In the present paper we consider a 
operator matrix acting in the direct sum of zero-particle and one-particle subspaces of Fock space. It is shown that this operator has no more than one positive and no more than two negative simple eigenvalues.
Keywords: Operator matrix, Fock space, eigenvalue, annihilation and creation operators, polynom, essential and discrete spectrum.
Block operator matrices are matrices the entries of which are linear operators between Banach or Hilbert spaces. Such operators often arise in mathematical physics, e.g. in fluid mechanics (see [1]), magnetohydrodynamics (see [2]) and quantum mechanics (see [3]). Spectral properties of the block operator matrices are studied in detail in [4]. One of the special class of block operator matrices are Hamiltonians associated with a system describing 
-particles in interaction without conservation of the number of particles. Here off-diagonal entries of such block operator matrices are annihilation and creation operators. The study of systems describing -particles in interaction without conservation of the number of particles is reduced to the study of the spectral properties of self-adjoint operators acting in the zero-particle, one-particle,…, 
-particle subspaces of a Fock space.
In the present paper we consider a 
block operator matrix acting in the direct sum of zero-particle and one-particle subspaces of a Fock space. We prove that this operator has no more than one positive and no more than two negative simple eigenvalues.
Let 
be the field of complex numbers and 
be the Hilbert space of square integrable (complex) functions on 
. Denote by 
the direct sum of spaces 
and 
, that is, 
. The spaces 
and 
are zero- and one-particle subspaces of a Fock space 
over , respectively.
Let us consider the following operator matrix 
acting in the Hilbert space as
,
where the entries 
are defined by
.
Here 
; 
is a fixed read number, the functions 
and 
are real-valued continuous functions on 
and 
denotes the adjoint operator to 
.
Under this assumptions the operator is bounded and self-adjoint in .
We remark that the operators 
and 
are called annihilation and creation operators, respectively.
We denote by 
, 
and 
the spectrum, essential spectrum and discrete spectrum of a bounded self-adjoint operator.
Lemma 1. The relation holds.
Proof. Since the operator is a bounded self-adjoint operator whose rank does not exceed three, we have . We show that . To this end, we consider the equation 
for 
, which is equivalent to the system of equations
(1)
where 
is the scalar product in . It is easy to see that the elements of the subspace
are solutions of system of equations (1). Then the fact 
implies that . The lemma is proved.
By Lemma 1 the operator may have only positive and negative discrete eigenvalues. The following theorem describes the number and location of these eigenvalues.
Theorem 1. The operator has no more than one positive and no more than two negative simple eigenvalues.
Proof. Let us consider the equation 
or the system of equations
. (2)
Since 
from the second equation of (2) we find
, (3)
where
. (4)
Substituting the expression (3) for 
into the first equation of the system of equations (2) and the equality (4) we have that the system of equations (2) has a solution if and only if
,
where 
is the norm in .
We note that, if 
and 
are linear dependent, then 
. Therefore,
and
.
By the inequality 
we obtain that
.
There are three cases are possible: 1) and are orthogonal; 2) and are parallel; 3) and are neither orthogonal and nor parallel.
Let and be orthogonal. Then
.
In this case the numbers
are zeroes of 
, i.e., the eigenvalues of .
We remark that the numbers 
are also zeroes of 
in the case where and are not orthogonal.
Let and be parallel. Then
.
In this case the polynomial 
can be written in the form
.
For convenience we assume that 
. From here it follows that the numbers
and
are zeroes of 
, i.e., the eigenvalues of . In the case where 
we have 
and 
.
We remark that the numbers 
are also zeroes of 
in the case where and are not parallel.
Let and be neither orthogonal and nor parallel. Then
.
Set 
and 
. Without loss of generality (otherwise we would be prove the following facts in the same way) we assume that the inequalities 
hold. Then it follows that 
. Since the numbers 
and 
are zeroes of and , respectively, we have
and
,
i.e. on the boundary of 
the polynomial 
has a different sign. Hence, there exists a point 
, such that 
and 
. Analogously one can prove that there exist the numbers 
and 
, which are zeroes of the polynomial .
Since is a polynomial of degree 3 these zeroes are simple.
One can see that 
. Theorem 1 is completely proved.
Notice that Theorem 1 plays important role in the study the number of eigenvalues corresponding generalized Friedrichs model.
References:
1. S. Chandrasekhar. Hydrodynamic and hydromagnetic stability. The International Series of Monographs on Physics. Clarendon Press Oxford University Press, New York, 1998. Reprint of the 1992 edition.
2. E. Lifschitz. Magnetohydrodynamic and spectral theory. Vol. 4 of Developments in Electromagnetic Theory and Applications. Kluwer Academic Publishers Group, Dordrecht, 1989.
3. Thaller. The Dirac equation. Texts and Monographs in Physics. Springer, Berlin, 1992.
4. Tretter. Spectral theory of block operator matrices and applications, Impe. Coll. Press, 2008.
