A Generalization of Ince s Equation

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1 Joural of Applied Matheatics ad Physics 7-8 Published Olie Deceber i SciRes. A Geeralizatio of Ice s Equatio Ridha Moussa Uiversity of Wiscosi Waukesha USA Eail: roussa@uw.edu Received October ; revised Noveber ; accepted 7 Noveber Copyright by author ad Scietific Research Publishig Ic. This work is licesed uder the Creative Coos Attributio Iteratioal Licese (CC BY). Abstract We ivestigate the Hill differetial equatio ( A ) y B y ( D ) y At B( t ) ad λ = D t are trigooetric polyoials. We are iterested i solutios that are eve or odd ad have period π or sei-period π. The above equatio with oe of the above coditios costitutes a regular Stur-Liouville eigevalue proble. We ivestigate the represetatio of the four Stur-Liouville operators by ifiite baded atrices. Keywords Hill Equatio Ice Equatio Stur-Liouville Proble Ifiite Baded Matrix Eigevalues Eigefuctios. Itroductio The first kow appearace of the Ice equatio ( a ) y ( b ) y ( λ d ) y cos si cos = is i Whittaker s paper ([] Equatio (5)) o itegral equatios. Whittaker ephasized the special case a = ad this special case was later ivestigated i ore detail by Ice [] [3]. Magus ad Wikler s book [] cotais a chapter dealig with the coexistece proble for the Ice equatio. Also Arscott [5] has a chapter o the Ice equatio with a =. Oe of the iportat features of the Ice equatio is that the correspodig Ice differetial operator whe applied to Fourier series ca be represeted by a ifiite tridiagoal atrix. It is this part of the theory that akes the Ice equatio particularly iterestig. For istace the coexistece proble which has o siple solutio for the geeral Hill equatio has a coplete solutio for the Ice equatio (see [6]). Whe studyig the Ice equatio it becae apparet that ay of its properties carry over to a ore geeral class of equatios the geeralized Ice equatio. These liear secod order differetial equatios describe How to cite this paper: Moussa R. () A Geeralizatio of Ice s Equatio. Joural of Applied Matheatics ad Physics

2 iportat physical pheoea which exhibit a proouced oscillatory character; behavior of pedulu-like systes vibratios resoaces ad wave propagatio are all pheoea of this type i classical echaics (see for exaple [7]) while the sae is true for the typical behavior of quatu particles (Schrödiger s equatio with periodic potetial [8]).. The Differetial Equatio We cosider the Hill differetial equatio ( A ) y B y ( λ D ) y = (.) A = a cos( t) = B = b si ( t) = D = d cos( t). = Here is a positive iteger the coefficiets a b d for = are specified real ubers. The real uber λ is regarded as a spectral paraeter. We further assue that a <. Uless stated otherwise solutios y t are defied for t. We will at ties represet the coefficiets a b d for = i the vector for: a = a a a b = b b b d = d d d. The polyoials d Q ( µ ) : = aµ bµ = (.) will play a iportat role i the aalysis of (.). For ease of otatio we also itroduce the polyoials Q µ : = Q µ =. (.3) Equatio (.) is a atural geeralizatio to the origial Ice equatio acos t y t bsi t y t λ dcos t y t =. (.) ( ) ( ) ( ) Ice s equatio by itself icludes soe iportat particular cases if we choose for exaple a = b = d = q we obtai the faous Mathieu s equatio y t λ qcos t y t = (.5) with associated pzlyoial ( ) Q q. = q ad d q( ν ) µ = (.6) If we choose a = b = q ν are real ubers Ice s equatio becoes Whittaker-Hill equatio y t q si t y t λ q ν cos t y t = (.7) with associated polyoial ( ) Q ( µ ) q( µ ν ) =. (.8) Equatio (.) ca be brought to algebraic for by applyig the trasforatio whe = ad a = b = we obtai ( ) d y ξ dy 8dξ d 8d ξ d d λ y. dξ = ξ( ξ) dξ ξ ξ = ξ = cos t. For exaple (.9) 7

