This paper is a contribution to certain algebraic and analytic aspects of higher order self-adjoint operators with polynomial coefficients

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

HIGHER ORDER SELF-ADJOINT OPERATORS WITH POLYNOMIAL COEFFICIENTS

HASSAN AZAD, ABDALLAH LARADJI, MUHAMMAD TAHIR MUSTAFA

Abstract. We study algebraic and analytic aspects of self-adjoint operators of order four or higher with polynomial coefficients. As a consequence, a sys- tematic way of constructing such operators is given. The procedure is applied to obtain many examples up to order 8; similar examples can be constructed for all even order operators. In particular, a complete classification of all order 4 operators is given.

1. Introduction

The classification of self-adjoint second order operators with polynomial coeffi- cients is a classical subject going back to Brenke [7]. This paper is a contribution to certain algebraic and analytic aspects of higher order self-adjoint operators with polynomial coefficients. Its main aim is to construct such operators. This involves determining the explicit differential equations for the polynomial coefficients of the operators and the boundary conditions which ensure self-adjointness. These oper- ators are not in general iterates of second order classical operators - as stated in [17]; (cf. [22]). As the weight function which makes these operators self-adjoint depends only on the first two leading terms of the operator, therefore, if one can find a second operator with the same weight function, the eigenpolynomials for both operators would be the same; see Section 4.

We should point out that some of the most important recent contributions to this subject are due to Kwon, Littlejohn and Yoon [18], Bavnick [3, 4, 5], Koekoek [12] and Koekoek-Koekoek [13]; see also the references therein.

A classical reference for higher order Sturm-Liouville theory is the book of Ince [10, Chap.IX]. This theory was revived by Everitt in [8]; see also Everitt et al. [9].

Classical references that deal with various aspects of polynomial solutions of differential equations are the references [6, 7, 20, 16, 25]. More recent papers that deal with the same subject are [1, 2, 14, 15, 21, 27]. A reference for the related topic of orthogonal polynomials is [26]. A more recent reference for this topic which also has an extensive bibliography is the book [11]. We should point out that the classification problem we address is related to the Bochner-Krall problem of finding all families of orthogonal polynomials that are eigenfunctions of a differential operator (see [6, 7, 9, 15, 19]).

2010Mathematics Subject Classification. 33C45, 34A05, 34A30, 34B24, 42C05.

Key words and phrases. Self-adjoint operators; polynomial coefficients.

c

2016 Texas State University.

Submitted October 21, 2016. Published January 18, 2017.

1

Here is a more detailed description of the results of this paper. We consider linear differential operators with polynomial coefficients that map the space of polynomi- als of degree at most k into itself - for allk. Proposition 2.1 gives necessary and sufficient conditions for such an n^th order linear operator to be self-adjoint. The integrability and asymptotic properties of the weight function and its derivatives near the zeroes of the leading term given in Propositions 3.3 and 3.4 carry enough information to determine the form of the first two terms of the operator in spe- cific cases. This is used further to determine the full operator, using the boundary conditions and the determining equations of Proposition 2.1. Although such deter- mining equations are known (see for example [23]), boundary conditions involving all polynomial coefficients of the linear operator do not seem to have been consid- ered earlier in the general case and these are equally crucial to the construction of operators given here. We should point out that the (n+ 1) determining equations in Proposition 2.1 are equivalent to a system of ⁿ₂ equations, as shown in [23].

The classical second order operators are completely determined by the integra- bility of the associated weight function.

In general, if the operator is of the formL(y) =ay⁽ⁿ⁾+by⁽ⁿ⁻¹⁾+. . . then, as shown in Proposition 3.5, forn >2, the multiplicity of each root of ais at least 2 and its multiplicity inbis less than its multiplicity inaand it is of multiplicity at least 1 inb. In particular, if the operator is of fourth order and the leading term has distinct roots, then every root occurs with multiplicity 2 and therefore the leading term must have exactly two distinct roots and their multiplicity in the next term is 1 - as shown in Proposition 3.5.

In [22], fourth order Sturm-Liouville systems were given for ordinary weights and for weights involving distributions. In particular, for ordinary weights, the authors found operators that are iterates of second order ones. In this paper, a systematic way of producing Sturm-Liouville systems with ordinary weights for all even orders is given. Although it is believed that all fourth order operators with classical weights are iterates of second order ones (see [17, 22]), the results we obtain show this is in fact incorrect. Indeed, the classification of fourth order operators we provide here (see Section 4.2) includes examples of operators that cannot be iterates of second order ones, as an eigenvalue argument shows.

The examples II.6, II.7 of fourth order operators in [22] with weights involving delta and Heaviside functions and the sixth order operator in Example 4.1 in [18]

with weight involving delta functions are solutions of the determining equations given in Section 4. For these 4th order operators of [22], the 3rd boundary condition in the sense of (4.6) fails at one or both the boundary points. For the sixth order operator of Example 4.1 in [18] the last three boundary conditions in (4.12) do not hold. Many such examples can be constructed but the solutions of the determining equations are too many to be listed efficiently (cf. Example 4.1, Section 4.3).

We also obtain examples of sixth and eighth order operators where all the bound- ary conditions hold. Due to space constraint, these examples are provided in the expanded online version of the paper at http://arxiv.org/abs/1409.2523. In fact similar examples can be constructed for any even order; such constructions involve increased computational complexity.

All the solutions presented in the examples have been first verified using Mathe- matica and then directly generated (from the same file) by Mathematica as LaTeX output for the paper.

