Sturm-Liouville Theory

Transcription

1 LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory an utility of special functions. Before preceing with the general theory, let me state clearly the basic properties of Fourier series we inten to generalize: i The Fourier sine functions { sin nπ L x n =, 2, 3,... } an cosine functions { cos } nπ L x n =,, 2,... are solutions of a secon orer linear homogeneous ifferential equation y = λy 2a satisfying certain linear homogenous bounary conitions: viz, the Fourier-sine functions satisfy an the Fourier-cosine functions satisfy y = = y L 2b y = = y L. ii The Fourier sine an cosine functions obey certain orthogonality relations 2 L nπ mπ { if n = m sin L L x sin L x = if n m 2 L nπ mπ { if n = m cos L L x cos L x = if n m iii Any continuous, ifferentiable function on, L] can be expresse in terms of a Fourier-sine or Fourier-cosine expansion f x = a 2 + nπ a n cos L x = n= nπ b n sin L x n= Moreover, the orthogonality relations ii allow us to etermine the coeffi cients a n an b n a n = 2 L nπ f x cos L L x b n = 2 L nπ f x sin L L x. Sturm-Liouville Problems D 5.. A Sturm-Liouville problem is a secon orer homogeneous linear ifferential equation of the form 3 p x y ] q x y + λr x y =, x, ]

2 . STURM-LIOUVILLE PROBLEMS 2 together with bounary conitions of the form 4a 4b a y + a 2 y = b y + b 2 y = In what follow, λ is to be regare as a constant parameter an we shall always assume that p x an r x are positive functions on the interval, ] 5 p x > x, ] r x > x, ] E 5.2. Taking p x = r x =, q x =, a = b =, a 2 = b 2 =, we see that the corresponing Sturm-Liouville problem has as its solutions y + λy = y = y = λ = nπ 2 n =, 2, 3,... an y x = sin nπx Note, in particular, that the solution of the Sturm-Liouville problem only exists for certan values of λ; these values are eigenvalues of the Sturm-Liouville problem N 5.3 Linear ifferential operator notation. In what follows we shall enote by L the linear ifferential operator 6 L = p x ] q x which acts on a function y x by L y] = p x y ] q x y In terms of L, the ifferential equation of a Sturm-Liouville problem can be expresse L y] = λr x y T 5.4 Lagrange s ientity. Suppose u an v are two functions satisfying bounary conitions of the form 7a 7b a y + a 2 y = b y + b 2 y = then L u] v ul v] =

3 . STURM-LIOUVILLE PROBLEMS 3 Proof. We have * L u] v = = = p u v = p u v = p u v p u p u ] v quv ] v p u u up v p v v + = p u v uv + ul v] v where we have twice utilize the integration by parts formula b Now consier the first term in the last line a f g = fg b a b p u v uv a quv u quv quv p v f g Since u an v satisfy 7a an 7b we have assuming a, a 2, b, b 2 quv an Hence u = a 2 a u, v = a 2 a v u = b 2 b u, v = b 2 b v u v u v = a 2 u v u a 2 a u v u v = b 2 b u v u v a b 2 b v p u v uv = p u v u v p u v u v = p p = We can hence conclue from * that an the theorem follows. L u] v = ul v] = = Recall that, for example, the Fourier-sine functions sin λx not only satisfy certain a secon orer linear ifferential equation f = λ 2 f but they also satisfy certain linear homogeneous bounary conitions f = = f L

4 . STURM-LIOUVILLE PROBLEMS 4 exactly when the parameter λ is tune to the bounary conitions { = f = λ 2 f f = f L = λ = nπ L f x = sin nπ L x What may seem a little surprising at first is that the fact that the Fourier-sine functions are solutions to a Sturm-Liouville problem is also responsible for their orthogonality properties 2 L nπ mπ { if n = m sin L L x sin L x = if n m T 5.5. If φ an ψ are two solutions of a Sturm-Liouville problem corresponing to ifferent values of the parameter λ then Proof. Suppose φ x ψ x r x =. L φ] = λ r x φ L ψ] = λ 2 r x ψ with λ λ 2. Then since φ, ψ satisfy the Strum-Liouville bounary conitions, we have by Theorem. L φ] ψ = φ L ψ] an because they satisfy Sturm-Liouville-type ifferential equations or λ r x φ x ψ x = λ λ 2 By hypothesis, λ λ 2 an so we must conclue φ x λ 2 r x ψ x φ x ψ x r x = φ x ψ x r x = As remarke above, it is common to write in shorthan the ifferential equation of a Sturm-Liouville problem as 8 L y] = λr x y an refer to the parameter λ corresponing to a particular solution y x as an eigenvalue of the ifferential operator L. This language, of course, is erive from the nomenclature of linear algebra where if A is an n n matrix, v is an n column vector an λ is a number such that Av = λv then v is calle an eigenvector of A an λ is the corresponing eigenvalue of A. In fact, it is common to use such linear-algebraic-like nomenclature throughout Sturm-Liouville theory; an henceforth we shall refer to the functions y x that satisfy 8 as eigenfunctions of L an the numbers λ as the corresponing eigenvalues of L. T 5.6. All the eigenvalues of a Sturm-Liouville problem are real.

