Exceptional regular singular points of second-order ODEs. 1. Solving second-order ODEs

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1 (May 14, 2011 Exceptional egula singula points of second-ode ODEs Paul Gaett gaett/ 1. Solving second-ode ODEs 2. Examples 3. Convegence Fobenius method fo solving u + b(x x is slightly moe complicated when the indicial equation u + c(x u = 0 (with b, c analytic nea 0 x2 α(α 1 + b(0α + c(0 = 0 has epeated oots o oots diffeing by an intege. [1] An impotant example in which this occus is the diffeential equation in adial coodinates fo spheical functions on hypebolic n-space: u + (n 1 coth u λ u = 0 The point 0 is a egula singula point fo this equation. The indical equation is α(α 1 + (n 1α = 0 When n = 2, the indicial equation has a double oot 0. When n 3, the oots ae 0 and 2 n, which diffe by an intege. 1. Solving second-ode ODEs [1.1] Fobenius method Let E = be the diffeential opeato and let d2 dx 2 + b(x x d dx + c(x x 2 (with b, c C[[x]] p(α = α(α 1 + b(0α + c(0 C[α] Conside f(x, α = x α a n x n (on x > 0, with a 0 0 n 0 with a n depending on α. We can solve Ef(x, α = p(α a 0 x α by ecusively solving fo the coefficients a n, as follows. Expand [1] Fo example, see [Coddington 1961], chapte 4, section 6. 1

2 Paul Gaett: Exceptional egula singula points of second-ode ODEs (May 14, 2011 Ef(x, α = p(α a 0 x α + + [ ] p(α + 1 a 1 + (something involving a 0 x α+1 [ ] p(α + 2 a 2 + (something involving a 0, a 1 x α+ That is, the coefficient of x α+n 2 in Ef is [ ] (α + n(α + n 1 + b(0(α + n + c(0 a n + (linea combination of a l s with l < n The coefficient of x α is p(α a 0. Fo n 1, as long as the coefficient p(α + n C of a n is non-zeo, the condition that the coefficient of x α+n vanish detemines a n in tems of a l with l < n. Thus, when the two oots of p(α = 0 do not diffe by an intege, the diffeential equation has a solution of the fom x α n 0 a nx n with eithe oot α, since the ecusion fo the coefficients succeeds. It is not difficult to pove that the seies has positive adius of convegence: we pove this late. [1.1.1] Remak: In any case, a fist solution can be obtained by taking α to be the solution of the indicial equation with lage eal pat, thus avoiding failue of the ecusion. The poblem is to obtain a second solution when the oots of the indicial equation ae equal o diffe by an intege. [1.2] Simple exceptional case The simple exceptional case is that p(α = (α α o 2. Then the ecusion fo the coefficients a n succeeds, but poduces just one solution f(x, α o. To obtain a second solution, apply / α to and evaluate at α = α o : Ef(x, α = p(α a 0 x α E f α (x, α o = p (α o a 0 x αo + p(α o a 0 x αo log x = 0 Given the expansion f(x, α = x α n a n x n, the expansion of the second solution is f α (x, α o = x αo n a n α xn + log x x αo a n x n = x αo n n a n α xn + log x f(x, α o Since a 0 does not depend on α, the diffeentiation in α annihilates this tem. Thus, the leading tem of the second solution is log x x α a 0 : taking a 0 = 1, (fist solution = x αo +... (fo p(α = (α α o 2 (second solution = log x x αo +... We pove convegence late. [1.3] Less-simple exceptional case In the less-simple exceptional case, the oots of the indicial equation ae α o and α o n o with a positive intege n o. The ecusion fo the coefficients of a solution of the fom x α n 0 a nx n may fail at n o, because it appaently equies division by p((α o n o + n o = p(α o = 0 That is, f(x, α is of the fom f(x, α = A(x, α + B(x, α α (α o n o 2