2. Algebraic aspects of higher order Sturm-Liouville theory Consider, on the space C^∞, thenth-order linear operator L =Pn

k=1ak(x)D^k, whereD is the usual differential operator and eacha_k :=a_k(x) is a polynomial of degree at mostk. In this way, for each natural numberN, the vector spacePN of all polynomials of degree at mostNisL-invariant. Our first objective is to obtain con- ditions on the polynomialsa_kfor the existence of an inner producthu, vi=R

Ipuvdx onC^∞ for which Lis self-adjoint, and where the weight pis sufficiently differen- tiable in a real interval I where an does not vanish. This smoothness assumption is reasonable since it is satisfied by all weights of classical orthogonal polynomials.

For a function f and an interval J with (possibly infinite) endpoints α < β, ∂J denotes the boundary {α, β} ofJ, and [f]J means lim_x→β−f(x)−lim_x→α+f(x), where both limits are finite. The statement that a function vanishes at an endpoint of the interval is to be understood in the sense of limits. For notational convenience, letbj:=paj (1≤j≤n), andb0be the zero function.

Our main result in this section is the following proposition. As mentioned in the introduction, the determining equations in (i) are known (see for example [23]).

Proposition 2.1. With the above notation, forL to be self-adjoint with respect to the inner product hu, vi=R

Ip(x)u(x)v(x)dx, it is necessary that p(x) =

exp _n²R an−1(x) an(x) dx

|a_n(x)|

onIand thatnbe even. Conversely ifp(x) =^exp

2 n

R an−1 (x) an(x) dx

|a_n(x)| andbj=paj, it is necessary and sufficient for Lto be self adjoint that the following conditions hold.

(i) (−1)^{j j}_j

bj+· · ·+ (−1)^{n n}_j

b^(n−j)n =bj onI for0≤j ≤n;

(ii)









n−1 j−1

− ⁿ⁻²_j−1

. . . (−1)^n−j

n−2 j−2

− ⁿ⁻³_j−2

. . . (−1)^n−j . . . .

n−j 0

− ^n−j−1₀

. . . (−1)^n−j















 b^(n−j)n

b^(n−j−1)_n−1 . . .

b_j









= 0

on∂I for1≤j ≤n. In particular,n must be even.

To prove the above proposition we need the following lemmas. The first is a formula for repeated integration by parts, while the last one may be of independent interest.

Lemma 2.2([24]). Letf andybe functionsktimes differentiable on some interval I. Then

Z

I

f y^(k)dx= (−1)^k Z

I

f^(k)y dx+ [

k−1

X

j=0

(−1)^jf^(j)y^(k−1−j)]I.

Lemma 2.3. Let f_j (1≤j ≤r) be functions continuous on some interval (a, b), wherebmay be infinite. If there exists a non-singular square matrixA= [a_ij]_1≤i,j≤r such that lim_x→b−Pr

j=1a_ijf_j(x) = 0 for 1 ≤i ≤r, then lim_x→b−f_j(x) = 0 for 1≤j ≤r.

Proof. Putgi(x) =Pr

j=1aijfj(x) (1≤i≤r), so that for allx∈(a, b), A(f₁(x), f₂(x), . . . , f_r(x))^T = (g₁(x), g₂(x), . . . , g_r(x))^T.

The conclusion follows from the fact that lim_x→b⁻gi(x) = 0 and that (f1(x), f2(x), . . . , fr(x))^T =A⁻¹(g1(x), g2(x), . . . , gr(x))^T.

Lemma 2.4. Let vi (0≤i≤r)be functions continuous on a real interval I such that for all polynomials u, [Pr

i=0viu⁽ⁱ⁾]I = 0. Then vi = 0 (0 ≤i ≤r) at each endpoint ofI.

Proof. Suppose first that the endpointsα, β (α < β) ofI are finite. We need only show thatvr= 0 at each endpoint of I, the statement for the remaining vi would then follow by straightforward (reverse) induction. Fromu= (x−α)^r(x−β)^rz (z a polynomial) we get [r!vrz]I = 0 i.e. [vrz]I = 0. Then, from z = 1 and z =x respectively, we get vr(β)−vr(α) = 0 andβvr(β)−αvr(α) = 0. These equations implyvr(β) =vr(α) = 0, as required.

Suppose now that β = ∞(with αpossibly infinite). Put uj =x^r+j (1 ≤j ≤ r+ 1). Then, since limx→∞x^j =∞and limx→∞Pr

i=0 (r+j)!

(r+j−i)!x^r+j−ivi(x) is finite for each j (1 ≤ j ≤ r+ 1), we obtain Pr

i=0

(r+j)!x^r−ivi(x)

(r+j−i)! → 0 as x → ∞. If we now put f_i(x) =x^r−iv_i(x), 0 ≤ i ≤ r, we get lim_x→∞Pr

i=0a_ijf_i(x) = 0 for 1≤j ≤r+ 1, wherea_ij= _(r+j−i)!^(r+j)! . In view of Lemma 2.3, we need only show that the matrixA= [a_ij]0≤i≤r,1≤j≤r+1 is non-singular. We havea_ij =i! ^r+j_i

henceA is non-singular if and only if the matrixB = [ ^r+j_i

]0≤i≤r,1≤j≤r+1 is non-singular.

If, more generally, we letD(m, n) := det[ ^m+j−1_i−1

]1≤i,j≤n+1 form≥n≥0, then it is easy to show that D(m, n) =D(m, n−1). This gives D(m, n) =D(m,1) = 1, so that detB=D(r+ 1, r)6= 0, i.e. Ais non-singular. The case whenαis infinite

is dealt with in a similar way.

Lemma 2.5. Let c andcj (0≤j ≤n) be functions continuous on an interval I.