5 . STURM-LIOUVILLE PROBLEMS 5 Proof. Consier the following inner prouct on the space of complex-value solutions of a Sturm-Liouville problem 8 for various λ u, v = u x v x Suppose u x = v x = R x + ii x that is, suppose we set u equal to v an split it up into its real an imaginary parts. Then u, u = = R x + ii x R x ii x R x 2 + I x 2] Note now that the integran is always non-negative. By a well-known result from Calculus, the integral of a continuous non-negative function over a finite interval is always non-negative, an zero only if the function is ientically equal to zero. We conclue that for any continuous, non-zero, complex-value function u x u x u x > Now consier Lagrange s ientity Theorem.4 when we set v x = u x, ] 9 u x L u x = u xl u x] an suppose u x is a Sturm-Liouville eigenfunction: L u] = λru for some λ C Taking the complex conjugate of this equation, we have, because p x, q x, an r x are all assume to be real-value functions, or L u] = λrv = = L u] = λru = L u] = λru p x u ] q x u = λru p x u ] q x u = λru In other wors, if u x is a solution of an S-L problem corresponing to eigenvalue λ, its complex conjugate function will be a solution of the same S-L problem, but with eigenvalues λ. Now plugging the right han sies of an into 9, we can conclue that or 2 λ ru u = λu ru λ λ ruu = We have argue above that u x u x is a non-negative function, an by assumption r x is a positive function on, ]. Thus, the integran is a non-negative function an < ruu so long as u x is not ientically zero. We can therefore conclue from 2 that λ λ = = λ R.

6 . STURM-LIOUVILLE PROBLEMS 6 In Linear Algebra, we say that an eigenvalue λ of a matrix A has multiplicity m if the imension of the corresponing eigenspace is m; that is to say, m = im NullSp A λi In Sturm-Liouville theory, we say that the multiplicity of an eigenvalue λ of a Sturm-Liouville problem a φ + a 2 φ = b φ + b 2 φ = L φ] = λr x φ x if there are exactly m linearly inepenent solutions for that value of λ. T 5.7. The eigenvalues of a Sturm-Liouville problem are all of multiplicity one. Moreover, the eigenvalues form an infinite sequence an can be orere accoring to increasing magnitue: {eigenvalues} = {λ, λ 2, λ 3,...}, λ < λ 2 < λ 3 < Proof. By "all eigenvalues have multiplicity one", we mean that for each Sturm-Liouville eigenvalue λ, there is only one linearly inepenent solution φ x of the Sturm-Liouville problem. Suppose there were two, φ an φ 2, linearly inepenent solutions of p x φ q x φ = λr x φ x a φ + a 2 φ = b φ + b 2 φ = Then, as solutions of a secon orer, homogeneous, linear ODE, linear inepenence implies W φ, φ 2 ] for all t, ] To see this explicitly, nee some properties of the Wronskian. L 5.8. If W y, y 2 ] x y x y 2 x y x y 2 x =, then y 2 x = Cy x, wtih C a constant. Proof. We can regar the hypothesis of the lemma as a ifferential equation for y 2. stanar form of a first orer linear ODE, it becomes y y 2 y 2 = which has as its general solution ] y y 2 x = C exp x y x y Putting it in the = C exp ln y x = C exp ln y x = Cy x We also have the following lemma which is in fact known as Abel s Theorem L 5.9. If y an y 2 are two solutions of y + p x y + qy = then 3 W y, y 2 ] x = c exp p x. Proof. Since y an y 2 are solutions y + p x y + qy = y 2 + p x y 2 + qy 2 =

