SEMIFACTORIZATION OF THE MULTIDIMENSIONAL SCHROEDINGER OPERATOR

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SEMIFACTORIZATION OF THE

MULTIDIMENSIONAL SCHROEDINGER OPERATOR

A. Csizmazia

To cite this version:

A. Csizmazia. SEMIFACTORIZATION OF THE MULTIDIMENSIONAL SCHROEDINGER OPERATOR. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-87-C3-88.

�10.1051/jphyscol:1989313�. �jpa-00229454�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n°3. Tome 50, mars 1989 C3-87

SEMIFACTORIZATION OF THE MULTIDIMENSIONAL SCHROEDINGER OPERATOR

A. CSIZMAZIA

1955 Ixora Rd., N. Miami, Fi 33181, U.S.A.

Résumé - Des semi-factorisations des opérateurs de Schroedinger en dimensions quatre et huit sont présentées dans cet article.

Abstract - Semifactorizations of the Schroedinger operator in dimensions four and eight are presented in this article.

1-INTRODUCTION

The most important problem in soliton theory is for n > 3 to (*) find strongly n+l-D analogues of the I+l-D soliton equations. No positive results on this problem have been announced. The author solved (*) for n=2 using novel semifactorizations of the 2-D Schroedinger operator 11,2/. Semifactorizations of the 4-D and 8-D Schroedinger operators are obtained herein . The algebra of quaternions and the (non-associative ) Cayley algebra of octonions are used.

2-SF,MTFACTORI7,M'ION IN 4-D

A semifactorization of the 2-D Schroedinger operator is provided by Eq (9) of III (or Section 6 of 111 ). This is generalized to 4-D in Lemma 1 below. Let M^v be the plane or torus of dimension v=2a . Say \\ , yi ... xa , ya are the standard coordinates for M. Set A v a £ h d ce a + 3 p" * with a =xn » P =yh • Assume u , m are real valued functions on M. Take L<v> to be the Schroedinger operator -A ^v + u. Suppose L<v>(e^m ) = 0 . This amounts to u = A (m) + Z (met )² + (mjj )². A 4 factors via quaternions: A 4 = dd = dd . d=di+d2 j with dn = 9 ct + >3 p" , j2 = -1 and ji = -ij. This generalizes to a semifactorization of L<4> . Take w = d(m) and D = d-w . "Re" means "real part o f . The differential operators on M with quaternion ( respectively : complex ) valued coefficients constitute a real associative algebra Q ( respectively : C). Say Ai , A2 are in C . Allow c = ±1. Define < A i , A 2 > c = (c*2 Ai). Ai + A2 j is identified with < Ai,A2>-i> Thus A* = Ai* -A21 j.Here l is the transpose without conjugation.

LEMMA 1. L<4> = Re (D*D).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989313

JOURNAL DE PHYSIQUE

P r o o f . Say A = A1 + A2 j

,

B = B1 + B2 j with Ah

,

Bh in C

.

We define A o B = A l ~ 1 - T i 2B 2+ (A2 B1 + B2 )j. ThisHtwisted product" is obtained from BA by replacing each subproduct Bh An

,

Bh

xn

with AnBh

,

& B ~ respectively. If a,b a r e quaternion valued functions on M ,then aOb = ba.

AB =

x 0 B

.

^So

L < 4 > = - A

+

Lwt 2 +Re A with A d . w-wO

if =

a(w). I

Note that (AOB)*=B*O(A*).

The semifactorization of L<2> given by Eq.(ll) of 111 is generalized to Lc4> by Eq.(l) together with Theorem 1 below. Let

4

over the real valued infinitely differentiable functions defined o n M4

. ⁴

^{C ) 4} is a n isomorphism of real

vector spaces.

'""7,

Later we factor V in the 8-D real algebra 0 1 described next. Allow c = 1

.

^{Say A}

^,

A', B, B' a r e in Q

.

Let A = <A,A1>=, B = <B,B'>c

.

^Define

When A, A', B, B' a r e quaternion valued functions Eq. (2) gives the usual product for octonions. ~2 with the product in Eq. (2) is a n algebra Oc

.

^Take

^{g =}

^<O,bc

.

A = A+A,&. A

*

=A*+C(X')*JL. (AB )*=B *A

*.

Say c=l. Set D'= w-d%

.

We consider operators defined on Mg

.

^In Eq. (2) take c=-1. Set d = d+d09_ with d 1 = d 3 +dq j

. ^{- A}

^{g=d*d = dd*}

.

^{Set w = d} (m) a n d D = d

-

^w

.

REFERENCES.

/ I / Csizmazia, A.,"Soliton Equations Generalized t o Higher Dimensions", in

"Nonlinear Evolutions, Proceedings of the ZVth Workshop O R Nonlinear Evolution Equations and Dynamical Systems" (Balaruc-les Bains, France, June 11-25

,

1987) 339-348

,

World Scientific Publishing Co.

,

Singapore

,

(1988)

121 Csizmazia, A.

,

"The Heisenberg

-

^Lax Equation Generalized", in

"

Some Topics on Inverse Problems

,

Proceedings of the XVIth Workshop on the Interdisciplinary Study of Inverse Problems" (Montpellier, France

,

Nov.30-Dec.4.

,

1987)

316-326, World Scientific Publishing Co.

,

Singapore , (1988).