If [Pn

j=0cjy^(j)]I =R

Icy dx for all infinitely differentiable functions y, thenc = 0 onI and each c_j vanishes at each endpoint of I.

Proof. We first prove thatc= 0 on I. Suppose on the contrary thatc(γ)6= 0 for some γ in I. We can assume thatc >0 on some subinterval [δ, ε] ofI containing γ. Let

φ(x) =

(exp _{(x−δ)(x−ε)}¹ ) ifδ≤x≤ε

0 otherwise

Then, putting y=φ(x), we get [Pn

j=0c_jy^(j)]_I = 0 and so R

Icy dx=Rε

δ cφ(x)dx= 0. This is impossible since the integrand in this last integral is positive. This shows thatc= 0 onI. By Lemma 2.4, eachcj equals zero at each endpoint ofI, and the

proof is complete.

Proof of Proposition 2.1. By Lemma 2.2, for any functionsyanduinC^∞we have hLy, ui=

Z

I

p(Ly)udx=

n

X

k=1

Z

I

pa_ky^(k)udx=

n

X

k=1

Z

I

(pa_ku)y^(k)dx

=

n

X

k=1

h^k−1X

j=0

(−1)^j(pa_ku)^(j)y^(k−1−j)i

I

+ (−1)^k Z

I

(pa_ku)^(k)y dx .

(2.1)

Suppose first thatLis self-adjoint. ThenhLy, ui=hy, Lui, i.e.

n

X

k=1

h^k−1X

j=0

(−1)^j(pa_ku)^(j)y^(k−1−j)i

I

+ (−1)^k Z

I

(pa_ku)^(k)y dx

=

n

X

k=1

Z

I

paku^(k)y dx.

(2.2)

Fixuand put c=

n

X

k=1

(−1)^k(bku)^(k)−

n

X

k=1

bku^(k), cj =

n

X

k=j+1

(−1)^k−1−j(bku)^(k−1−j) (0≤j ≤n−1). Then (2.2) gives [Pn−1

j=0 cjy^(j)]I =R

Icy dx, which by Lemma 2.5 implies c = 0 on I and cj = 0 (0≤ j ≤ n−1) at each endpoint of I. Applying Leibniz rule to the terms (bku)^(k) ofcwe obtain

n

X

k=1

bku^(k)=

n

X

k=1

(−1)^k

k

X

j=0

k j

b^(k−j)_k u^(j). (2.3) Since this is true for alluin C^∞, we can equate coefficients of u^(k) and get from k=nthe classical fact thatnis even and that for 0≤k≤n−1,

2bk = k+ 1

k

b⁰_k+1− k+ 2

k

b⁰⁰_k+2+· · ·+ n

k

b^(n−k)_n ifkis odd 0 =−

k+ 1 k

b⁰_k+1+

k+ 2 k

b⁰⁰_k+2− · · ·+ n

k

b^(n−k)_n ifkis even

(2.4)

From k = n−1, we obtain the equation 2pa_n−1 = n(pa_n)⁰, which gives the well-known form of the weightp= exp(_n²R an−1

an dx)/|an|.

Applying Leibniz rule again to cj = Pn

k=j+1(−1)^k−1−j(bku)^(k−1−j) (0 ≤ j ≤ n−1), we obtain in a similar manner, but this time on∂I (i.e. at the endpoints of I) the following equations for 1≤j≤n,

n−1 j−1

b^(n−j)_n − n−2

j−1

b^(n−j−1)_n−1 +· · ·+ (−1)^n−jbj = 0 n−2

j−2

b^(n−j)_n − n−3

j−2

b^(n−j−1)_n−1 +· · ·+ (−1)^n−jbj = 0 . . .

n−j 0

b^(n−j)_n −

n−j−1 0

b^(n−j−1)_n−1 +· · ·+ (−1)^n−jbj= 0 This can be put in matrix formA_jh

b^(n−j)n b^(n−j−1)_n−1 . . . bj

iT

= 0 (1≤j≤n), whereAj is thej×(n−j+ 1) matrix

A_j =









n−1 j−1

− ⁿ⁻²_j−1

. . . (−1)^n−j

n−2 j−2

− ⁿ⁻³_j−2

. . . (−1)^n−j . . . .

n−j 0

− ^n−j−1₀

. . . (−1)^n−j







 .

We thus have (−1)^{j j}_j

b_j+· · ·+ (−1)^{n n}_j

b^(n−j)n =b_j onI for 0≤j≤nand

A_j







 b^(n−j)n

b^(n−j−1)_n−1 . . .

b_j









= 0

on∂I for 1≤j≤n.

Conversely, it is clear that if (−1)^{j j}_j

b_j+· · ·+ (−1)^{n n}_j

b^(n−j)n = b_j on I for 0≤j ≤nand

Aj







 b^(n−j)n

b^(n−j−1)_n−1 . . .

bj









= 0

on ∂I, then equation (2.2) holds for all functions y, uin C^∞, and therefore L is

self-adjoint.

The following observation is particularly useful. When, in the above proof, j ≥n−j+ 1 i.e. j ≥1 +n/2, we get more equations than “unknowns” b^(n−j)n , b^(n−j−1)_n−1 , . . .,bj. Thus, deleting the first 2j−n−1 rows ofAjand puttingk=n−j (0 ≤ k ≤ n/2−1), we obtain the equations Bk

h

b^(k)n b^(k−1)_n−1 . . . bn−k

iT

= 0 whereBk is the (k+ 1)×(k+ 1) matrix

Bk=









2k k

− ^2k−1_k−1

. . . (−1)^k

2k−1 k

− ^2k−2_k−1

. . . (−1)^k . . . .

k k

− ^k−1_k−1

. . . (−1)^k







 .