3 Paul Gaett: Exceptional egula singula points of second-ode ODEs (May 14, 2011 whee A(x, α is the polynomial in x consisting of tems of degee below n o, and B(x, α is the pat aboveo-equal that degee. Application of the diffeential opeato E to (α (α o n o f(x, α gives ( E ((α (α o n o f(x, α = ((α (α o n o p(α a 0 x α = 0 α α=αo n o α α=αo n o That is, a second solution is ((α (α o n o f(x, α α α=αo n o = A(x, α o n o + B α (x, α o n o Tem-wise, witing a second solution is f(x, α = x α n<n o a n x n + x α α (α o n o n n o b n x n ((α (α o n o f(x, α = x αo no a n x n + log x x αo no b n x n b + x αo no n α α=αo n o n<n o n n o n n o In paticula, the log tems only appea in the highe-ode tems: taking a 0 = 1, (fist solution = x αo +... (second solution = x αo no +... We pove convegence late. α xn (fo p(α = (α α o (α (α o n o and 0 < n o Z 2. Examples [2.1] Euclidean n-space: adial eigenfunctions nea 0 In adial coodinates, the Laplacian on Euclidean n-space is The coesponding eigenvalue equation is u + n 1 u u + n 1 u λu = 0 The equation fo a fundamental solution is also of inteest: u + n 1 u λu = δ (with Diac delta Rewitten to appaise the egula singula point at 0, the opeato is u + n 1 u 2 λ 2 u also showing that the indicial equation is independent of λ: it is α(α 1 + (n 1α = 0 3

4 Paul Gaett: Exceptional egula singula points of second-ode ODEs (May 14, 2011 with oots 0, 2 n, notably diffeing by an intege. Fo n = 2, the oot is double-0. With λ = 0, the equation u + n 1 u = 0 is of Eule-Cauchy type, explicitly solvable: the two solutions ae 1 and 2 n (fo n 2 fist and second solutions = 1 and log (fo n = 2 (fo λ = 0 In this example, the fist solution 1 is analytic at 0. The second solution is not analytic at 0. Fo geneal λ, the fist solution is analytic at 0, the second solution is not analytic at 0, and should be expected to have logaithmic tems even fo n > 2, although these will not appea in the leading tem: fist and second solutions = and 2 n +... (fo n and log +... (fo n = 2 (fo λ 0 [2.1.1] Remak: Computation of asymptotics at the iegula singula point at infinity eveals that fo odd n the two solutions of the homogeneous equation have elementay expessions of the fom e ± λ (polynomial in 1 Fo example, e ± λ (on R 1 e ± λ ( e ± λ ( ± 3 λ e ± λ ± 5 λ λ (on R 3 (on R 5 (on R 7 Thus, in fact, logaithmic tems do not appea in the asymptotic nea 0 fo n odd, although this is not obvious fom the expansion at 0. [2.1.2] Remak: Futhe, in the elementay expessions fo odd n, fo the blow-ups at 0 to cancel, up to a constant the fist solution, analytic at 0, must be the sum of the two expessions. Thus, the fist solution can neve have exponential decay at infinity. Unless λ is puely imaginay, it will have exponential blow-up. Fo λ puely imaginay, it will have polynomial decay. [2.1.3] Remak: Since the suppot of δ is just {0}, a solution of u + n 1 u λu = δ satisfies the homogeneous fom of the equation away fom 0, so is a linea combination of the fist and second solution, with the second solution being necessay to poduce δ at 0. Fo Re(λ > 0, Fouie tansfom methods on R n poduce an exponentially deceasing fundamental solution. 4

