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Kernel Inspired Factorizations of Partial Differential Operators
Journal of Mathematical Analysis and Applications 234, 580᎐591 Ž1999. Article ID jmaa.1999.6383, available online at http:rrwww.idealibrary.com on
Kernel Inspired Factorizations of Partial Differential Operators Alex Kasman Department of Mathematics, College of Charleston, 66 George Street, Charleston, South Carolina 29424-0001 E-mail: [email protected] Submitted by Colin Rogers Received Noveber 24, 1998
It is elementary to factor an ordinary differential operator given a nonzero function in its kernel. Here a higher dimensional generalization of this fact is developed and utilized to determine nontrivial factorizations of constant coefficient partial differential operators. Through Darboux transformations, this result is applied to the solution of linear partial differential equations, commutativity, quantum integrability and the ‘‘bispectral problem.’’ 䊚 1999 Academic Press
1. INTRODUCTION The factorization of ordinary differential operators ŽODOs. is quite similar to the factorization of polynomials in a single variable. If one knows a root g ⺓ of some polynomial pŽ x . g ⺓w x x then it immediately follows that pŽ x . s q Ž x .Ž x y . for some polynomial q g ⺓w x x. Note that just as x y is the only monic linear polynomial that vanishes at , K s ( ⭸⭸x ( y1 Ž x . is the only monic first order ODO with Ž x . in its kernel. Analogously, the knowledge that Ž x . is in the kernel of an ODO implies that K is a factor of that operator. Ž . i THEOREM 1.1. If / 0 is in the kernel of the ODO L s Ý m is1 f i x ⭸ ⭸ Ž ⭸ [ ⭸ x . then it factors as L s Q( K for some ODO Q and the first order differential operator K s Ž x .( ⭸ ( y1 Ž x .. Moreover, since any ODO ‘‘completely factors’’ into first order terms, knowing the entire kernel is sufficient to determine all possible factorizations. ŽIn general, one can say that L s Q( K if and only if ker K ; ker L.. This property has proved to be useful in constructing Darboux transforma580 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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tions for ordinary differential operators with many applications in mathematical physics w2, 8, 9, 17, 18, 20x. For the same reasons, there is interest in being able to perform such Darboux transformations for partial differential operators ŽPDOs. w1, 3, 5, 6, 12, 15, 22, 23x. However, in keeping with the analogy to polynomials, the fact that one cannot necessarily completely factor a PDO complicates matters in the higher dimensional situation. Even though the analogous statement: ‘‘If the PDO L1 annihilates e¨ ery function in the kernel of the PDO L2 then L1 s Q( L2 for some PDO Q’’ is true w21x, it does not seem to be useful in determining previously unknown factorizations. Here we will prove a result which, although not completely general, leads to novel factorizations of constant coefficient partial differential operators. For any differentiable function Ž x 1 , . . . , x n . and irreducible polynomial q Ž z1 , . . . , z n . we will introduce below a partial differential operator K q and a proper subset Fq ; ker K q. The main theoretical result of this paper is to prove that any PDO whose kernel contains Fq has K q as a right factor. Using this fact, which has the well known result Theorem 1.1 as a corollary, we are able to determine new, explicit, nontrivial factorizations of constant coefficient partial differential operators. The application of such factorizations to the solution of linear PDEs, bispectrality and quantum integrability through Darboux transformations is discussed. 1.1. Notation Let D be the ring of partial different operators in the variables x s Ž x 1 , . . . , x n . and coefficients in some differential field K of functions. ª Ž The shorthand notations ⭸ [ ⭸ r ⭸ x and ⭸ s ⭸ , . . . , ⭸n . will be used. ŽSo i i 1 ª D s Kw ⭸ x is made up of polynomials in the ⭸ i ’s with coefficients that are functions in K.. Differential operators act on functions by differentiation and multiplication in the obvious way, and the action of the operator L g D on a function f Ž x 1 , . . . , x n . will be denoted Lw f x. Moreover, the composition of operators L1 , L2 g D will be written L1 ( L2 . We will say that a function f Ž x 1 , . . . , x n . is polynomial-exponential if it is of the form
ª
N
f Ž x1 , . . . , x n . s
ª ª
Ý pi Ž x 1 , . . . , x n . e ² x , ␣ : i
Ž 1.2.
is1 ª n Ž ²,: for some N g ⺞, polynomials pi g ⺓w x 1 , . . . , x n x and ␣ i g ⺓ . Here n . denotes the usual inner product on ⺓ . Note that the set of polynomialexponential functions forms a ring which is moreover closed under differentiation. Similarly, the quotient field of ratios of such functions with
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ALEX KASMAN
nonzero denominators is also closed under differentiation. Let us call this quotient field the rational-exponential functions. This paper is primarily concerned with the ring of partial differential operators in the variables x i with coefficients that are rational-exponential functions as defined above. More specifically, the main result involves the factorization of constant coefficient differential operators in this ring. In addition, in order to prove the results below it will be useful to introduce the ring ⺤ of rational coefficient translational-differential operators in ª z s Ž z1 , . . . , z n . Žcf. w19x which considers the one dimensional case.. An element ⌳ g ⺤ is an operator of the form N
⌳s
Ý Li (T␣
Ž 1.3.
ª
i
is1
where T␣ª is the translation operator ª T␣ª f Ž ª z. s f Žª zq␣ .
ª ␣ g ⺓n
Ž 1.4.
and L i are rational coefficient differential operators in the variables z i m
Li s
Ý ri j Ž ªz . ( pi j ž ⭸ˆ1 , . . . , ⭸ˆn / ,
js1
⭸ˆi s
⭸ ⭸ zi
,
ri j g ⺓ Ž ª z. ,
pi j g ⺓ w ª zx.
Ž 1.5.
Note that ⺓w ª z x ; ⺤ by associating p(Tª0 g ⺤ to the polynomial p. Special attention should be paid to the subring ⺤ 0 of constant coefficient translational-differential operators Ži.e., those for which L i g ⺓w ⭸ˆ1 , . . . , ⭸ˆn x.. The main significance of these operators here comes from the following observation: LEMMA 1.6. The map I from the ring ⺤ 0 of constant coefficient translational-differential operators in ª z to the ring of polynomial-exponential functions in ª x gi¨ en by ⌳ w e² x , z: x s I Ž ⌳ . w e² x , z: x ª ª
ª ª
⌳ g ⺤0
Ž 1.7.
is an isomorphism. ªª
In particular, e ² x, z : is an eigenfunction for every element of ⺤ 0 and the eigenvalues are polynomial-exponential functions.