Clearly, detBk =±detEk where

Ek=









0 0

1 0

. . . ^k₀

1 1

2 1

. . . ^k+1₁ . . . .

k k

k+1 k

. . . ^2k_k







 .

As in the proof of Lemma 2.4, if we let

E(m, k) =

m−k 0

_m−k+1

0

. . . ^m₀

m−k+1 1

m−k+2

1

. . . ^m+1₁ . . . .

m k

m+1 k

. . . ^m+k_k ,

then elementary column operations giveE(m, k) =E(m, k−1) =· · ·=E(m,1) = 1. Therefore detEk =E(k, k)6= 0, i.e. Bk is non-singular, and we obtain

b^(k)_n =b^(k−1)_n−1 =· · ·=bn−k= 0 on∂I for 0≤k≤n/2−1. (2.5) An interesting consequence of this is that if the weightp= 1, thena^(k)n =a^(k−1)_n−1 = 0 for 0 ≤ k ≤ n/2−1 on ∂I, and so, if an is not constant, I must be a finite interval [α, β] withan=A(x−α)^n/2(x−β)^n/2 for some non-zero constantAand a_n−1 = ^An₂²(x−(α+β)/2)(x−α)^n/2−1(x−β)^n/2−1 (recall that the degree of a_k is at most k and that 2pa_n−1 = n(pa_n)⁰). We thus obtain the form of the

two leading polynomial coefficients of what may be considered as the n-th order Legendre differential equation.

Proposition 2.1 gives a necessary and sufficient set of conditions under which the operatorLis self-adjoint with respect to an inner product of the formhu, vi= R

Ipuvdx. This was achieved under the assumption thatpis an admissible weight, that isR

Ipudxis finite for allC^∞functionsu, and thatpsatisfies certain differen- tiability conditions. It is therefore highly desirable that we obtain conditions under which this assumption holds. The next section is devoted to such an analysis.

3. Analytic aspects of Sturm-Liouville theory Keeping the same notation as before, let p= exp _n²R b

adx

/|a| be the weight function, where, for brevity,a=anis a polynomial of degree at mostnandb=a_n−1 is a polynomial of degree at mostn−1. Without loss of generality, we may assume ato be monic. The weight functionpis a priori defined on an interval whereadoes not vanish. However, on an intervalI whereamay have roots, it is clear that we can take pto be defined piecewise, with p= exp(²_nR _b

adx)/|a|except at the roots of a. Thus it is important to discuss the integrability of the weight function near the roots ofaand the differentiability at these roots. The integrability is basically a consequence of the following lemma.

Lemma 3.1. For >0 andd, α integers withα >0,R 0

e^kxd

x^α dx exists if and only if k <0 andd <0.

Proof. If d = 0 and α = 1, then clearly the integral is +∞. So assume that d > 0. Then for sufficiently small positive x, 1/2 < e^kxd < 3/2 and therefore

1

2x^α < ^ekxd_xα < _2x³α. Integrating over the interval [0, ], clearly the integral is infinite forα= 1, whereas forα >1,

0< η <

Z η

x^−αdx= ^−α+1

−α+ 1 + 1

α−1η^−(α−1), so the integralR

0 e^kxd

x^α dxis infinite.

Hence a necessary condition for the integral to be finite is that d <0. Let us writed=−δwithδ >0. Therefore,

Z 0

e^kxd x^α dx=

Z 0

e^k(1x)δ x^α dx.

Ifk= 0, this is integrable if and only if −α+ 1>0, which is not the case.

Ifk >0, then Z

0

e^{k( 1x)}

δ

x^α dx= Z 1/

∞

u^αe^kuδ(−1)u⁻²du= Z ∞

1/

u^α−2e^kuδdu >

Z ∞

1 +1

e^kuu^α−2du (asδis a positive integer).

Ifα−2≥0, this is clearly infinite.

For the remaining caseα= 1, substitutingku=v, the integralR∞

1

+1e^kuu^α−2du becomesR∞

k(¹+1) e^v

vdv.

As ^e_v^v >^v₂ (forv >0), this integral is clearly infinite.

Thus for > 0, a necessary condition forR 0

e^kxd

x^α dx to be finite is that k <0, d <0. It remains to show that in this case, the above integral is finite.

Let d = −δ, k = −l, δ > 0, l > 0. So ^ekxd_xα = u^αe^−luδ, where u = _x¹. Now lim_u→∞^e_u^luδ_α = ∞, so limu→∞u^αe^−luδ = 0. Thus R

0 e^kxd

x^α dx is bounded near 0.

Hence the functionf(η) =R η

e^kxd

x^α dx, with 0< η < is monotonic bounded, hence its limit asη→0⁺exists. This completes the proof of the lemma.

Corollary 3.2. For >0,d, α integers with α >0,R0

− e^kxd

|x^α|dx exists if and only if k(−1)^d <0 andd <0.

The proof of the above corollary follows by substituting u=−xand applying Lemma 3.1.

We say that a functionf(x) defined and continuous on an open intervalI con- taining 0 as a left end point is left integrable at 0 if for anyη ∈I, lim_→0+Rη

f(x)dx exists. Similarly if f(x) is a function defined and continuous on an an open inter- val I containing 0 as a right end point is right integrable at 0 if for any η ∈ I, lim_→0−R

ηf(x)dx exists. Clearly this is equivalent to saying that the function g(x) =f(−x) is left integrable at 0. By suitable translations, one can define the concept of left and right integrability at the end points of an intervalI on which the given function is defined and continuous. Let r be a zero of a(x) = an(x) and let ma(r) =α andmb(r) = β be the multiplicities of ras a root of a(x) and b(x) =b_n−1(x). Thus _a(x)^b(x) = (x−r)^β−αφ(x) whereφ(x) is a rational function with φ(r)6= 0. Hence _a(x)^b(x) =φ(r)(x−r)^β−αψ(x) whereψ(r) = 1.