5 Paul Gaett: Exceptional egula singula points of second-ode ODEs (May 14, 2011 [2.2] Hypebolic n-space: adial eigenfunctions nea 0 The diffeential equation in adial coodinates fo zonal spheical functions on hypebolic n-space is u + (n 1 coth u λ u = 0 Rewitten to highlight the egula singula point at = 0, it is u + (n 1 coth u 2 λ 2 u = 0 The indical equation is the same as that fo Euclidean space, and is independent of λ: α(α 1 + (n 1α = 0 When n = 2, the indicial equation has a double oot 0. When n 3, the oots ae 0 and 2 n, which diffe by an intege. Since the solution of the indicial equation with lage eal pat is 0, the fist solution is egula at = 0. This is the zonal spheical function fo that eigenvalue. When n = 2, the second solution has a logaithmic leading tem. Fo n > 2, the second solution has logaithmic tems futhe out in the expansion, but not in the leading tem: fist and second solutions = and 2 n +... (fo n and log +... (fo n = 2 (nea 0 [2.2.1] Remak: Away fom 0, a solution to u + (n 1 coth u λ u = δ (Diac delta at base point satisfies the coesponding homogeneous equation. Being a linea combination of the fist and second solutions, it must always non-tivially include the second solution to obtain δ at 0. In paticula, the full asymptotics of a fundamental solution nea 0 should be expected to include logaithmic tems fo n > 2, although not in the leading tem. [2.3] Hypebolic n-space: special adial eigenfunctions nea + Unlike the Euclidean case, in hypebolic n-space the adial (spheically symmetic Laplacian has a egula singula point at infinity, in suitable coodinates. With y = e, it suffices to look at y 0 + o, equivalently y +. The y-coodinate is aleady adapted to the case y 0 +, so we conside this. The equation becomes y 2 u + y(1 + (n 1 y + 1/y y 1/y u λu = 0 (as y 0 + The singula point at 0 is visibly egula, with indicial equation α(α (n 1 λ = 0 Paametizing the eigenvalue by λ = s(s (n 1, the oots ae α = s (n 1 s (with eigenvalue λ = s(s (n 1 5

6 Paul Gaett: Exceptional egula singula points of second-ode ODEs (May 14, 2011 The special case s = (n 1/2 plays a cental ole in HaishChanda s teatment of Schwatz functions, and in that case (n 1/2 is a double oot. Thus, the two solutions ae (fist solution = y n 1 (second solution = log y y n 1 (as y 0 + The spheical symmety gives symmety unde y 1/y, so (fist solution = y n 1 (second solution = log y y n 1 In tems of the oiginal adius, (fist solution = e n (second solution = e n (as y + (as + 3. Convegence [3.1] Fist solutions We ecall the elatively easy agument fo convegence of the seies fo the fist solution. Let A, M 1 be lage enough so that b(x = b n x n (with b n A M n n 0 c(x = n 0 c n x n (with c n A M n Inductively, suppose that a l (CM l, with a constant C 1 to be detemined in the following. Then n n(n+α α a n A n i+α M i (CM n n i +A M i (CM n i AM n C n 1( n(n + 1 +n α +n 2 i=1 Dividing though by n n + α α, this is This motivates the choice i=1 a n AM n n 1 (n α + 2 C 2 n + α α C (n α + 2 sup 1 n Z 2 n + α α which gives a n A(CM n, and a positive adius of convegence. [3.2] Simple second solution Poof of convegence of the second solution when p(α = (α α o 2 equies fine attention to the dependence on α. The ecusion fo the coefficients a n of f(x, α is p(α + n a n = (α + la l b n l + a l c n l 6

7 Paul Gaett: Exceptional egula singula points of second-ode ODEs (May 14, 2011 Diffeentiating with espect to α, p (α + n a n p(α + n a n α = a l b n l + (α + l a l α b n l + Invoking the estimates fom above, and inductively supposing a n / α (DM n with D C, a l α c n l Thus, Taking + a n α (CM l AM n l p(α + n a n α p (α + n (CM n + α + l (DM l AM n l + (DM l AM n l p (α + n (CM n p(α + n + A(n + n α n(n n Dn 1 M n p(α + n ( p (α + n C D sup + A(3n + n2 + n α n 1 p(α + n p(α + n gives the inductive step, poving that the second solution conveges absolutely on some non-tivial inteval 0 < x < x o. [3.3] Less-simple second solution A simila but messie agument succeeds in the less-simple case, as well, poving convegence on some non-tivial inteval. Bibliogaphy [Coddington 1961] E.A. Coddington, An intoduction to odinay diffeential equations, Dove,