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
583
2. FACTORING PARTIAL DIFFERENTIAL OPERATORS Let Ž x 1 , . . . , x n . g K be an arbitrary nonzero differentiable function and q Ž z1 , . . . , z n . be an irreducible polynomial. We will associate to this choice a differential operator and a set of functions. DEFINITION 2.1. Let K q g D be the differential operator 1
ª
K q [ Ž ª x . ( q Ž ⭸ .(
Žª x.
.
Note in particular that this operator annihilates every function in the set of functions ª ª
Fq [ Ž ª x . e² x , z: < ª z g ⺓,
qŽ ª z . s 04
parametrized by the irreducible algebraic variety qy1 Ž0.. THEOREM 2.2.
L g D factors as L s Q( K q for some Q g D
if and only if Fq ; ker L. Proof. Clearly if L has a right factor of K q then Fq ; ker L. So, let us assume the latter and prove the former. ª Consider first the case that Ž ª x . ' 1 and so K q s q Ž ⭸ .. Note that fŽª x, ª z . [ Lw e ² x , z : x s Ý fi Ž ª x . pi Ž ª z . e² x , z: ª ª
ª ª
for some polynomials pi and functions f i . Then, for any fixed value of ª x in its domain, this function must vanish for all ª z g qy1 Žª0.ª. Consequently, by the Nullstellensatz, the polynomial Žin ª z . f Žª x, ª z . e ²yx, z : has a factor of ª q Ž z . and so we may factor ª ª
fŽª x, ª z. s qŽ ª z . Ý fˆi Ž ª x . qˆi Ž ª z . e² x , z: .
Ž 2.3.
ª
Then, L factors as L( Q( q Ž ⭸ . with Qs
ª
Ý fˆi Ž ªx . qˆi Ž ⭸ . .
Ž 2.4.
ˆ s L( g D; then by the precedMore generally, for arbitrary let L ª ˆ ˆ ˆ 1 s Q( ˆ q Ž ⭸ª.( 1 . Ž ing paragraph we have that L s Q( ⭸ .. Hence L s L( ˆ 1 we get the desired factorization. Finally, letting Q s Q(
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ALEX KASMAN
Remark 1. Note that if n s 1 and q Ž z . s z g ⺓w z x then Fz Ž x . s Ž x .4 and K z Ž x . s ( ⭸ ( y1 . In particular, in this case one finds Theorem 1.1 as a consequence of Theorem 2.2. Remark 2. Theorem 2.2, though quite simple once it has been read and understood, is particular surprising since it does not require that L annihilate all of ker K q but only the Žproper. subspace spanned by the elements of Fq. This is a stronger statement than one might expect. ŽFor instance, note that if ' 1 and q Ž x 1 , x 2 . s x 12 y x 2 then x 1 g ker K q but it cannot be written as a linear combination of the functions in Fq ; ker K q.. Remark 3. Of course, most differential operators will not have any set of the form Fq in their kernels, and so this theorem is not entirely general. However, it has the advantage of being useful. The special instance of this theorem in the case that q is linear was used to construct Darboux transformations in w5x and the case in which Ž ª x . g ⺓w ª x x is a polynomial is implicitly contained in w3x. It will be further applied in the next section to determine new factorizations of constant coefficient differential operators.
3. CONSTANT COEFFICIENT DIFFERENTIAL OPERATORS 3.1. One Dimensional Case In the case of constant coefficient ordinary differential operators, Theorem 1.1 implies the following: THEOREM 3.1. There exists a constant coefficient ODO with right factor of K s Ž x .( ⭸ ( y1 Ž x . iff Ž x . s Ý nis1 pi Ž x . e i x for some pi g ⺓w x x and i g ⺓. Proof. Let be of this form and pŽ x . s ŁŽ x y i .1q deg p i g ⺓w x x; then the operator L s pŽ ⭸ . has Ž x . in its kernel. By the discussion above, this implies that L has a right factor of the form K. Alternatively, given a constant coefficient operator L s pŽ ⭸ . Ž p g ⺓w x x. let i Ž1 F i F n. denote the distinct roots of p and m i be their multiplicities. Then it is simple to check that x j e i x g ker L for 0 F j F m i y 1 and 1 F i F n. Over ⺓, these functions span a Ždeg p .-dimensional space which must therefore be all of ker L. Consequently, every function in the kernel of a constant coefficient ODO is of the given form. It will be seen that the situation is much the same in n dimensions if one replaces with any similar function of n variables and replaces ⭸ with any irreducible constant coefficient partial differential operator.
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
585
3.2. Higher Dimensional Case If one were to pick an arbitrary differential operator K, it would be unusual to be able to find another operator Q such that Q( K is a constant coefficient operator. The following theorem, analogous to Theorem 3.1, provides a class of nonconstant coefficient operators which are right factors of constant coefficient differential operators. THEOREM 3.2. Let Ž ª x . / 0 be a polynomial-exponential function and q Ž x . g ⺓w ª x x be an irreducible polynomial; then there exists a partial differenª tial operator Q / 0 such that Q( K q g ⺓w ⭸ x is a constant coefficient operator. ª
Proof. Let
Žª x. s
m
ª ª
Ý pi Ž ªx . e ² x , ␣ :
Ž 3.3.
i
is1 ª n for arbitrary polynomials pi and vectors ␣ the translationali g ⺓ . Then ªª ªª y1 Ž . differential operator T s I g ⺤ satisfying Te ² x, z : s Ž ª x . e ² x, z : Žcf. Lemma 1.6. is given by
m
Ts
ª
Ý pi ž ⭸z / T␣ .
Ž 3.4.
