AMS 212B Perturbation Methods Lecture 14 Copyright by Hongyun Wang, UCSC. Example: Eigenvalue problem with a turning point inside the interval

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1 AMS B Perturbtion Methods Lecture 4 Copyright by Hongyun Wng, UCSC Emple: Eigenvlue problem with turning point inside the intervl y + λ y y = =, y( ) = The ODE for y() hs the form y () + λ f() y() = with f() = (-/). The WKB epnsion for > nd < : φ ( ) = ±i f ( s)ds = ±i s s 3 ds = ±i = ±it +, > ±T, < where T + = > T = < φ ( ) = 4 log f A generl solution t lrge λ is y( ) = = 4 log ( /) ( /) cos λt /4 ( + )+ b sin( λt + ), > c ep λt /4 ( )+ d ep( λt ), < Suppose y(t) is the eigenfunction of lrge eigenvlue λ. y(-) = (T ) = = 3/5 - -

2 AMS B Perturbtion Methods c ep λ d ep λ 5 = c d 6 = ep λ 5 Using the connection formuls, we hve = d + c b = d c + c /d = Constrint from y(-) = b c /d On the other hnd, using the boundry condition t =, we obtin y() =, (T+) = = 7/5 cos λ b sin λ 7 5 = b = tn λ 7 5 Constrint from y() = Combining the two constrints on /b from two boundry conditions, we rrive t tn λ 7 5 = λ(7/5) = (n /4)π λ = (5/7)(n /4)π Lrge eigenvlues re λ n 5 7 n 4 π, n = integer, n lrge. Net we consider the cse where one boundry condition flls on the turning point. Emple (Skip this emple in lecture): Eigenvlue problem with turning point t boundry - -

3 AMS B Perturbtion Methods y + λ =, y( ) = y y = The WKB epnsion for > : ( ) = y WKB ( /) where T + = The inner solution ner = : ( ) = α Ai λ y inn At =, we hve 3 cos λt /4 ( + )+ b sin( λt + ), > +βbi ( λ ) y() = α Ai() + β Bi() = Property of Airy functions: = Bi ( ) Ai 3 = 3 /3 Γ /3 3 (We will derive this property when we discuss symptotic epnsions of integrls). Using this property of Airy function, we hve α β = Bi Ai = 3 Recll the reltion between (α, β) nd (, b) = b = α +β π λ /6 α β π λ /6 b = α +β α /β + = α β α /β = Constrint from y() = On the other hnd, using the boundry condition t =, we obtin - 3 -

4 AMS B Perturbtion Methods y() =, (T+) = = 7/5 cos λ b sin λ 7 5 = b = tn 7 5 λ Constrint from y() = Combining the two constrints on /b from two boundry conditions, we rrive t tn λ 7 5 = λ = 3 tn + nπ + 3 λ = tn + nπ + 3 Lrge eigenvlues re λ n tn + nπ + 3, n: integer, n A more generl cse of turning point: y + λ g( )+ λh( ) y =, g = Gol: Find WKB pproimtions for > nd <, find inner solution, nd mtch them. Prt B: Find the solution ner = Ner =, we do scling u = λ α, α = /3. The eqution becomes d y λ α du + λ λ α u g λ α u + λh λ α u y = d y du + λ 3α u g( λ α u)+ λ α h( λ α u ) y u d y + u+o λ /3 du y u = = - 4 -

5 AMS B Perturbtion Methods The leding term eqution is the sme s in the specil cse: d y du u y ( u ) =. Thus, the (leding term) inner solution ner = is unffected by the term λh() y. y ( inn) ( ) = α Ai( u)+βbi( u), u = λ Prt A: Find the WKB epnsions for > nd <. The differentil eqution is y + λ g( )+ λh( ) Let y y =, g = ep δ n λ n= φ n Substituting into eqution yields ( λ)φ n ( ) + δ n ( λ)φ n ( ) δ n n= Clculting ϕ(): δ(λ) = λ δ φ ( λ) φ ( ) = g n= ( ) = λ g φ ( ) = ±i g( ) φ ( ) = ±i s g( s)ds = = λ g( ) λh( ) Tht is, ϕ() is unffected by the presence of λh() y. Clculting ϕ(): δ(λ) = λδ ( λ) φ φ ( ) φ φ φ ( ) = λ h φ ( ) = h( ) φ ( ) ( ) = h ( ) φ ( ) φ φ ( ) = ±i h( ) g( ) log φ

6 AMS B Perturbtion Methods φ ( ) = ±i ds h s s g s 4 log g ( ) Prt C: Mtching We consider the region of = O λ ds ds. h s = s g s s h s = g s = h 3 +O φ ( ) = 4 log g ( ) +O λ 4 = O ( λ 4 ) ( s h ( )+O( s) )ds Tht is, in the region of = O(λ -/ ), y (WKB) () is unffected by the term λh() y. Therefore, the mtching reltions re unffected by the presence of λh() y. = b = α +β π λ /6 α β π λ /6 c = d = π λ /6 β α π λ /6 Note: = ( d + c ) b = ( d c ) *) The connection formul is unffected by the term λh() y. *) In the rnge of = O(), y (WKB) () will be ffected by the term λh() y - 6 -

