n n y n (x m + q(xy(x+ (xy(x 0, where r(x p(x, p, q n r re rel functions n (x > 0 is clle weight or ensity function It is cler tht this eqution cn be written s L y(x (xy(x, where L (L is clle the Sturm-Liouville opertor pplep(x r(x + q(x pple p(x + q(x It turns out the with n pproprite bounry conitions the Sturm-Liouville opertor is Hermitin over the intervl [, b] These bounry conitions re [y i (xp(xy 0 j(x] x [y i (xp(xy 0 j(x] xb for ll i, j, where y i n y j stn for two eigenfunctions of the opertor L To check it let us put the Sturm-Liouville
7 form L (py 0 0 qy into the Eq (2 y i (x[l y j (x] ( ( yi (py 0 j 0 + qy j u yi v 0 (pyj 0 0 u 0 (y i 0 y i pyj 0 b + vnishes u (y i 0 p u 0 ((y i 0 p 0 [(y i 0 py j ] b vnishes v py 0 j (y i 0 py 0 j v 0 y 0 j v y j ((y i 0 p 0 y j [(p(y i qy i ] y j y i (py 0 j 0 y i qy j y i qy j [L y i (x] y j (x y i qy j [((y i 0 p 0 y j + y i qy j ] Why is the Sturm-Liouville eqution importnt? It is becuse ll secon-orer liner orinry ifferentil equtions cn be recst in the form of the LHS of the Sturm-Liouville eqution How? Consier n orinry ifferentil eqution of the form We wnt to rewrite this eqution s P (x 2 y(x 2 + Q(x y(x + R(xy(x 0 pple p(x + q(x y(x (xy(x To o so, one must fin p, q n in the following wy: Exmple 2 Let us consier Bessel s eqution p(x x Q(t e P (t t, (x q(x p(x R(x P (x x 2 y 00 + xy 0 +( 2 x 2 v 2 y 0 Here x t p(x e t 2 t e ln x x (x q(x x ( 2 x 2 v 2 x 2 ( 2 x 2 v 2 x Thus we get (xy 0 0 +( x v 2 y 0 x
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