ª
i
is1
Now let pŽ z . g ⺓w z x be the polynomial m
pŽ z . [
Ł Ty␣ Ž q Ž ªz . . ª
is1
m iq1
i
m i [ deg pi
Ž 3.5. ª
ª
and L be the constant coefficient differential operator L s pŽ ⭸ . g ⺓w ⭸ x. Then L e ² x , z : s T ( Lw e ² x , z : x s T ( p Ž z . w e ² x , z : x ª ª
ª ª
ª ª
Ž 3.6.
and pŽ z . is constructed specifically so that T ( pŽ z . s q Ž z .(T X for some other translational-differential operator T X with polynomial coefficients. ªª ² x, z : x ' 0 for all ª Consequently, Lw e z g qy1 Ž0. and then Theorem 2.2 implies that L has a right factor of K q . 4. EXAMPLES, APPLICATIONS AND COMMENTS Without knowledge of its kernel, factorization of even an ordinary differential operator can be difficult Žcf. w13, 14x.. If one does know functions in the kernel, the problem of factorization becomes trivial but
586
ALEX KASMAN
such factorizations have application in the construction of Darboux transformations w2, 8, 9, 16, 17, 18, 20x, which is essentially conjugation in the ring of differential operators. Despite the inherent difficulty in factoring PDOs there has recently been much interest in the higher dimensional generalizations of the Darboux transformation w1, 3, 12, 15, 22, 23x. By using the factorizations determined above to perform Darboux transformations for constant coefficient PDOs one may construct new, explicit, higher dimensional examples of commutativity, quantum integrability, and bispectrality. Pick an irreducible polynomial q g ⺓w ª z x, polynomials pi g ⺓w ª x x and ª vectors ␣ i for 1 F i F m g ⺞. Then defining by Ž3.3. and p by Ž3.5., it ª ª follows that the constant coefficient differential operator L s pŽ ⭸ . g ⺓w ⭸ x factors as L s Q( K q for some Q. Knowing this, one may then compute the coefficients of Q explicitly Žfor instance, with a computer algebra package. since L and K q are known. EXAMPLE. For n s 2, consider the case that q Ž ª z . s z12 y z 2 , p1 s 1, p 2 ª ª ª w x Ž . Ž . Ž s ␥ g ⺓ x , ␣ 1 s 1, 0 and ␣ 2 s 0, 1 . Using 3.3. and Ž3.5. one finds that 2
L s Ž Ž ⭸ 1 y 1 . y ⭸ 2 . Ž ⭸ 12 y ⭸ 2 q 1 .
Žª x . s e x1 q ␥ e x 2 and K q s ⭸ 12 y ⭸ 2 y
2 e x1
⭸1 q 1 y
2␥ e x 1qx 2
2
.
ªª
As predicted, one may check that Lw e ² x, z : x vanishes for any ª z g qy1 Ž0. and so by Theorem 2.2 L s Q( K q . It is then merely an elementary calculation to check that Q s ⭸ 12 y ⭸ 2 y
2␥ e x 2
⭸1 q
2␥ e x 1qx 2
2
q 1.
Darboux Transformations for Linear PDEs Given the factorization L s Q( K q determined above, we have a Darboux transformation relating the solutions to the equation Lw f x s f Ž g ˜w f˜x s f˜ for L ˜ s K q ( Q. In particular, it is ⺓. and the new equation L clear that the application of the operators K q and Q provides maps ˜ y . between ker L y and ker L
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
THEOREM 4.1.
587
˜ s K q ( Q then we ha¨ e the maps If L s Q( K q and L ˜y K q : ker L y ª ker L f ¬ K q w f x
˜ y ª ker L y Q: ker L f˜¬ Q f˜ between the kernels of the operators. Proof. Suppose f g ker L y ; then
˜ y . f˜ s K q ( Q( K q w f x y K q w f x ŽL s K q ( L w f x y K q w f x s K q w f x y K q w f x s 0 Žand similarly for the map in the other direction.. As in other applications of Darboux transformations to differential operators w1, 3, 8, 9, 15, 20, 23x, these are most useful if they can be shown to preserve some form or property of interest. Here, for instance, the bispectral property and quantum integrability are considered although it seems likely that the transformations described here could be used as well for other purposes. Unfortunately, since the procedure described above depends explicitly on the fact that the initial operator has constant coefficients, it is not clear how one can iterate these transformations except in trivial circumstances. EXAMPLE Žcontinued.. Continuing the example begun above, we have the new differential operator
˜ s ⭸ 22 y 2 ⭸ 12⭸ 2 q 2 ⭸ 1 ⭸ 2 y 2 ⭸ 2 q ⭸ 14 y 2 ⭸ 13 L q q q
2 Ž e 2 x 1 q 6␥ e x 1qx 2 q e 2 x 2␥ 2 .
Ž e x 1 q e x 2␥ .
2
⭸ 12
y2 Ž e 3 x 1 q 9 e 2 x 1qx 2␥ y 3e x 1q2 x 2␥ 2 q e 3 x 2␥ 3 .
Ž e x 1 q e x 2␥ .
3
⭸1
e 4 x 1 q 8␥ e 3 x 1qx 2 y 10 e 2Ž x 1qx 2 .␥ 2 q 8 e x 1q3 x 2␥ 3 q e 4 x 2␥ 4
Ž e x 1 q e x 2␥ .
4
588
ALEX KASMAN
and the relationship Theorem 4.1 relating the solutions of the corresponding differential equations. For instance, it is trivial to find a solution to Lw f x s 0 by assuming f Ž x 1 , x 2 . s f 1Ž x 1 . q f 2 Ž x 2 .. One such function is f s 1 x 1 e x 1 q 2 x 2 e x 2 g ker L. However, the transformation f˜[ K q w f x s Ž 2 1 ␥ y 2 . e x 2 y q
2␥ 2 e 3 x 2 Ž 1 ␥ x 1 y 2 x 2 .
2
2␥ e 2 x 2 Ž 1 ␥ Ž x 1 y 1 . y 2 x 2 .