Definition. We say that a rootrofa(x) is an ordinary root ifmb(r)−ma(r)+16= 0, and it is a logarithmic root ifmb(r)−ma(r) + 1 = 0.

Using Lemma 3.1 we have the following result which is one of the main tools for explicit determination of self-adjoint operators.

Proposition 3.3. Letβ =mb(r), α=ma(r), so that_a(x)^b(x) = (x−r)^β−αφ(x), where φ(x)is a rational function withφ(r) = lim_x→r(x−r)^{α−β b}_a(x)^(x) 6= 0. Then

(i) For an ordinary rootr ofa, the weight functionp(x)is integrable from the right atr if and only if α−β ≥2 andφ(r)>0. It is integrable from the left at r if and only if α−β ≥2 and (−1)^α−βφ(r)<0. In this case, the weight function p(x)is respectively right/left C^∞ differentiable at r andp and all its (one sided) derivatives vanish at r.

(ii) For a logarithmic rootrofa, the weight functionp(x)is right/left integrable near r if and only if ^|x−r|

2 nφ(r)

|x−r|^α is integrable near r if and only if _n²φ(r)− α+ 1>0.

Proof. Using the notation in this proposition, we can write a(x) = (x−r)^α(a0+a1(x−r) +. . .),

b(x) = (x−r)^β(b₀+b₁(x−r) +. . .),

wherea0,b0are not zero. For convenience of notation, we may suppose thatr= 0.

The weight function, near the rootr= 0 can then be written as p(x) =

exp(R ₂

n b(x) a(x)dx)

|a₀||x^α| ψ(x),

where ψ(0) = 1 and ψ is infinitely differentiable near 0. Hence the order of left or right differentiability at 0 ofp(x) is the same as that of ^exp(

R ₂

n b(x) a(x)dx)

|x^α| . We will prove differentiability at 0 for all orders after the proof of Proposition 3.4.

For the integrability, we may suppose that near 0, 1/2< ψ(x)<3/2 and there- fore near 0, we have the estimate

1 2 e

R 2 n

b(x) a(x)dx

|a0||x^α| < p(x)<3 2 e

R 2 n

b(x) a(x)dx

|a0||x^α| .

This means thatp(x) is left or right integrable at 0 if and only if exp(R ₂

n b(x)

a(x)dx)/|x^α| is left or right integrable at 0. Now

2 n

b(x) a(x) = 2

nd0

(1 +d1x+. . .) 1 +c₁x+. . . = 2

nd0x^β−αψ(x),

withψ(0) = 1. Thus, as long asxdoes not change sign, there are positive constants k1, k2 such that

k1

2

nd0x^β−α≤ 2 n

b(x) a(x) ≤k2

2

nd0x^β−α.

Therefore, as long asxdoes not change sign, and interchanging k1, k2for negative values ofx, we have the estimate - for a base pointp0

e

Rx

p0(k₁²_nd₀t^β−α)dt

|x^α| ≤e

Rx

p0(²_n^b(t)_a(t))dt

|x^α| ≤e

Rx

p0(k_2n²d₀t^β−α)dt

|x^α|

Assuming that 0 is an ordinary root - that isβ−α6=−1 and integrating we get A1e^(k1n2d0)(^x_β−α+1^β−α+1)

|x^α| ≤e

Rx

p0(²_n^b(t)_a(t))dt

|x^α| ≤A2e^(k2n2d0)(^x_β−α+1^β−α+1)

|x^α| whereA1, A2 are positive constants.

Thus by Lemma 3.1, the weight function is integrable from the right if and only if _β−α+1^d0 <0 andβ−α+ 1<0. Asα, βare integers, we getd0>0 andα−β ≥2.

It is integrable from the left if and only if ^d0(−1)_β−α+1^β−α+1 <0 andβ−α+ 1<0. Thus, the requirement for left integrability becomesd0(−1)^β−α+1>0 andα−β ≥2.

Hence the weight is both left and right integrable if and only ifd0>0,β−α+1<

0, (−1)^β−α+1 = 1. Thusβ−α+ 1 is an even negative integer andd0>0 are the requirements for both integrability from the left and right.

Recall that _a(x)^b(x) =d0x^β−αψ(x), withψ(0) = 1. Thusx^{α−β b}_a(x)^(x) =d0ψ(x). Hence lim_x→0x^{α−β b}_a(x)^(x) =d₀. This gives part (i) of the Proposition - as far as integrability is concerned.

Now assume, with the same notation as above and withr= 0, thatβ−α+1 = 0.

In this case, we need to investigate the integrability of exp(Rx p₀

2

nt⁻¹ψ(t)d)/|x^α|. As ψ(0) = 1, given any >0, for sufficiently small x, 1− < ψ(x)<1 +. Therefore

2

nd₀(1−)|x|⁻¹< 2

nd₀|x|⁻¹< 2

nd₀(1 +)|x|⁻¹.

Integrating - from the right near 0, we get K1

e^{n2d0(1−)ln|x|}

|x^α| <e

Rx p0

2

nd0t⁻¹ψ(t)dt

|x^α| < K2

e^{n2d0(1+)ln|x|}

|x^α| , whereK1, K2 are positive constants. This gives

K₁|x|^{n2d0(1−)−α}< e

Rx

p0(_n²_a(t)^b(t))dt

|x^α| <2K₂|x|^{n2d0(1+)−α.}

If the weight function is right integrable near 0, then necessarily _n²d0(1−)−α+1>

0. Hence_n²d0−α+1>0. If this holds then the displayed inequalities above establish the integrability of the weight function.