3 3. Eigevalues Equatio (.) is a eve Hill equatio with period π. We are iterested i solutios which are eve or odd ad have period π or sei period π i.e. y( tπ ) =± y. We kow that y( t ) is a solutio to (.) the y( tπ ) ad y( t) are also solutios. Fro the geeral theory of Hill equatio (see [9] Theore.3.); we obtai the followig leas: Lea 3.. Let y t be a solutio of (.) the y y y( t ) is eve with sei period π if ad oly if y y y( t ) is odd with sei period π if ad oly if y y y( t ) is odd with period π if ad oly if y y Equatio (.) ca be writte i the self adoit for y t is eve with period π if ad oly if = π = ; (3.) = π = ; (3.) = π = ; (3.3) = π =. (3.) (( A ) ω y ) D ω y λω y = (3.5) ω A At B t = exp d t. (3.6) B Note that ω is eve ad π -periodic sice the fuctio ( t ) A ( t ) At periodic. Proof. Let rt ( At ) ω which is equivalet to ad Notig that we see that =. (3.5) ca be writte as Therefore (3.8) ca be writte as Sice ( r y ) D ω y λω y is cotiuous odd ad π - = (3.7) ω λω. r t y t r t y t D t t y t = t y t (3.8) = ( ) ω ω r t At t A t t B t ω = A At ω. r t = B t t ω ω ( ) ω ω λω B t t y t A t t y t D t t y t = t y t. (3.9) ω is strictly positive the lea follows. I the case of Ice s Equatio (.) we have the followig forula for the fuctio ω 73

4 ω b a acos t if a : = b exp cos t if a =. (3.) Whe the fuctio ca be coputed explicitly usig Maple. For exaple let us cosider the case = with = 8 a = [ ] b. Applyig (3.6) we obtai ω = ( 7 cos t ( cos t) ) 3 Equatio (.) with oe of the boudary coditios i lea 3. is a regular Stur-Liouville proble. Fro the theory of Stur-Liouville ordiary differetial equatios it is kow that such a eigevalue proble has a sequece of eigevalues that coverge to ifiity. These eige values are deoted by α α β ad β = to correspod to the boudary coditios i lea 3. respectively. This otatio is cosistet with the theory of Mathieu ad Ice s equatios (see [] []). Lea 3. iplies the followig theore. Theore 3.. The geeralized Ice equatio adits a otrivial eve solutio with period π if ad oly if λ = α ( abd ) for soe ; it adits a otrivial eve solutio with sei-period π if ad oly if. λ = α abd for soe ; it adits a otrivial odd solutio with sei-period π if ad oly if λ = β abd for soe ; it adits a otrivial odd solutio with period π if ad oly if λ = β abd for soe. Exaple 3.3. To gai soe uderstadig about the otatio we cosider the alost trivial copletely solvable exaple the so called Cauchy boudary value proble λ y t y t = (3.) subect to the boudary coditios of lea 3.. We have the followig for the eigevalues λ i ters of =. ) Eve with period π we have λ = α =. ) Eve with sei-period π we have λ = α =. 3) Odd with sei-period π we have λ = β =. ) Odd with sei-period π we have λ = β =. The foral adoit of the geeralized Ice equatio is By itroducig the fuctios (( A ) y ) ( B y ) ( λ D ) y =. (3.) * B t A t B t a b t = = = ( ) si * D t D t A t B t a b d t = = = ( ) cos we ote that the adoit of (.) has the sae for ad ca be writte i the followig for: Lea 3.. If equatio if ad oly if y ( ) λ A t y t B t y t D t y t =. (3.3) y t is twice differetiable defied o ω is a solutio to its adoit. the y t is a solutio to the geeralized Ice 7

