Sturm-Liouville Theory

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1 LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory n utility of specil functions. Before preceing with the generl theory, let me stte clerly the sic properties of Fourier series we inten to generlize: i The Fourier sine functions { sin nπ L x n = 1,, 3,... } n cosine functions { cos } nπ L x n =, 1,,... re solutions of secon orer liner homogeneous ifferentil eqution 1 y = λy stisfying certin liner homogenous ounry conitions: viz, the Fourier-sine functions stisfy n the Fourier-cosine functions stisfy y = = y L y = = y L. ii The Fourier sine n cosine functions oey certin orthogonlity reltions L nπ mπ { 1 if n = m sin L L x sin L x = if n m L nπ mπ { 1 if n = m cos L L x cos L x = if n m iii Any continuous, ifferentile function on, L] cn e expresse in terms of Fourier-sine or Fourier-cosine expnsion f x = + nπ n cos L x = n=1 nπ n sin L x n=1 Moreover, the orthogonlity reltions ii llow us to etermine the coefficients n n n n = L nπ f x cos L L x n = L nπ f x sin L L x 1. Sturm-Liouville Prolems Definition 1.1. A Sturm-Liouville prolem is secon orer homogeneous liner ifferentil eqution of the form 3 p x y ] q x y + λr x y =, x, ] 55

2 1. STURM-LIOUVILLE PROBLEMS 56 together with ounry conitions of the form 4 4 y + α y = y + β y = In wht follow, λ is to e regre s constnt prmeter n we shll lwys ssume tht p x n r x re positive functions on the intervl, ] 5 p x > x, ] r x > x, ] Exmple 1.. Tking p x = 1 r x = 1 q x = = = L = = 1 α = β =, we see tht the corresponing Sturm-Liouville prolem y = λy y = y L = hs s its solutions nπ y x = sin L x nπ n λ = n = 1,, 3,... L Note, in prticulr, tht the solution of the Sturm-Liouville prolem only exists for certn vlues of λ; these vlues re eigenvlues of the Sturm-Liouville prolem Nottion 1.3 Liner ifferentil opertor nottion. In wht follows we shll enote y L the liner ifferentil opertor 6 L = p x ] q x which cts on function y x y L y] = p x y ] q x y In terms of L, the ifferentil eqution of Sturm-Liouville prolem cn e expresse L y] = λr x y Theorem 1.4 Lgrnge s ientity. Suppose φ n ψ re two functions stisfying ounry conitions of the form 7 7 then y + α y = y + β y = L φ] ψ φl ψ] =

3 1. STURM-LIOUVILLE PROBLEMS 57 Proof. We hve * L φ] ψ = = = p φ ψ = p φ ψ = p φ ψ p φ p φ ] ψ qφψ ] ψ p φ φ φp ψ p ψ ψ + qφψ φ = p φ ψ φψ + φl ψ] v where we hve twice utilize the integrtion y prts formul Now consier the first term in the lst line f g = fg qφψ qφψ p ψ f g qφψ p φ ψ φψ Since φ n ψ stisfy 7 n 7 we hve ssuming, α,, β n Hence φ = α φ, ψ = α ψ φ = β φ, ψ = β ψ φ ψ φ ψ = α φ ψ φ α φ ψ φ ψ = β φ ψ φ ψ β ψ p u v uv 1 = p 1 u 1 v 1 u 1 v 1 p u v u v = p 1 p = We cn hence conclue from * tht n the theorem follows. L φ] ψ = + φl ψ] = = Recll tht, for exmple, the Fourier-sine functions sin λx not only stisfy certin secon orer liner ifferentil eqution f = λ f ut they lso stisfy certin liner homogeneous ounry conitions f = = f L

4 1. STURM-LIOUVILLE PROBLEMS 58 exctly when the prmeter λ is tune to the ounry conitions { = f = λ f f = f L = λ = nπ L f x = sin nπ L x Wht my seem little surprising t first is tht the fct tht the Fourier-sine functions re solutions to Sturm-Liouville prolem is lso responsile for their orthogonlity properties L nπ mπ { 1 if n = m sin L L x sin L x = if n m Theorem 1.5. If φ n ψ re two solutions of Sturm-Liouville prolem corresponing to ifferent vlues of the prmeter λ then Proof. Suppose φ x ψ x r x =. L φ] = λ 1 r x φ L ψ] = λ r x ψ with λ 1 λ. Then since φ, ψ stisfy the Strum-Liouville ounry conitions, we hve y Theorem 11.1 L φ] ψ = φ L ψ] n ecuse they stisfy Sturm-Liouville-type ifferentil equtions or λ 1 r x φ x ψ x = λ 1 λ By hypothesis, λ 1 λ n so we must conclue φ x λ r x ψ x φ x ψ x r x = φ x ψ x r x = As remrke ove, it is common to write in shorthn the ifferentil eqution of Sturm-Liouville prolem s 8 L y] = λr x y n refer to the prmeter λ corresponing to prticulr solution y x s n eigenvlue of the ifferentil opertor L. This lnguge, of course, is erive from the nomenclture of liner lger where if A is n n n mtrix, v is n n 1 column vector n λ is numer such tht Av = λv then v is clle n eigenvector of A n λ is the corresponing eigenvlue of A. In fct, it is common to use such liner-lgeric-like nomenclture throughout Sturm-Liouville theory; n henceforth we shll refer to the functions y x tht stisfy 8 s eigenfunctions of L n the numers λ s the corresponing eigenvlues of L. Theorem 1.6. All the eigenvlues of Sturm-Liouville prolem re rel.