˜w f˜x s 0 which would be difficult to find using provides a solution to L other methods. Commutati¨ ity and Quantum Integrability A factorization of one constant coefficient operator L provides a means ª for producing a set of commutating operators. Let Mi g ⺓w ⭸ x be constant coefficient partial differential operators. Then the Žgenerically noncon˜ i [ K q ( Mi ( Q mutually commute. Explicitly, one sees stant. operators L that
˜i , L ˜ j s K q ( Mi ( Q( K q ( M j ( Q y K q ( M j ( Q( K q ( Mi ( Q L s K q ( Mi ( L( M j ( Q y K q ( M j ( L( Mi ( Q s 0 ª
by the commutativity of L and Mi g ⺓w ⭸ x. Consequently, letting M0 s 1 ˜0 , . . . , L ˜ n is a commutaand Mi s ⭸ i Ž1 F i F n. the ring generated by L tive ring with Krull dimension n which cannot be generated by fewer than n q 1 generators. Viewing the commutativity of these operators as the quantum analogue of the Poisson commutativity of functions on the cotangent bundle to a symplectic manifold, such ring may be considered as an example of a quantum integral system w4᎐6, 12x. Bispectrality ªª
The function e ² x, z : is, of course, an eigenfunction for the ring of constant coefficient partial differential operators in ª x with eigenvalues depending on the parameters ª z. Moreover, this same function is also an eigenfunction for the constant coefficient translational-differential operators ⺤ 0 in ª z with eigenvalues depending on ª x. This is an instance of ‘‘bispectrality’’ w7, 10, 11, 19, 22x. Then, it is interesting to note that the function K q wexp² ª x, ª z :x is also a common eigenfunction for differential ª operators in x and translational-differential operators in ª z. The one
589
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
dimensional case Ž n s 1. is particularly interesting since it indicates a sort of bispectrality for solitons of the KP hierarchy and will be considered separately in w19x. ŽSee also w7x which considers such translational-differential bispectrality in the context of quantum duality.. THEOREM 4.2. The function [ K q wexp² ª x, ª z :x satisfies the eigen¨ alue equations
˜ Žª L x, ª z . s pŽ ª z. Ž ª x, ª z.
⌳ Žª x, ª z. s mŽª x. Ž ª x, ª z . Ž 4.3.
˜ s K q ( Q g D, m s deg q q 1 and some ⌳ g ⺤. for L Proof. The key observation here is that
Žª x, ª z. s Žª x . ( q Ž ⭸ ⭈. ( s
ˆ Ž ª x.
1
w e² x , z: x ª ª
ª
Ž x.
ª ª
deg q Ž ª x.
e² x , z:
where ˆŽ ª x . is another polynomial-exponential function of the form
ˆ Ž ª x. s
m ˆ
ª
Ý ˆpi Ž ªx . e ² x , ␣ˆ : . i
is1
Then, letting ⌳s
ž
m ˆ
Ý ˆpi ž ⭸ˆ1 , . . . , ⭸ˆn / (T␣ˆk (
/
is1
1 qŽ ª z.
m
(
žÝ
is1
pi ⭸ˆ1 , . . . , ⭸ˆn (T␣ªi
ž
/
/
it follows that ⌳ satisfies Ž4.3.. EXAMPLE Žcontinued.. Returning to the example above, oneªmay check ª that Ž ª x, ª z . [ K q wexp² ª x, ª z :x is also given by s Ž1r 2 . ⌫w e ² x, z : x where ⌫ g ⺤ is the translational-differential operator ⌫ s ␥ 2 Ž 1 q z12 y z 2 . TŽ0 , 2. q Ž 1 y 2 z1 q z12 y z 2 . TŽ2 , 0. q 2␥ Ž z12 y z1 y z 2 . TŽ1 , 1. . Then if ⌳ s ⌫ ( q Ž1z . (T with T s Iy1 Ž . s TŽ1, 0. q ␥ TŽ0, 1. one finds that ⌳w Ž ª x, ª z .x s Ž ª x . 3 Ž ª x, ª z .. Since this bispectrality is not only a consequence of the transformation but is also used in the construction itself, this result is similar to w12x where bispectrality of partial differential operators is utilized to perform Darboux ª
590
ALEX KASMAN
transformations. In particular, this paper extends that same idea to the case of translational bispectrality where the proof presented in w12x fails.
ACKNOWLEDGMENTS I am indebted to M. Singer and S. Tsarev for helpful discussions and to Yu. Berest and E. Horozov for continuing assistance and prior collaboration w3, 12x.
REFERENCES 1. A. Andrianov, N. Borisov, and M. Ioffe, The factorization method and quantum systems with equivalent energy spectra, Phys. Lett A 105, No. 1, 2 Ž1984., 19᎐22. 2. B. Bakalov, E. Horozov, and M. Yakimov, Backlund-Darboux transformations in Sato’s ¨ grassmannian, Serdica Math. J. 22 Ž1996., 571᎐588. 3. Yu. Berest and A. Kasman, D-modules and Darboux Transformations, Lett. Math. Phys. 43 Ž1998., 279᎐294. 4. A. Braverman, P. Etingof, and D. Gaitsgory, Quantum Integrable Systems and Differential Galois Theory, Transformation Groups 2 Ž1997., 31᎐56. 5. O. A. Chalykh and A. P. Veselov, Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 125 Ž1990., 597᎐611. 6. O. A. Chalykh and A. P. Veselov, Integrability in the theory of the Schrodinger operators ¨ and harmonic analysis, Comm. Math. Phys. 152 Ž1993., 29᎐40. 7. O. A. Chalykh, Duality of the generalized Calogero and Ruijsenaars problems, Uspekhi Mat. Nauk 52, No. 6Ž318. Ž1997., 191᎐192. 8. G. Darboux, Sur une proposition relative aux equation lineares, Compt. Rend. 94 Ž1882., 1456. 9. G. Darboux, ‘‘Lec¸ons sur la theorie generale des surfaces,’’ 2eme ´ ´ ´ ´ partie, Gauthiers-Villars, Paris, 1889. 10. J. J. Duistermaat and F. A. Grunbaum, Differential equations in the spectral parameter, ¨ Comm. Math. Phys. 103 Ž1986., 177᎐240. 11. F. A. Grunbaum, Bispectral musings, in ‘‘CRM Proceedings and Lecture Notes,’’ Vol. 14, ¨ pp. 31᎐46, Amer. Math. Soc., Providence, 1988. 12. E. Horozov and A. Kasman, ‘‘Darboux Transformations for Bispectral Quantum Integrable Systems,’’ submitted for publication. Available as math-phr9806002. 13. M. van Hoeij, Formal solutions and factorization of differential operators with power series coefficients, J. Symbolic Comput. 24, No. 1 Ž1997., 1᎐30. 14. M. van Hoeij, Factorization of differential operators with rational functions coefficients, J. Symbolic Comput. 24, No. 5 Ž1997., 537᎐561. 15. A. Gonzalez-Lopez and N. Kamran, ‘‘The Multidimensional Darboux Transformation,’’ hep-thr961200. 16. A. Kasman and M. Rothstein, Bispectral darboux transformations: the generalized Airy case, Physica D 102 Ž1997., 159᎐173. 17. A. Kasman, Darboux transformations from n-KdV to KP, Acta Appl. Math. 49, No. 2 Ž1997., 179᎐197. 18. A. Kasman, Darboux transformations and the bispectral problem, in ‘‘CRM Proceedings and Lecture Notes,’’ Vol. 14, pp. 81᎐92, Amer. Math. Soc., Providence, 1988.
