The main purpose of the present paper¹ is to construct the semi-commutative differential operators associated with the 1-dimensional Dirac operator

where the potential is the infinitely differentiable function. Define the 1st order ordinary differential operartors by

then, the oprtator is expressed as

Note that the variable x can be regarded as both real or complex throughout the paper.

The differential operators A, B are said to be semicommutative, if the commutator is the multiplicative operator. As for the 1-dimensional Schrödinger operator

the identities

are well known for the differential operator of the order difined by

where are the KdV polynomials which will be explained precisely in Section 2.

On the other hand, in [1], R. M. Miura discovered the following interesting fact; if solves the mKdV (−) equation

then both functions defined by

solve the KdV equation

where the subscript denotes the partial differentiation. The transformation defined by (7) is the Miura transformation which plays the crucial role in the soliton theory. By (7), we have immediately the relation

The transformation defined by (9) is nothing but the Darboux transformation. Using the KdV polynomial, the relation (9) can be expressed as

Considering the above facts, we investigate the problem to express the differential polynomials in terms of the function for the stationary case, i.e., when the function v is independent of the time variable t, i.e., in what follows are defined by

As a result, we obtain the new formulation of the stationary mKdV (−) hierarchy.

In our previous works [2,3], it is clarified that the semi-commutative operators and Darboux transformation are deeply related to the spectral theory of the differential operator when the potential is algebrogeometric. The aim of our work is to extend these results concerned with the 1-dimensional Schrödinger operator to the 1-dimensional Dirac operator. The present work can be regarded as the first step of it. By applying the results of the present paper, we can obtain the various transformation formulas concerned with the algebro-geometric elliptic potential. These results will be reported in the forthcoming paper.

The contents of the present paper are as follows. In Section 2, we explain the fundamental materials which are necessary for the present work. In Section 3, calculation of the commutator of the Dirac operator and the differential operator constructed from the semi-commutative operator of the KdV hierarchy is carried out, and the main theorem of the present paper is stated. Section 4 is devoted to the proof of the main theorem. In Section 5, we construct the recursion operator associated with the mKdV hierarchy.

The KdV polynomials are differential polynomials defined by the recurrence relation

with the condition, where is the formal pseudo-differential operator defined by

Then turn out to be the differential polynomials in. For examples, we have

If depends also on the time variable t, then the evolution equation

is nothing but the KdV equation. Hence, we call them the KdV polynomials. See [3] for more details of the KdV polynomials.

In what follows, we will often use the higher order derivatives of the differential polynomials. So, for the brevity, we will use the following notations of derivatives of the KdV polynomials defined by

where. Thus we have

From now on, we restrict ourselves to the stationary problem, i.e., the function under consideration depends only on the space variable x.

One immediately verifies the identities

Define the multi-component operator by

By (13), one can show immediately the operator identity

On the other hand, define the multi-component differential operator by

Then, by (5), (14), and (16), we have immediately

Therefore the operator are semi-commutative with the operator.

Define the scalar differential operators by

respectively. By direct calculation, we have immediately

By the definition (6) of the operator, we have immediately

By (15), we have

Furthermore, by the definition (2) of, one verifies

where we used the identity

where

The identity (22) is derived in [4], and is called the fundamental identity of the Darboux transformation in it, By the fundamental identity (22), we have immediately

Put

then we can express the identity (21) in terms of as

For, one verifies easily

,

.

Thus, for hold, and they are not differential operators, but are the multipricative operator. Thus, the operator and the Dirac operator turns out to be semi-commutative for.

For general n, we have the following theorem which is the main result of the present paper.

Theorem 1. The multi-component differential operators and are semi-commutative, i.e., are the multiplicative operators, and they coincide with each other, i.e., the equality

hold for all n. Moreover, if we denote them as, then the identities

hold for all.

If the potential v depends also on the time variable t, i.e., , the evolution equation

is nothing but the mKdV (−) equation. Hence, we call the differential polynomials, the mKdV (−) polynomials.

We prove the theorem by induction. Firstly, for, one verifies

On the other hand, by (10), we have

Secondly, for, we assume that

holds. Then, for, we have

From (21) and (23), we have

This implies that are the multiplicative operators, i.e., and are semi-commutative.

Similarly, by induction, we can show that coincide with each other.

Next we show the identity (26). By straightforward calculation, one can show

Put

then, by (27), we have

where we used the relation

which are derived immediately from (22).

Thus, we have

By (7), we have

By (12) which is the definition of, we have

This completes the proof of Theorem 1.

In the preceding section, we have shown the relation

hold for all n. Therefore, by (28), we have

Then, by (22), we have

The relation (29) defines the recursion operators of

. Therefore, we have the following theorem.

Theorem 2. The formal pseudo-differential operator defined by

is the recursion operator associated with the mKdV (−) polynomials, , i.e.,

hold.

Using this recursion operator, one can calculate easily the hierarchy of the mKdV (−) polynomials.