Similar arguments give the same condition for integrability from the left- namely

2

nd0−α+ 1 > 0. This completes the proof of the proposition, except for the differentiability of the weight, which is discussed after Proposition 3.4 Using lower and upper bounds on the asymptotic form of the weight function (asx→ ∞), or partial fraction decomposition ofb/a, we have the following result.

Proposition 3.4. (i) If a has no real roots and p(x) = exp(_n²Rx p₀

b(t)

a(t)dt)/|a(x)|

then the weight function p(x) gives finite norm for all polynomials if and only if degb−degais an odd positive integer and the leading term of b is negative.

(ii) If a has only one root, say 0, then p(x) = exp(²_nRx p₀

b(t)

a(t)dt)/|a(x)| gives a finite norm for all polynomials on(0,∞) if and only if

(a) degb−dega≥0 and the leading term ofb is negative.

(b) If a=x^α(A0+A1x+. . .) andb=x^β(B0+B1x+. . .), where A0 andB0

are nonzero constants, thenα−β ≥1, and ^B_A⁰

0 >0 forα−β≥2 whereas

2B₀

nA₀ −α+ 1>0 forα−β= 1.

Proof. Assume thata(x) has no real roots. We may assume that the leading term ofa(x) is 1. Thus

a(x) =xⁿ+an−1xⁿ⁻¹+. . . , b(x) =kx^m+. . . ,

wheren, mare the degrees ofa, b. Thus_a(x)^b(x) =kx^m−nψ(x), with lim_x→∞ψ(x) = 1.

Alsoa(x) = xⁿφ(x) and limx→∞φ(x) = 1. Hence for sufficiently large positivex, there are positive constantsd1, d2, c1, c2 with

1

|x|ⁿd₁e

Rx p0

2

nkc₁t^m−ndt

< p(x)< 1

|x|ⁿd₂e

Rx p0

2

nkc₂t^m−ndt

< p(x) (3.1) Ifm−n=−1, then forc >0,e

Rx p0

2

nkct^m−ndt

=A|x|^n2kc. ThereforeR∞ M Ax^{N x}

2 nkc

xⁿ dx cannot be finite ifN is large enough. Thusm−n6=−1. Therefore from (3.1) we obtain the estimate

A₁

|x|ⁿd1

e^n2kc1xm

−n+1

m−n+1 < p(x)< A₂

|x|ⁿd2

e^2nkc2xm

−n+1 m−n+1

whereA1, A2, c1, c2, d1, d2are positive constants. We wantR∞

M x^Np(x)dxto be finite for all monomialsx^N. Now for any positive c, R∞

M x^Nexp(_n²kc^x_m−n+1^m−n+1)dx is finite is equivalent to the finiteness of the integralR1/M

0 exp(_n²kc^{x−(m−n+1)}_m−n+1 )/x^N+2dx. By Lemma 3.1, this is finite if and only if _m−n+1^k < 0, −(m−n+ 1) < 0. Thus k < 0, (m−n+ 1) = l > 0, where k is the leading term of b(x): recall that we

have taken the leading term of a(x) to be 1. Similarly if we require finiteness of the integrals R−M

−∞ x^Ne^2nkcxm

−n+1

m−n+1 dx, then the requirements are ^k(−1)_m−n+1^m−n+1 <0,

−(m−n+ 1)<0 Therefore the conditions forp(x) = exp(R ₂

n b(x)

a(x)dx)/|x^α| to be a weight on (−∞,∞) are thatk <0, and (m−n+ 1) should be an even positive number. This completes the proof of (i). Part (ii) follows from the integrability of

the weight on (0,∞) and Proposition 3.3.

Differentiability properties of the weight function. The differentiability prop- erties of the weight function at zeroes of the leading term a(x) follow from the following observation: if d is a positive integer and k is a positive number, then lim_x→0+ e^−kx−d

P(x) = 0 for all polynomials P(x). Let N be the order of 0 in P(x).

Then P(x) can be written as P(x) = ax^NQ(x) where Q is a polynomial with Q(0) = 1. Thus it suffices to show that lim_x→0+ e^−kx−d

x^N = 0. This is the same as lim_u→∞ ^uN

e^kud = 0, which is obviously true. Therefore lim_x→0+e^−kx−d

P(x) Q(x) = 0 for polynomialsP, QwithP6= 0. As in the proof of proposition 3.3, the weight func- tion, near the rootr= 0 can then be written asp(x) =ψ(x) exp(R ₂

n b(x)

a(x)dx)/(|a0||x^α|) withψ(0) = 1. Thus, if α, βare the multiplicities of the root 0 ina, brespectively andβ−α+ 16= 0, we have the estimate

A₁e^(k1n2d0)(xβ

−α+1 β−α+1)

|x^α| ≤e

Rx

p0(²_n^b(x)_a(x))dt

|x^α| ≤A₂e^(k22nd0)(xβ

−α+1 β−α+1)

|x^α|

where A₁, A₂, k₁, k₂ are positive constants. We discuss the right hand limit, as the left hand limit is treated similarly. As the weight is integrable, we must have β−α+1<0,d₀>0. Thus, by the above observation, lim_x→0+p(x) = 0, wherep(x) is the weight function. Moreover, by the same observation, lim_x→0+p(x)R(x) = 0 for any rational function. Let φ(x) =Rx

p₀(²_n^b(x)_a(x))dt. Therefore p(x) =e^φ(x)/|a(x)|.