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19. A. Kasman, Spectral Difference Equations Satisfied by KP Soliton Wavefunctions, In¨ erse Problems 14 Ž1998., 1481᎐1487. 20. V. B. Matveev and M. A. Salle, ‘‘Darboux Transformations and Solitons,’’ Springer Ser. Nonlinear Dynam., Springer-Verlag, BerlinrNew York, 1991. 21. M. Singer, private communication. 22. A. P. Veselov, Baker᎐Akhiezer functions and the bispectral problem in many dimensions, in ‘‘CRM Proceedings and Lecture Notes,’’ Vol. 14, pp. 123᎐129, Amer. Math. Soc., Providence, 1998. 23. B. N. Zakhariev and V. M. Chabanov, New situation in quantum mechanics, Žwonderful potentials from the inverse problem., In¨ erse Problems 13 Ž1997., R47᎐R79.
Kernel Inspired Factorizations of Partial Differential Operators Alex Kasman Department of Mathematics, College of Charleston, 66 George Street, Charleston, South Carolina 29424-0001 E-mail: [email protected] Submitted by Colin Rogers Received Noveber 24, 1998
It is elementary to factor an ordinary differential operator given a nonzero function in its kernel. Here a higher dimensional generalization of this fact is developed and utilized to determine nontrivial factorizations of constant coefficient partial differential operators. Through Darboux transformations, this result is applied to the solution of linear partial differential equations, commutativity, quantum integrability and the ‘‘bispectral problem.’’ 䊚 1999 Academic Press
1. INTRODUCTION The factorization of ordinary differential operators ŽODOs. is quite similar to the factorization of polynomials in a single variable. If one knows a root g ⺓ of some polynomial pŽ x . g ⺓w x x then it immediately follows that pŽ x . s q Ž x .Ž x y . for some polynomial q g ⺓w x x. Note that just as x y is the only monic linear polynomial that vanishes at , K s ( ⭸⭸x ( y1 Ž x . is the only monic first order ODO with Ž x . in its kernel. Analogously, the knowledge that Ž x . is in the kernel of an ODO implies that K is a factor of that operator. Ž . i THEOREM 1.1. If / 0 is in the kernel of the ODO L s Ý m is1 f i x ⭸ ⭸ Ž ⭸ [ ⭸ x . then it factors as L s Q( K for some ODO Q and the first order differential operator K s Ž x .( ⭸ ( y1 Ž x .. Moreover, since any ODO ‘‘completely factors’’ into first order terms, knowing the entire kernel is sufficient to determine all possible factorizations. ŽIn general, one can say that L s Q( K if and only if ker K ; ker L.. This property has proved to be useful in constructing Darboux transforma580 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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tions for ordinary differential operators with many applications in mathematical physics w2, 8, 9, 17, 18, 20x. For the same reasons, there is interest in being able to perform such Darboux transformations for partial differential operators ŽPDOs. w1, 3, 5, 6, 12, 15, 22, 23x. However, in keeping with the analogy to polynomials, the fact that one cannot necessarily completely factor a PDO complicates matters in the higher dimensional situation. Even though the analogous statement: ‘‘If the PDO L1 annihilates e¨ ery function in the kernel of the PDO L2 then L1 s Q( L2 for some PDO Q’’ is true w21x, it does not seem to be useful in determining previously unknown factorizations. Here we will prove a result which, although not completely general, leads to novel factorizations of constant coefficient partial differential operators. For any differentiable function Ž x 1 , . . . , x n . and irreducible polynomial q Ž z1 , . . . , z n . we will introduce below a partial differential operator K q and a proper subset Fq ; ker K q. The main theoretical result of this paper is to prove that any PDO whose kernel contains Fq has K q as a right factor. Using this fact, which has the well known result Theorem 1.1 as a corollary, we are able to determine new, explicit, nontrivial factorizations of constant coefficient partial differential operators. The application of such factorizations to the solution of linear PDEs, bispectrality and quantum integrability through Darboux transformations is discussed. 1.1. Notation Let D be the ring of partial different operators in the variables x s Ž x 1 , . . . , x n . and coefficients in some differential field K of functions. ª Ž The shorthand notations ⭸ [ ⭸ r ⭸ x and ⭸ s ⭸ , . . . , ⭸n . will be used. ŽSo i i 1 ª D s Kw ⭸ x is made up of polynomials in the ⭸ i ’s with coefficients that are functions in K.. Differential operators act on functions by differentiation and multiplication in the obvious way, and the action of the operator L g D on a function f Ž x 1 , . . . , x n . will be denoted Lw f x. Moreover, the composition of operators L1 , L2 g D will be written L1 ( L2 . We will say that a function f Ž x 1 , . . . , x n . is polynomial-exponential if it is of the form
ª
N
f Ž x1 , . . . , x n . s
ª ª
Ý pi Ž x 1 , . . . , x n . e ² x , ␣ : i
Ž 1.2.
is1 ª n Ž ²,: for some N g ⺞, polynomials pi g ⺓w x 1 , . . . , x n x and ␣ i g ⺓ . Here n . denotes the usual inner product on ⺓ . Note that the set of polynomialexponential functions forms a ring which is moreover closed under differentiation. Similarly, the quotient field of ratios of such functions with
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ALEX KASMAN
nonzero denominators is also closed under differentiation. Let us call this quotient field the rational-exponential functions. This paper is primarily concerned with the ring of partial differential operators in the variables x i with coefficients that are rational-exponential functions as defined above. More specifically, the main result involves the factorization of constant coefficient differential operators in this ring. In addition, in order to prove the results below it will be useful to introduce the ring ⺤ of rational coefficient translational-differential operators in ª z s Ž z1 , . . . , z n . Žcf. w19x which considers the one dimensional case.. An element ⌳ g ⺤ is an operator of the form N
⌳s
Ý Li (T␣
Ž 1.3.