5 ad the Proof. We Kow that For ease of otatio let Substitutig for p q B A = = ω = ω A B A B D D A B ω = ( B A )( A) A ( B A ) ( B A ) ( A) ω. ( B A )( A) A ( B A ) ( B A ) ( A) B A p = q = A ( A)( ωy) B ( ωy) ( λ D )( ωy) ( A)( ω y ω y ωy ) B ( ω y ωy ) ( λ D )( ωy) = ( A)( qωy pωy ωy ) B ( pωy ωy ) ( λ D )( ωy) =. B * ad * D ad siplifyig we obtai ( A)( ωy) B ( ωy) ( λ D )( ωy) ω( ( A) y By ( λ D) y) =. Fro lea 3. we kow that if y is twice differetiable y is a solutio to the geeralized Ice s equatio with paraeters λ a b ad d if ad oly if ω y is a solutio to its foral adoit. Sice the fuctio ω is eve with period π the boudary coditio for y ad ω y are the sae. Therefore we have the followig theore. Theore 3.5. We have for α a b d = α a a b d a b = (3.) β a b d = β a a b d a b =. (3.5) Fro Stur-Liouville theory we obtai the followig stateet o the distributio of eigevalues. Theore 3.6. The eigevalues of the geeralized Ice equatio satisfy the iequalities α α α 3 α < < < < (3.6) β β β 3 The theory of Hill equatio [] gives the followig results. Theore 3.7. If λ α or λ belogs to oe of the closed itervals with distict edpoits α β = the the geeralized Ice equatio is ustable. For all other real values of λ the equatio is stable. I the case α = β for soe positive iteger ad the paraeters a istability iterval: The geeralized Ice equatio is stable if. Eigefuctios abd abd (3.7) ( abd) ( abd ) λ = α = β. By theore 3. the geeralized Ice s equatio with λ α b d the degeerate iterval [ ] α β is ot a = abd adits a o trivial eve solutio with period π. It is uiquely deteried up to a costat factor. We deote this Ice fuctio by Ic = Ic ( t; abd ) whe it is oralized by the coditios Ic > ad 75

6 π ( Ic ( t )) d t π =. (.) The geeralized Ice s equatio with λ α ( ) π. It is uiquely deteried up to a costat factor. We deote this Ice fuctio by Ic = Ic ( t; abd ) whe it is oralized by the coditios Ic > ad = abd adits a o trivial eve solutio with sei-period π ( ) π Ic t d t =. (.) The geeralized Ice equatio with λ β ( ) It is uiquely deteried up to a costat factor. We deote this Ice fuctio by Is = Is ( t abd ) whe it is oralized by the coditios Is = abd adits a o trivial odd solutio with sei-period π. d dt > ad ; π ( ) π Is t d t =. (.3) The geeralized Ice equatio with λ β ( ) uiquely deteried up to a costat factor. We deote this Ice fuctio by Is Is ( t ) d dt it is oralized by the coditios Is = abd adits a o trivial odd solutio with period π. It is > ad = ; abd whe π ( ) π Is t d t =. (.) Fro Stur-Liouville theory ([] Chapter 8 Theore.) we obtai the followig oscillatio properties. Theore.. Each of the fuctio systes { Ic } { Ic } = { Is } = { Is } is orthogoal over [ π ] with respect to the weight = (.5) (.6) (.7) = (.8) ω that is for π ω t Ic t Ic t d t = (.9) π ω t Ic t Ic t dt = (.) π ω t Is t Is t dt = (.) π ω t Is t Is t dt =. (.) π. Usig the trasforatios that led to Theore 3.5 we obtai the followig result. Theore.. We have Moreover each of the previous syste is coplete over [ ] c ( abd ) ad ( ) ( ; ) = ( ) ω ( ; ) ( ; ) ( ; * * ) = ( * * ) ω ( ; ) ( ; ) Ic t ab d c abd t ab Ic t abd (.3) Is t ab d s ab d t ab Is t abd (.) s abd are positive ad idepedet of t ad 76

7 with b = b b b d = d d d b = a b d = d a b =. The adopted oralizatio of Ice fuctios is easily expressible i ters of the Fourier coefficiets of Ice fuctios ad so is well suited for uerical coputatios [6]; However it has the disadvatage that Equatios (.3) ad (.) require coefficiets c ad s which are ot explicitly kow. Of course oce the geeralized Ice fuctios Ic ad Is are kow we ca express c ad s i the for c s ( abd) ( ab) ( ; ab d ) Ic = ω ; Ic ; ( abd) ( ab) ( abd) ( ; ab d ) Is =. ω ; Is ; ( abd) If we square both sides of (.3) ad (.) ad itegrate we fid that If ω ( t; ) ( ω ) π (.5) (.6) c t; ab Ic t; abd dt = π (.7) ( ω ) π s t; ab Is t; abd dt = π. (.8) ab is very siple the it is possible to evaluate the itegrals i (.7) (.8) i ters of the Fourier coefficiets of the geeralized Ice fuctios. This provides aother way to to calculate c ad s. Oce we kow c ad s we ca evaluate the itegrals o the left-had sides of the followig equatios * = π * ω c t; ab Ic t; abd d t Ic t; abd Ic t; ab d d t. (.9) * = π * ω s t; ab Is t; abd d t Is t; abd Is t; ab d d t. (.) The itegrals o the right-had sides of (.9) ad (.) are easy to calculate oce we kow the Fourier series of Ice fuctios. 5. Operators ad Baded Matrices I this sectio we itroduce four liear operators associated with Equatio (.) ad represet the by baded atrices of width. It is this siple represetatio that is fudaetal i the theory of the geeralized Ice equatio. We assue kow soe basic otios fro spectral theory of operators i Hilbert space. Let H be the Hilbert space cosistig of eve locally square-suable fuctios f : with period π. The ier product is give by By restrictig fuctios to [ ] cosider a secod ier product We cosider the differetial operator π π f g = f t g t d. t (5.) H is isoetrically isoorphic to the stadard ( ) π L π. We also f g = ω t f t g t d. t (5.) ω ( S y) ( A ) y B y D y =. (5.3) D S of defiitio of cosists of all fuctios y H for which y ad y are absolutely cotiuous ad this correspods to the usual doai of a Stur- Liouville operator associated with the boudary coditios (3.). It is kow ([] Chapter V Sectio 3.6) that The doai ( ) y H by restrictig fuctios to [ π ] 77