Then all derivatives of p(x) are of the form e^φ(x)R(x), where R(x) is a rational function. Therefore the right-hand limits at 0 of all the derivatives of the weight function are 0.

In caseβ−α+ 1 = 0, the weight can be written near a zero of the leading term - by change of notation- as

p(x) = c

|a||x|^(n2d0−α) e^φ(x) 1 +ψ(x), whereφ, ψ are analytic functions near 0, and

a(x) = (x)^α(a₀+a₁(x) +. . .), b(x) = (x)^β(b0+b1(x) +. . .), where a₀, b₀ are not 0 and d₀ = _a^b0

0. Here ²_nd₀−α+ 1 > 0 - by the assumption of integrability of the weight. This completes our discussion of the differentiability properties of the weight function near a zero of the leading term.

The differentiability properties of ordinary roots have already been discussed.

We now assume that the multiplicity of a root r of a(x) = an(x) is α and its multiplicity inb(x) =a_n−1(x) isβ. For convenience of notation we assume that r is zero.

As above, we have an(x) = x^α(A0+A1x+. . .), an−1(x) = x^β(B0+B1x+ . . .) with (β −α) = −1. Thus near x = 0, the weight is of the form p(x) =

1

|A0||x|^{(n2BA00−α)}_1+ψ(x)^eφ(x) whereφ,ψare analytic near zero andψ(0) = 0. When there is no danger of confusion we will writep(x)∼ |x|^{(2nBA00−α)}. Nowp⁰ =p(²_n^b_a −^a_a⁰).

Therefore, all higher derivatives ofpare of the formpρwhereρis a rational function and all higher derivatives of pρ are also multiples of pby rational functions. For later use we record the asymptotic behavior ofp⁰ near a zero ofa_n.

p⁰(x) = 1

|A0||x|^{(2nBA00−α)} e^φ(x) 1 +ψ(x)

2 n b a−a⁰

a

The weight p is integrable near zero if and only if (_n^2B_A⁰

0 −α+ 1) > 0. More- over lim_x→0p(x)an(x) = 0 if and only if ²_n^B_A⁰

0 > 0. By the integrability of the weight _n^2B_A⁰

0 > α−1≥0. Thus the boundary condition lim_x→0p(x)a_n(x) = 0 is a consequence of the integrability of the weight near zero. Similarlyp(x)an−1(x) =

1

|A0||x|^(n2BA00−1)_1+ψ(x)^eφ(x) (B0+B1x+. . .), keeping in mind that α−β = 1. Hence lim_x→0p(x)a_n−1(x) = 0 if and only if (_n^2B_A⁰

0 −1)>0.

3.1. Higher order operators. The principal aim of this section is to prove the following result.

Proposition 3.5. Let L=an(x)y⁽ⁿ⁾+a_n−1(x)y⁽ⁿ⁾+· · ·+a2(x)y⁰⁰+a1(x)y⁰ be a self-adjoint operator of ordernwithn >2. Ifanhas a real root then the multiplicity of the root is at least2 and the multiplicity of the same root ina_n−1is positive and less than its multiplicity inan.

Proof. Letrbe a real root of an and assume that it is a simple root. It is then a logarithmic root. Therefore, nearr, we have

p(x)∼ |x−r|^(n2BA00−1),

where an(x) = (x−r)(A0+A1(x−r) +. . .),a_n−1(x) = (B0+B1(x−r) +. . .) andA₀,B₀ are not zero. Nowp⁰=p _n^2an−1_a

n −^a_a⁰ⁿ

n

. Therefore p⁰(x)∼ |x−r|^(n2BA00−1)2

n

a_n−1(x)

a_n(x) −a⁰_n(x) a_n(x)

. Similarly

p(x)a_n−1(x)∼ |x−r|^(n2BA00−1)(B0+B1(x−r) +. . .)

The boundary conditions and the determining equations in Proposition 2.1 imply that (anp), (a_n−1p) and (a_n−1p)⁰ vanish on the boundary.

Now lim_x→ran(x)p(x) = 0 is a consequence of the integrability of the weight near r. Similarly lim_x→ra_n−1(x)p(x) = 0 if and only if (²_n^B_A⁰

0 −1) > 0. Let lr = limx→r(x−r)^{α−β an−1}_a ^(x)

n(x) . Clearly lr = ^B_A⁰

0 = ^an−1_a0 ^(r)

n(r) , as α−β = 1. Since limx→rpan−1= 0 andan−1(r)6= 0 we see thatpmust vanish atrin the sense that its limit atris zero. The boundary condition lim_x→r(a_n−1p)⁰= 0 now implies that lim_x→rp⁰(x) = 0. Nowp⁰ =p(_n^2an−1_a

n −^a_a⁰ⁿ

n). Thus near the rootr, p⁰∼ |x−r|^(n2lr−2α)2

na_n−1−a⁰_n .

Ifa_n−1−ⁿ₂a⁰_n ≡0 then in particular (_n^2an−1_a0 ^(r)

n(r) −1) = 0. This means that lim_x→r(x−

r)²_n^an−1_a ^(x)

n(x) −1 = 0 i.e. (_n^2B_A⁰

0 −1) = 0. This contradicts the boundary condition lim_x→ra_n−1(x)p(x) = 0.

Let (a_n−1−ⁿ₂a⁰_n) = (x−r)^λH(x) where λ≥0 and H(r) 6= 0. If λ > 0 then lim_x→r(x−r)_n^2an−1_a ^(x)

n(x) −1 = 0 which again contradicts the boundary condition limx→ran−1(x)p(x) = 0. Hencep⁰ ∼ |x−r|^(2nlr−2α)H(x) sop⁰→0 atrif and only if (_n²l_r−2α)>0.