ª
i
is1
where T␣ª is the translation operator ª T␣ª f Ž ª z. s f Žª zq␣ .
ª ␣ g ⺓n
Ž 1.4.
and L i are rational coefficient differential operators in the variables z i m
Li s
Ý ri j Ž ªz . ( pi j ž ⭸ˆ1 , . . . , ⭸ˆn / ,
js1
⭸ˆi s
⭸ ⭸ zi
,
ri j g ⺓ Ž ª z. ,
pi j g ⺓ w ª zx.
Ž 1.5.
Note that ⺓w ª z x ; ⺤ by associating p(Tª0 g ⺤ to the polynomial p. Special attention should be paid to the subring ⺤ 0 of constant coefficient translational-differential operators Ži.e., those for which L i g ⺓w ⭸ˆ1 , . . . , ⭸ˆn x.. The main significance of these operators here comes from the following observation: LEMMA 1.6. The map I from the ring ⺤ 0 of constant coefficient translational-differential operators in ª z to the ring of polynomial-exponential functions in ª x gi¨ en by ⌳ w e² x , z: x s I Ž ⌳ . w e² x , z: x ª ª
ª ª
⌳ g ⺤0
Ž 1.7.
is an isomorphism. ªª
In particular, e ² x, z : is an eigenfunction for every element of ⺤ 0 and the eigenvalues are polynomial-exponential functions.
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
583
2. FACTORING PARTIAL DIFFERENTIAL OPERATORS Let Ž x 1 , . . . , x n . g K be an arbitrary nonzero differentiable function and q Ž z1 , . . . , z n . be an irreducible polynomial. We will associate to this choice a differential operator and a set of functions. DEFINITION 2.1. Let K q g D be the differential operator 1
ª
K q [ Ž ª x . ( q Ž ⭸ .(
Žª x.
.
Note in particular that this operator annihilates every function in the set of functions ª ª
Fq [ Ž ª x . e² x , z: < ª z g ⺓,
qŽ ª z . s 04
parametrized by the irreducible algebraic variety qy1 Ž0.. THEOREM 2.2.
L g D factors as L s Q( K q for some Q g D
if and only if Fq ; ker L. Proof. Clearly if L has a right factor of K q then Fq ; ker L. So, let us assume the latter and prove the former. ª Consider first the case that Ž ª x . ' 1 and so K q s q Ž ⭸ .. Note that fŽª x, ª z . [ Lw e ² x , z : x s Ý fi Ž ª x . pi Ž ª z . e² x , z: ª ª
ª ª
for some polynomials pi and functions f i . Then, for any fixed value of ª x in its domain, this function must vanish for all ª z g qy1 Žª0.ª. Consequently, by the Nullstellensatz, the polynomial Žin ª z . f Žª x, ª z . e ²yx, z : has a factor of ª q Ž z . and so we may factor ª ª
fŽª x, ª z. s qŽ ª z . Ý fˆi Ž ª x . qˆi Ž ª z . e² x , z: .
Ž 2.3.
ª
Then, L factors as L( Q( q Ž ⭸ . with Qs
ª
Ý fˆi Ž ªx . qˆi Ž ⭸ . .
Ž 2.4.
ˆ s L( g D; then by the precedMore generally, for arbitrary let L ª ˆ ˆ ˆ 1 s Q( ˆ q Ž ⭸ª.( 1 . Ž ing paragraph we have that L s Q( ⭸ .. Hence L s L( ˆ 1 we get the desired factorization. Finally, letting Q s Q(
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ALEX KASMAN
Remark 1. Note that if n s 1 and q Ž z . s z g ⺓w z x then Fz Ž x . s Ž x .4 and K z Ž x . s ( ⭸ ( y1 . In particular, in this case one finds Theorem 1.1 as a consequence of Theorem 2.2. Remark 2. Theorem 2.2, though quite simple once it has been read and understood, is particular surprising since it does not require that L annihilate all of ker K q but only the Žproper. subspace spanned by the elements of Fq. This is a stronger statement than one might expect. ŽFor instance, note that if ' 1 and q Ž x 1 , x 2 . s x 12 y x 2 then x 1 g ker K q but it cannot be written as a linear combination of the functions in Fq ; ker K q.. Remark 3. Of course, most differential operators will not have any set of the form Fq in their kernels, and so this theorem is not entirely general. However, it has the advantage of being useful. The special instance of this theorem in the case that q is linear was used to construct Darboux transformations in w5x and the case in which Ž ª x . g ⺓w ª x x is a polynomial is implicitly contained in w3x. It will be further applied in the next section to determine new factorizations of constant coefficient differential operators.
3. CONSTANT COEFFICIENT DIFFERENTIAL OPERATORS 3.1. One Dimensional Case In the case of constant coefficient ordinary differential operators, Theorem 1.1 implies the following: THEOREM 3.1. There exists a constant coefficient ODO with right factor of K s Ž x .( ⭸ ( y1 Ž x . iff Ž x . s Ý nis1 pi Ž x . e i x for some pi g ⺓w x x and i g ⺓. Proof. Let be of this form and pŽ x . s ŁŽ x y i .1q deg p i g ⺓w x x; then the operator L s pŽ ⭸ . has Ž x . in its kernel. By the discussion above, this implies that L has a right factor of the form K. Alternatively, given a constant coefficient operator L s pŽ ⭸ . Ž p g ⺓w x x. let i Ž1 F i F n. denote the distinct roots of p and m i be their multiplicities. Then it is simple to check that x j e i x g ker L for 0 F j F m i y 1 and 1 F i F n. Over ⺓, these functions span a Ždeg p .-dimensional space which must therefore be all of ker L. Consequently, every function in the kernel of a constant coefficient ODO is of the given form. It will be seen that the situation is much the same in n dimensions if one replaces with any similar function of n variables and replaces ⭸ with any irreducible constant coefficient partial differential operator.