5 1. STURM-LIOUVILLE PROBLEMS 59 Proof. Consier the following inner prouct on the spce of complex-vlue solutions of Sturm-Liouville prolem 8 for vrious λ Suppose u, v = u x v x u x = v x = R x + ii x tht is, suppose we set u equl to v n split it up into its rel n imginry prts. Then u, u = = R x + ii x R x ii x R x + I x ] Note now tht the integrn is lwys non-negtive. By well-known result from Clculus, the integrl of continuous non-negtive function over finite intervl is lwys non-negtive, n zero only if the function is ienticlly equl to zero. We conclue tht for ny continuous, non-zero, complex-vlue function u x u x u x > Now consier eqution otine from the conclusion of Theorem 11.4 y setting v = u, 9 L u] v = n suppose u is Sturm-Liouville eigenfunction: ul u] 1 L u] = λru for some λ C Tking the complex conjugte of this eqution, we hve, ecuse p x, q x, n r x re ll ssume to e rel-vlue functions, or L u] = λrv = = 11 L u] = λru = L u] = λru p x u ] q x u = λru p x u ] q x u = λru An so plugging the right hn sies of 1 n 11 into 9 we cn conclue tht or 1 λ ru u = λu ru λ λ ruu = We hve rgue ove tht u x u x is non-negtive function, n y ssumption r x is positive function on, 1]. Thus, the integrn is non-negtive function n < ruu so long s u x is not ienticlly zero. We cn therefore conclue from 1 tht λ λ = = λ R.

6 1. STURM-LIOUVILLE PROBLEMS 6 In Liner Alger, we sy tht n eigenvlue λ of mtrix A hs multiplicity m if the imension of the corresponing eigenspce is m; tht is to sy, m = im NullSp A λi In Sturm-Liouville theory, we sy tht the multiplicity of n eigenvlue λ of Sturm-Liouville prolem 1 φ + φ = 1 φ 1 + φ 1 = L φ] = λr x φ x if there re exctly m linerly inepenent solutions for tht vlue of λ. Theorem 1.7. The eigenvlues of Sturm-Liouville prolem re ll of multiplicity one. Moreover, the eigenvlues form n infinite sequence n cn e orere ccoring to incresing mgnitue: {eigenvlues} = {λ 1, λ, λ 3,...}, λ 1 < λ < λ 3 < The orering of the eigenvlues λ, n the fct tht they multiplicity free, estlishes certin cnonicl orering of the corresponing Sturm-Liouville eigenfunctions. We enote y φ n the solution of the Sturm- Liouville prolem with λ = λ n. Actully, this only etermines φ n x up to constnt multiple ecuse if φ x stisfies 3, 4, 4 then so oes ny constnt multiple of φ x. It is common prctise to remove this miguity y emning in ition which fixes φ x up to sclr fctor of the form e iα. φ n x φ n xr x = 1 Moreover, the fcts tht the eigenvlues re ll rel n multiplicity free, lso implies tht we cn choose the φ n x to e rel-vlue functions of x ecuse the complex spn of the functions φ n x must coincie with tht of φ n x. Thus, we cn choose the eigenfunctions of Sturm-Liouville prolem to e rel-vlue functions φ n x φ n xr x = 1 Theorem 1.8. Let φ 1, φ, φ 3,... e the normlize eigenfunctions of Sturm-Liouville prolem n suppose f is piecewise continuous function on, 1]. Then if the series converges to t ech point on, 1. c n := f x φ n x r x c n φ n x n=1 f x + + f x Here f x ± := lim f x ± ε ε + Note tht t ny point x where f is continuous f x + + f x = f x.

7 1. STURM-LIOUVILLE PROBLEMS 61 In summry, ny time you hve ifferentil eqution of the form p x y ] q x y = λr x y with homogeneous ounry conitions of the form Then: 1 y + y = = 1 y 1 + y 1 Solutions will exist for only iscrete ut otherwise infinite set of vlues for the prmeter λ, ll of which re rel numers. There is unique lowest eigenvlue λ n the other eigenvlues cn e totlly orere λ < λ 1 < λ < λ 3 < For ech λ n, there is solution φ n of p x y ] q x y = λ n r x y tht is unique up to sclr fctor. The solutions φ n, n =, 1,,... cn e normlize such tht { 1 if n = m φ n x φ m x r x = otherwise Any continuous function f x on the intervl, 1] cn e expne in terms of the Sturm-Liouville eigenfunctions φ n, n =, 1,,,... f x = n φ n x n= with the coefficients n eing etermine y n = f x φ n x r x If f x is merely piece-wise continuous on, 1], one hs inste f x + + f x = n φ n x n= with the coefficients n etermine y the sme formul.