By Proposition 2.1, the operator must satisfy - beside other equations - the determining equations

n(a_np)⁰= 2(a_n−1p), (3.2)

(n−1)(n−2)

6 (an−1p)⁰⁰−(n−2)(an−2p)⁰+ 2(an−3p) = 0 (3.3) Equation (3.3) is equivalent to

C₁(a_n−1

a_n )³+C₂(a_n−1 a_n )(a_n−1

a_n )⁰+C₃(a_n−1 a_n )⁰⁰ +C₄(a_n−2

a_n )⁰+C₅a_n−1 a_n

a_n−2

a_n +C₆a_n−3 a_n = 0

(3.4)

where

C₁=2(n−1)(n−2)

3n² , C₂= (n−1)(n−2)

n , C₃=(n−1)(n−2)

6 ,

C4=−(n−2), C5=−2(n−2)

n , C6= 2.

This implies the identity

an−1 an−1−na⁰_n

an−1−n 2a⁰_n

≡0 modan (3.5)

Using this identity, as (x−r) dividesanbut it does not dividean−1nor (an−1−ⁿ₂a⁰_n), it must divide (an−1−na⁰_n). But then limx→r(an−1−na⁰_n) = 0. This means that lim_x→rn^2an−1_a0

n −2 = 0 i.e. _n^2B_A⁰

0 −2 = 0. As seen abovep⁰ →0 at rif and only if (²_nl_r−2α)>0. Sinceα= 1 we have a contradiction.

Thereforea_n cannot have a simple root and its multiplicity αin a_n is at least 2. Suppose that the multiplicityβofrina_n−1is zero. By considering the order of poles ofan in (3.4) we see thatβ cannot be zero. This completes the proof.

This result has an important consequence for fourth order self-adjoint operators.

Corollary 3.6. Let L be a self-adjoint operator of order 4 and a₄ be its leading term. If a4 has more than one real root then it has exactly two real roots with multiplicity 2. Moreover the multiplicity of each real root ofa4 in a3 is1.

Proposition 3.7. Let n >2 and suppose that a =a_n has at most one real root.

Then2 degb−dega≤n−2 or3 degb−2 dega≤n−3, whereb=a_n−1. (i) If ahas no real root thendega <degb≤n−3;

(ii) Supposea has only one real root r with multiplicity α, let β be the multi- plicity of ras a root ofb, and leta= (x−r)^αu,b= (x−r)^βv. Then

2≤dega≤degb≤n−2 and 1 + degu≤degv≤n−3

Proof. First, in all cases, degb≥1. This is because ifahas no real root then degb is odd and ifa has (at least) one real root then this will also be a root for b (by Proposition 3.5). If we multiply by a³ both sides of (3.4), then the six terms on the left-hand side will be polynomials with respective degrees

3 degb, 2 degb+ dega−1, 2 dega+ degb−2,

2 dega+ dega_n−2−1, dega+ degb+ dega_n−2, 2 dega+ dega_n−3 A comparison of degrees shows thata_n−2anda_n−3 cannot be both zero and that

2 degb≤dega+ degan−2, or 3 degb≤2 dega+ degan−3

Using the fact that dega_j≤j, we obtain

2 degb−dega≤n−2 or 3 degb−2 dega≤n−3

Ifahas no real roots then, by Proposition 3.4 (i), degb−dega≥1 and hence dega <degb≤n−3

Ifahas only one real rootrwith multiplicity α, then α≥2 andb hasr as a root with multiplicityβ, where 1≤β < α(by Proposition 3.5). Since degb≥dega, we obtain that

2≤dega≤degb≤n−2

Leta= (x−r)^αu, b= (x−r)^βv. Then dega=α+ deguand degb =β+ degv, and we obtain degv−degu≥α−β ≥1. Now β+ degv≤n−2, so 1 + degu≤

degv≤n−3 and thus degu≤n−4.

4. Examples of higher order operators, their eigenvalues and orthogonal eigenfunctions

Let L be an operator of the form L(y) = Pn

k=0ak(x)y^(k), where degak ≤ k;

then the eigenvalues ofLare the coefficients of xⁿ in L(xⁿ),n= 0,1,2, . . .. Proposition 4.1. Let L be a linear operator that maps the space Pn of all poly- nomials of degree at most n into itself for all n≤N. If the eigenvalues of L are distinct or if L is a self-adjoint operator then there is an eigenpolynomial ofL in every degree≤N.

This means that if two operators leave the space of polynomials of degree at mostninvariant for allnand the weight function which makes the two operators self-adjoint is the same, then they have the same eigenfunctions. The eigenvalues in general are not simple. The proof of the above proposition is left to the reader.

Letλbe an eigenvalue ofLandPn(λ) the corresponding eigenspace in the space Pn of all polynomials of degree less than or equal ton.

If n0 is the minimal degree in Pn(λ), then there is, up to a scalar only one polynomial inPn(λ) of degree n0. Choose a monic polynomial Q1in Pn₀(λ).

Letn1be the smallest degree, if any, greater thann0inPn(λ). The codimension of Pn₀(λ) in Pn(λ) is 1. Therefore, in the orthogonal complement of Pn₀(λ) in Pn1(λ), choosing a monic polynomial Q2, which will necessarily be of degree n1, the polynomialsQ₁, Q₂give an orthogonal basis ofPn1(λ). Continuing this process, we eventually get an orthogonal basis ofPn(λ) consisting of monic polynomials.