A class of

In the chiral quark soliton (CQS) model in nuclear physics (see, e.g., [

Comparing

In a paper [

As is pointed out in [

In this paper, for each natural number

The present paper is organized as follows. We first recall some basic facts in operator theory in Section

Let

We denote by

Let

The next lemma summarizes some basic facts on strongly commuting (resp., anticommuting) self-adjoint operators.

Let

Let

Let

Part (i) is well known (e.g., [

Let

Let

We denote by

Every densely defined closable linear operator

We denote by

Let

To introduce a mass operator, let

The Hamiltonian

In this work, we do not intend to discuss essential self-adjointness of

The operator

In the abstract CQS model [

Condition

Condition

Condition

Condition

This follows from Lemma

Assume

This follows from Lemma

Simillar to the proof of part (ii).

We define

Assume

It is well known or easy to see that, for all

Assume

(i) Since

(ii) By

One of the other sufficient conditions for

As is well known, the standard free Dirac operator

Let

We define

Let

Since the subspace

Let

By (

Proposition

It may be difficult in general to show the existence of self-adjoint, unitary solutions

Let

Since

Additionally we make a remark on the converse of Lemma

Let

Equation (

The following lemma gives a sufficient condition for a solution to (

Let

As in the proof of Lemma

We have from Proposition

Let

In supersymmetric quantum mechanics with a supercharge

To investigate ker

(i) For each

For a linear operator

Assume

The self-adjointness of

Inequality (

By

Let

By the functional calculus, one has

Assume

Moreover, the constant

The operator

We set

Under the same assumption as in Theorem

We next consider a perturbation of

The quantity

Under the assumption of Theorem

Since

Assume

We write

In the papers [

We introduce a class of

If

For each

Since

For

Assume

Under condition

We note that, if one regards

Let

Assume

We have

The following theorem gives another sufficient condition for

Assume

We write

In this section, we consider the essential spectrum of

Let

The operator

The operator

For each

Let

We write

If

Let

By (

In this section, in view of Theorem

Let

We first note an elementary fact:

Suppose that the assumption of Lemma

Suppose that

In view of Theorem

For each

For a compact operator

Let

By the weak Hausdorff-Young inequality (e.g., [

In general, for all compact operators

As for

Let

We need only to consider the case where

As in Corollaries 1.2 and 1.3 in [

Under the same assumption as in Theorem

Suppose that the assumption of Theorem

Now we apply the results in the preceeding section to the Dirac operator

Let

If

Suppose that

Moreover, the number

(i) We can write

(ii) By (

We have from Corollary

Let

We can also use Theorems

Let

If

Suppose that

By Theorem

Theorem

Let

Asao Arai is supported by the Grant-in-Aid 24540154 for Scientific Research from JSPS.