8 S is self-adoit with copact resolvet whe cosidered as a operator i ( H ω ) ( ) Hilbert space ( ) ad its eigevalues are α abd =. All eigevalues of S are siple. If we cosider S as a operator i the o the sae doai ( ) ; H the its adoit * S is give by the operator but with b d replaced by b * values as. product (( ) ) ( ) y A t y t B t y t D t y t D S see ([] Chapter III Exaple 5.3). The adoit * S is of the sae for as S d * respectively. By Theore 3.5 we see that S * has the sae eige- S Let ( ) be the space of square-suable sequeces x { x } =. The = with its stadard ier defies a biective liear ap T ( T x) x x : = cos = : H. Cosider the operator M : = T ST defied o = = : <. (5.) D( M) T ( D( S) ) x x = Let e deotes the sequece with a i the th positio ad s i all other positios we also defie u : = ( Te ) i.e. u ( t ) = ad u = cos for =. We fid that the operator M ca be represeted i the followig way Me δ = ad δ k = if k ad with e. qe if = = = re q δ e qe if = = r =. M is self-adoit with copact resolvet i ier product geerates a or that is equivalet to the usual Note that the factor (5.5) should appear oly equipped with the ier product TxTy ω. This. The operator M has the eigevalues α abd ad the correspodig eigevectors for sequeces of Fourier coefficiets for the fuctios Ic. Now cosider the operator S that is defied as S i (5.3) but i the Hilbert space H cosistig of eve fuctios with sei-period π. This operator has eigevalues α ( abd ) with eigefuctios Ic =. cos t the Usig the basis defies a biective liear ap T Let ( ) = ( ) Tx t : x cos t = : H. Cosider the operator = = : <. D( M) T ( D( S) ) x ( ) x = M : = T ST defied o u t : = Te t = cos t for = we get the followig forula for M = = (5.6) M e = re q e q e 78

9 if q = q = Q = r = ( ) if. Now cosider the operator S 3 that is defied as S but i the Hilbert space H 3 cosistig of odd fuctios with sei-period π. This operator has the eigevalues β with eigefuctios Is =. si t. Usig the basis fuctios ( ) defies a biective liear ap T ( 3 ) = ( ) Tx t : x si t 3 3 = : H. Cosider the operator M : = T ST defied o Let ad 3 = = : <. D( M3) T3 ( D( S3) ) x ( ) x = u t : = Te t = si t for = we have the followig forula for M 3 3 = ε = = Me re q e q e (5.7) if q = q = Q = r = ( ) if if ε =. if < Fially cosider the operator S that is defied as S but i the Hilbert space H cosistig of odd fuctios with period π. This operator has the eigevalues β with eigefuctios Is =. si t Usig the basis defies a biective liear ap T : Let Tx t : x si t ( ) = ( ) = H. Cosider the operator = = : <. D( M) T ( D( S) ) x ( ) x = M : = T ST defied o u t : = Te t = si t for =. The the forula for M is i ( ) = ε = = = M e re q e q e q e (5.8) r = =. Exaple 5.. For the Whittaker-Hill Equatio (.7) i the followig for [8] the fuctio y λ αscos t α cos t y = α s (5.9) ω fro (3.6) is equal to therefore the operators S = 3 are self-adoit o the 79