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
585
3.2. Higher Dimensional Case If one were to pick an arbitrary differential operator K, it would be unusual to be able to find another operator Q such that Q( K is a constant coefficient operator. The following theorem, analogous to Theorem 3.1, provides a class of nonconstant coefficient operators which are right factors of constant coefficient differential operators. THEOREM 3.2. Let Ž ª x . / 0 be a polynomial-exponential function and q Ž x . g ⺓w ª x x be an irreducible polynomial; then there exists a partial differenª tial operator Q / 0 such that Q( K q g ⺓w ⭸ x is a constant coefficient operator. ª
Proof. Let
Žª x. s
m
ª ª
Ý pi Ž ªx . e ² x , ␣ :
Ž 3.3.
i
is1 ª n for arbitrary polynomials pi and vectors ␣ the translationali g ⺓ . Then ªª ªª y1 Ž . differential operator T s I g ⺤ satisfying Te ² x, z : s Ž ª x . e ² x, z : Žcf. Lemma 1.6. is given by
m
Ts
ª
Ý pi ž ⭸z / T␣ .
Ž 3.4.
ª
i
is1
Now let pŽ z . g ⺓w z x be the polynomial m
pŽ z . [
Ł Ty␣ Ž q Ž ªz . . ª
is1
m iq1
i
m i [ deg pi
Ž 3.5. ª
ª
and L be the constant coefficient differential operator L s pŽ ⭸ . g ⺓w ⭸ x. Then L e ² x , z : s T ( Lw e ² x , z : x s T ( p Ž z . w e ² x , z : x ª ª
ª ª
ª ª
Ž 3.6.
and pŽ z . is constructed specifically so that T ( pŽ z . s q Ž z .(T X for some other translational-differential operator T X with polynomial coefficients. ªª ² x, z : x ' 0 for all ª Consequently, Lw e z g qy1 Ž0. and then Theorem 2.2 implies that L has a right factor of K q . 4. EXAMPLES, APPLICATIONS AND COMMENTS Without knowledge of its kernel, factorization of even an ordinary differential operator can be difficult Žcf. w13, 14x.. If one does know functions in the kernel, the problem of factorization becomes trivial but
586
ALEX KASMAN
such factorizations have application in the construction of Darboux transformations w2, 8, 9, 16, 17, 18, 20x, which is essentially conjugation in the ring of differential operators. Despite the inherent difficulty in factoring PDOs there has recently been much interest in the higher dimensional generalizations of the Darboux transformation w1, 3, 12, 15, 22, 23x. By using the factorizations determined above to perform Darboux transformations for constant coefficient PDOs one may construct new, explicit, higher dimensional examples of commutativity, quantum integrability, and bispectrality. Pick an irreducible polynomial q g ⺓w ª z x, polynomials pi g ⺓w ª x x and ª vectors ␣ i for 1 F i F m g ⺞. Then defining by Ž3.3. and p by Ž3.5., it ª ª follows that the constant coefficient differential operator L s pŽ ⭸ . g ⺓w ⭸ x factors as L s Q( K q for some Q. Knowing this, one may then compute the coefficients of Q explicitly Žfor instance, with a computer algebra package. since L and K q are known. EXAMPLE. For n s 2, consider the case that q Ž ª z . s z12 y z 2 , p1 s 1, p 2 ª ª ª w x Ž . Ž . Ž s ␥ g ⺓ x , ␣ 1 s 1, 0 and ␣ 2 s 0, 1 . Using 3.3. and Ž3.5. one finds that 2
L s Ž Ž ⭸ 1 y 1 . y ⭸ 2 . Ž ⭸ 12 y ⭸ 2 q 1 .
Žª x . s e x1 q ␥ e x 2 and K q s ⭸ 12 y ⭸ 2 y
2 e x1
⭸1 q 1 y
2␥ e x 1qx 2
2
.
ªª
As predicted, one may check that Lw e ² x, z : x vanishes for any ª z g qy1 Ž0. and so by Theorem 2.2 L s Q( K q . It is then merely an elementary calculation to check that Q s ⭸ 12 y ⭸ 2 y
2␥ e x 2
⭸1 q
2␥ e x 1qx 2
2
q 1.
Darboux Transformations for Linear PDEs Given the factorization L s Q( K q determined above, we have a Darboux transformation relating the solutions to the equation Lw f x s f Ž g ˜w f˜x s f˜ for L ˜ s K q ( Q. In particular, it is ⺓. and the new equation L clear that the application of the operators K q and Q provides maps ˜ y . between ker L y and ker L
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
THEOREM 4.1.
587
˜ s K q ( Q then we ha¨ e the maps If L s Q( K q and L ˜y K q : ker L y ª ker L f ¬ K q w f x
˜ y ª ker L y Q: ker L f˜¬ Q f˜ between the kernels of the operators. Proof. Suppose f g ker L y ; then
˜ y . f˜ s K q ( Q( K q w f x y K q w f x ŽL s K q ( L w f x y K q w f x s K q w f x y K q w f x s 0 Žand similarly for the map in the other direction.. As in other applications of Darboux transformations to differential operators w1, 3, 8, 9, 15, 20, 23x, these are most useful if they can be shown to preserve some form or property of interest. Here, for instance, the bispectral property and quantum integrability are considered although it seems likely that the transformations described here could be used as well for other purposes. Unfortunately, since the procedure described above depends explicitly on the fact that the initial operator has constant coefficients, it is not clear how one can iterate these transformations except in trivial circumstances. EXAMPLE Žcontinued.. Continuing the example begun above, we have the new differential operator
˜ s ⭸ 22 y 2 ⭸ 12⭸ 2 q 2 ⭸ 1 ⭸ 2 y 2 ⭸ 2 q ⭸ 14 y 2 ⭸ 13 L q q q
2 Ž e 2 x 1 q 6␥ e x 1qx 2 q e 2 x 2␥ 2 .
Ž e x 1 q e x 2␥ .
2
⭸ 12
y2 Ž e 3 x 1 q 9 e 2 x 1qx 2␥ y 3e x 1q2 x 2␥ 2 q e 3 x 2␥ 3 .
Ž e x 1 q e x 2␥ .
3
⭸1
e 4 x 1 q 8␥ e 3 x 1qx 2 y 10 e 2Ž x 1qx 2 .␥ 2 q 8 e x 1q3 x 2␥ 3 q e 4 x 2␥ 4
Ž e x 1 q e x 2␥ .