7 AMS B Perturbtion Methods Asymptotic pproimtions of integrls f t b ep λh( t ) dt, λ + f t b cos λh( t ) dt, λ + f t b sin λh( t ) dt, λ + We strt with + e λt f ( t )dt, λ + Region of dominnt contribution Emple: I = e λt cos( πt )dt, λ + Since e -λ t decys very fst for lrge λ, the region of dominnt contribution is very smll region ner t =. So we only need to pproimte cos(πt) ner =. We use the Tylor epnsion of cos(πt) t t = I = e λt cos( πt )dt = e λt π t! + π4 t 4 4!! dt To finish the emple, we need to clculte t α e λt dt. We use chnge of vrible: s = λ t, t = s/λ, dt = ds/λ t α e λt dt = λ α sα e s λ ds = λ α+ where the Gmm function is defined s z e d Γ z We obtin very useful formul s α e s ds = Γ α + λ α

8 t α e λt dt = Γ α + λ α+ AMS B Perturbtion Methods Properties of the Gmm function Γ() = Γ(α + ) = α Γ(α) Γ(n + ) = n! n = integer Γ = π The lst property cn be derived using the probbility density of the Gussin distribution Derivtion: Γ = s e s ds Chnge of vrible s = u = e u du = πσ ep u πσ σ du σ = = π Bck to our emple Applying the formul to our emple, we hve I = e λt π t + π4 t 4! dt = Γ! 4! λ = λ π λ 3 + π4 λ 5! λ 3 Γ 3 π! Γ( 5) + π4! 4! λ 5 Integrl e λt f ( t )dt cn be treted in similr wy. Now we consider slightly more generl cse: b e λt f ( t )dt where nd/or b +. The emple below shows how to del with the sitution where the lower integrtion limit is not nd the upper integrtion limit is not infinity

9 AMS B Perturbtion Methods Emple: 3 I = e λt dt, λ + +t 3 First, we use chnge of vrible to shift the lower limit of integrtion to zero: s = t, t = s +, dt = ds 3 I = e λt λ +s dt = e = e λ e λs +t s + + s 3 ds Net, we del with the upper limit of integrtion Clim: 3 ds I = e λ e λs ds = e λ e λs ds e λs ds + ( + s) 3 + ( + s) 3 + ( + s) 3! ###" ### $!###" ### $ I I I = T.S.T. T.S.T. stnds for Trnscendentlly Smll Term, which mens it is symptoticlly smller thn ny powers of /λ. Derivtion: We use chnge of vrible: u = s, s = u + I = e λs λ u+ = e = e λ e λu + + s + u+3 + u+3 = T.S.T. 3 ds 3 du 3 du Since we re going to hve power series of /λ in the epnsion of I, we do not need to keep ny T.S.T. term. Summry: In generl, we hve b e λt f ( t )dt = e λt f ( t )dt + T.S.T, λ + Bck to our emple To clculte I, we epnd - 9 -

10 AMS B Perturbtion Methods 3 = + + s = +3s +3s +! + 3 s + 3 s +! / = 3 s + 3 s s Thus, we hve I ~e λ I = e λ e λs 3 4 s s +! ds = e λ Γ( ) λ 3 Γ + 3 Γ( 3) 4 λ 3 = e λ λ 3 4λ + 3 6λ +O 3 λ 3 +! = 3 4 s s +! +O λ 3 λ 3 Emple: (Skip this emple in lecture) 4 I = e λt t +t dt ~ e λt t +t dt = e λt t + t 8 t +! dt = Γ 3 + λ 3 = λ λ 4 3 λ 5 +! Γ( 4) Γ( 5) +! λ 4 8 λ 5 Emple (non-integer power): I = e λt t +t dt, λ + = e λt t +t! t dt = e λt t 3 t +t! dt = Γ ( / ) Here we hve used Γ 3/ + Γ 5/!= π λ / λ 3/ λ 5/ λ π / λ + 3 π 3/ 4 λ 5/! - -

11 AMS B Perturbtion Methods Γ = π Γ 3 = Γ = Γ 5 = 3 Γ 3 = 3 4 π π Emple: (Skip this emple in lecture) 4 I = e λt log( +t )dt ~ e λt t t log( +t)dt = e λt t t 4 t + t 6 3! dt = Γ 5/ λ 5/ Γ( 9/) + Γ( 3/)! λ 9/ 3 λ 3/ Γ 5 = 3 Γ = 3 4 π Γ 9 = 7 5 Γ 5 = 5 6 π Γ 3 = 9 Γ 9 = π Now we summrize the procedure discussed bove s lemm. Wtson s lemm: Consider the integrl T = e λt t α g t I λ Suppose ) α > ) g t Ke ct Then we hve dt - -

12 I( λ)~ N n= n! g n AMS B Perturbtion Methods Γ( α + n+) λ α+n+ Note: Condition ) is to mke sure tht the integrl is convergent t t =. Condition ) is to mke sure tht for t >, g(t) is dominted by e λ t so tht for lrge λ the region of dominnt contribution is smll region ner t =. Proof: T = e λt t α g t I λ dt ~ e λt t α g( t )dt ~ e λt N g ( n ) t α n! n= N g ( n ) = e λt t α+n dt n! +!= n= N n= n! g n Γ( α + n+) λ α+n+ t n dt Reling the condition α >. Suppose g() = = g (k-) () = but g (k) (). Then α is restricted by α + k >. Emple: 4 I = e λt log( t +t )dt ~ e λt t t t log( +t)dt = e λt t t 4 t t + t 6 3! dt = e λt t / t 3/ + 3 t 7/! dt = Γ ( / ) Γ( 5/) + Γ( 9/)! λ / λ 5/ 3 λ 9/ - -