10 Hilbert spaces H = 3 respectively. Hece the ifiite atrices S = 3 are syetric. They are represeted by M M M αs α αs α αs α α αs 6 αs α αs 36 = (5.) α αs α ( s ) αs α α α α( s α) 9 αs α α αs 5 αs α αs = α αs α 9 ( s ) αs α α α α( s α) 9 αs α α αs 5 αs α αs = α αs α 9 3 M α αs α αs 6 αs α α αs 36 αs s =. α αs α α α 6 (5.) (5.) (5.3) 6. Fourier Series The geeralized Ice fuctios adit the followig Fourier series expasios A Ic = A cos( t) (6.) = = ( ) Ic t A cos t (6.) = 8

11 = ( ) Is t B si t (6.3) = = ( ) Is t B si t. (6.) = We did ot idicate the depedece of the Fourier coefficiets o abd. The oralizatio of Ice fuctios iplies A = > A A = = (6.5) A A = = = > (6.6) B = ( ) B > (6.7) = = B = ( ) B >. (6.8) = = Usig relatios (.3) ad (.) we ca represet the geeralized fuctios i a differet way Therefore we ca write = ω Ic abd t; ab c abd Ic ab d (6.9) = ω Is abd t; ab s abd Is ab d (6.) b = a b d = d a b =. C Ic( abd ) = ( ω ( t; ab )) C cos( t) (6.) = Ic ( abd ) = ( ω ( t; ab )) C cos( t) (6.) = Is ( abd ) = ( ω ( t; ab )) D si ( t) (6.3) = Is ( abd ) = ( ω ( t; ab )) D si ( t) (6.) = ( ( abd) ) ( ( abd )) C = c A D = s B ad the Fourier coefficiets A ad B belog to the paraeters ab d. Properties of the coefficiets C ad D follow fro those of A ad B. A geeralized Ice fuctio is called a geeralized Ice polyoial of the first kid if its Fourier series (6.) (6.) (6.3) or (6.) teriate. It is called a geeralized Ice polyoial of the secod kid if its expasio (6.) (6.) (6.3) or (6.) teriate. If they exist these geeralized Ice polyoials ad their correspodig eigevalues ca be coputed fro the fiite subsectios of the atrices Sectio 5. Exaple 6.. Cosider the equatio M = 3 of cos t cos t y si t si t y λ y = (6.5) 8

12 oe ca check that if we set λ = ay costat fuctio y is a eigefuctio correspodig to the eigevalue α =. The adopted oralizatio of Sectio iplies that Ic ( t ) =. It is a geeralized Ice polyoial (eve with period π ). Refereces [] Whittaker E.T. (95) O a Class of Differetial Equatios Whose Solutios Satisfy Itegral Equatios. Proceedigs of the Ediburgh Matheatical Society [] Ice E.L. (93) A Liear Differetial Equatio with Periodic Coefficiets. Proceedigs of the Lodo Matheatical Society [3] Ice E.L. (95) The Real Zeros of Solutios of a Liear Differetial Equatio with Periodic Coefficiets. Proceedigs of the Lodo Matheatical Society [] Magus W. ad Wikler S. (966) Hill s Equatio. Joh Wiley & Sos New York. [5] Arscott F.M. (96) Periodic Differetial Equatios. Pergao Press New York. [6] Volker H. (3) Coexistece of Periodic Solutios of Ice s Equatio. Aalysis [7] Recktewald G. ad Rad R. (5) Coexistece Pheoeo i Autoparaetric Excitatio of Two Degree of Freedo Systes. Iteratioal Joural of No-Liear Mechaics [8] Heerey A.D. ad Veselov A.P. (9) Whittaker-Hill Equatio ad Seifiite Gap Schrödiger Operators. -. arxiv:96.697v [9] Eastha M. (973) The Spectral Theory of Periodic Differetial Equatios. Scottish Acadeic Press Ediburgh Lodo. [] Volker H. () Four Rearks o Eigevalues of Laé s Equatio. Aalysis ad Applicatios [] Coddigto E. ad Leviso N. (955) Theory of Ordiary Differetial Equatios. Robert E. Krieger Publishig Copay Malarbar. [] Kato T. (98) Perturbatio Theory for Liear Operators. Spriger-Verlag Berli Heidelberg New York. 8

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