4
588
ALEX KASMAN
and the relationship Theorem 4.1 relating the solutions of the corresponding differential equations. For instance, it is trivial to find a solution to Lw f x s 0 by assuming f Ž x 1 , x 2 . s f 1Ž x 1 . q f 2 Ž x 2 .. One such function is f s 1 x 1 e x 1 q 2 x 2 e x 2 g ker L. However, the transformation f˜[ K q w f x s Ž 2 1 ␥ y 2 . e x 2 y q
2␥ 2 e 3 x 2 Ž 1 ␥ x 1 y 2 x 2 .
2
2␥ e 2 x 2 Ž 1 ␥ Ž x 1 y 1 . y 2 x 2 .
˜w f˜x s 0 which would be difficult to find using provides a solution to L other methods. Commutati¨ ity and Quantum Integrability A factorization of one constant coefficient operator L provides a means ª for producing a set of commutating operators. Let Mi g ⺓w ⭸ x be constant coefficient partial differential operators. Then the Žgenerically noncon˜ i [ K q ( Mi ( Q mutually commute. Explicitly, one sees stant. operators L that
˜i , L ˜ j s K q ( Mi ( Q( K q ( M j ( Q y K q ( M j ( Q( K q ( Mi ( Q L s K q ( Mi ( L( M j ( Q y K q ( M j ( L( Mi ( Q s 0 ª
by the commutativity of L and Mi g ⺓w ⭸ x. Consequently, letting M0 s 1 ˜0 , . . . , L ˜ n is a commutaand Mi s ⭸ i Ž1 F i F n. the ring generated by L tive ring with Krull dimension n which cannot be generated by fewer than n q 1 generators. Viewing the commutativity of these operators as the quantum analogue of the Poisson commutativity of functions on the cotangent bundle to a symplectic manifold, such ring may be considered as an example of a quantum integral system w4᎐6, 12x. Bispectrality ªª
The function e ² x, z : is, of course, an eigenfunction for the ring of constant coefficient partial differential operators in ª x with eigenvalues depending on the parameters ª z. Moreover, this same function is also an eigenfunction for the constant coefficient translational-differential operators ⺤ 0 in ª z with eigenvalues depending on ª x. This is an instance of ‘‘bispectrality’’ w7, 10, 11, 19, 22x. Then, it is interesting to note that the function K q wexp² ª x, ª z :x is also a common eigenfunction for differential ª operators in x and translational-differential operators in ª z. The one
589
FACTORIZATION OF PARTIAL DIFFERENTIAL OPERATORS
dimensional case Ž n s 1. is particularly interesting since it indicates a sort of bispectrality for solitons of the KP hierarchy and will be considered separately in w19x. ŽSee also w7x which considers such translational-differential bispectrality in the context of quantum duality.. THEOREM 4.2. The function [ K q wexp² ª x, ª z :x satisfies the eigen¨ alue equations
˜ Žª L x, ª z . s pŽ ª z. Ž ª x, ª z.
⌳ Žª x, ª z. s mŽª x. Ž ª x, ª z . Ž 4.3.
˜ s K q ( Q g D, m s deg q q 1 and some ⌳ g ⺤. for L Proof. The key observation here is that
Žª x, ª z. s Žª x . ( q Ž ⭸ ⭈. ( s
ˆ Ž ª x.
1
w e² x , z: x ª ª
ª
Ž x.
ª ª
deg q Ž ª x.
e² x , z:
where ˆŽ ª x . is another polynomial-exponential function of the form
ˆ Ž ª x. s
m ˆ
ª
Ý ˆpi Ž ªx . e ² x , ␣ˆ : . i
is1
Then, letting ⌳s
ž
m ˆ
Ý ˆpi ž ⭸ˆ1 , . . . , ⭸ˆn / (T␣ˆk (
/
is1
1 qŽ ª z.
m
(
žÝ
is1
pi ⭸ˆ1 , . . . , ⭸ˆn (T␣ªi
ž
/
/
it follows that ⌳ satisfies Ž4.3.. EXAMPLE Žcontinued.. Returning to the example above, oneªmay check ª that Ž ª x, ª z . [ K q wexp² ª x, ª z :x is also given by s Ž1r 2 . ⌫w e ² x, z : x where ⌫ g ⺤ is the translational-differential operator ⌫ s ␥ 2 Ž 1 q z12 y z 2 . TŽ0 , 2. q Ž 1 y 2 z1 q z12 y z 2 . TŽ2 , 0. q 2␥ Ž z12 y z1 y z 2 . TŽ1 , 1. . Then if ⌳ s ⌫ ( q Ž1z . (T with T s Iy1 Ž . s TŽ1, 0. q ␥ TŽ0, 1. one finds that ⌳w Ž ª x, ª z .x s Ž ª x . 3 Ž ª x, ª z .. Since this bispectrality is not only a consequence of the transformation but is also used in the construction itself, this result is similar to w12x where bispectrality of partial differential operators is utilized to perform Darboux ª
590
ALEX KASMAN
transformations. In particular, this paper extends that same idea to the case of translational bispectrality where the proof presented in w12x fails.
ACKNOWLEDGMENTS I am indebted to M. Singer and S. Tsarev for helpful discussions and to Yu. Berest and E. Horozov for continuing assistance and prior collaboration w3, 12x.
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19. A. Kasman, Spectral Difference Equations Satisfied by KP Soliton Wavefunctions, In¨ erse Problems 14 Ž1998., 1481᎐1487. 20. V. B. Matveev and M. A. Salle, ‘‘Darboux Transformations and Solitons,’’ Springer Ser. Nonlinear Dynam., Springer-Verlag, BerlinrNew York, 1991. 21. M. Singer, private communication. 22. A. P. Veselov, Baker᎐Akhiezer functions and the bispectral problem in many dimensions, in ‘‘CRM Proceedings and Lecture Notes,’’ Vol. 14, pp. 123᎐129, Amer. Math. Soc., Providence, 1998. 23. B. N. Zakhariev and V. M. Chabanov, New situation in quantum mechanics, Žwonderful potentials from the inverse problem., In¨ erse Problems 13 Ž1997., R47᎐R79.