7 . STURM-LIOUVILLE PROBLEMS 7 If we multiply the first equation by y 2 x, the secon equation by y x an a the resulting equations, we have y y 2 y 2 y p x y y 2 y y 2 = or y y 2 y y 2 + p x y y 2 y 2 y = which says the Wronskian W x = y x y 2 x y x y 2 x of y an y 2 satisfies W x + p x W x =. This is a first orer, homogeneous, linear ifferential equation whose solution is well-known to be ] W x = C exp p x C 5.. If y x an y 2 x are two solutions of a secon orer, homogeneous, linear ODE, an W y, y 2 ] x = at any point x in their omain, then y x an y 2 x are linearly epenent i.e. y 2 x = cy x with c a constant. Proof. Note that the exponential factor in this formula 3 for W x is never equal to so long as its argument is efine. Therefore, if W x = at some point x, the constant C must equal zero, an hence W x must vanish at all points x. But if W x = for all x, we have the hypothesis of Lemma 5.8 an so y 2 x = Cy x. Let s now return to the setting of the theorem we re trying to prove, where φ an φ 2 are two solutions of a S-L problem with the same eigenvalue λ. In view of Corollary 5., if I can show that W φ, φ 2 ] =, I can conclue that W φ, φ 2 ] x = for all x, an hence φ an φ 2 are linearly epenent. Now since φ an φ 2 also satisfy the S-L bounary conitions a φ + a 2 φ = a φ 2 + a 2 φ 2 = or ] ] ] φ φ a φ 2 φ = 2 a 2 This a homogeneous 2 2 system. As such it either has a unique solution which is a = a 2 =, which will not occur if we have a legitimate S-L bounary conition at x = or φ φ = et φ 2 φ = φ 2 φ 2 φ φ 2 = W φ, φ 2 ] We can now argue W φ, φ 2 ] = W φ, φ 2 ] x = for all x φ 2 x = λφ x φ, φ 2 are linearly epenent The theorem also inclues the statement that the eigenvalues of a S-L can be liste in a natural orer: λ < λ < λ 2 < Since we alreay know the eigenvalues are real an istinct, the only thing we really have to show that there is a lowest eigenvalue λ. This is actually not too har to o, but the argument takes us well astray of the rest of the course. It involves the use of a sort of "secon erivative test" from variational calculus to show that a minimal solution exists. The orering of the eigenvalues λ, an the fact that they multiplicity free, establishes a certain canonical orering of the corresponing Sturm-Liouville eigenfunctions. We enote by φ n the solution of the Sturm- Liouville problem with λ = λ n. Actually, this only etermines φ n x up to a constant multiple because if

8 . STURM-LIOUVILLE PROBLEMS 8 φ x satisfies 3, 4a, 4b then so oes any constant multiple of φ x. It is common practise to remove this ambiguity by emaning in aition which fixes φ x up to a scalar factor of the form e iα. φ n x φ n xr x = Moreover, the facts that the eigenvalues are all real an multiplicity free, also implies that we can choose the φ n x to be real-value functions of x because the complex span of the functions φ n x must coincie with that of φ n x. Thus, we can choose the eigenfunctions of a Sturm-Liouville problem to be real-value functions φ n x φ n xr x = T 5.. Let φ, φ 2, φ 3,... be the normalize eigenfunctions of a Sturm-Liouville problem an suppose f is a piecewise continuous function on, ]. Then if the series converges to at each point on,. c n := f x φ n x r x c n φ n x n= f x + + f x 2 Here f x ± := lim f x ± ε ε + Note that at any point x where f is continuous f x + + f x 2 = f x. In summary, any time you have a ifferential equation of the form p x y ] q x y = λr x y with homogeneous bounary conitions of the form Then: a y + a 2 y = = b y + b 2 y Solutions will exist for only a iscrete but otherwise infinite set of values for the parameter λ, all of which are real numbers. There is a unique lowest eigenvalue λ an the other eigenvalues can be totally orere λ < λ < λ 2 < λ 3 < For each λ n, there is a solution φ n of p x y ] q x y = λ n r x y that is unique up to a scalar factor.

9 . STURM-LIOUVILLE PROBLEMS 9 The solutions φ n, n =,, 2,... can be normalize such that { if n = m φ n x φ m x r x = otherwise Any continuous function f x on the interval, ] can be expane in terms of the Sturm-Liouville eigenfunctions φ n, n =,, 2,,... f x = a n φ n x n= with the coeffi cients a n being etermine by a n = f x φ n x r x If f x is merely piece-wise continuous on, ], one has instea f x + + f x = a n φ n x 2 n= with the coeffi cients a n etermine by the same formula.