Structure of Essential Spectrum and Discrete Spectra of the Energy Operator of Five-Electron Systems in the Hubbard Model. Third and Fourth Doublet States ()
1. Introduction
In the early 1970s, three papers [1] [2] [3], where a simple model of a metal was proposed that has become a fundamental model in the theory of strongly correlated electron systems, appeared almost simultaneously and independently. In that model, a single nondegenerate electron band with a local Coulomb interaction is considered. The model Hamiltonian contains only two parameters: the matrix element t of electron hopping from a lattice site to a neighboring site and the parameter U of the on-site Coulomb repulsion of two electrons. In the secondary quantization representation, the Hamiltonian can be written as
(1)
where
and
denote Fermi operators of creation and annihilation of an electron with spin
on a site m and the summation over
means summation over the nearest neighbors on the lattice.
The model proposed in [1] [2] [3] was called the Hubbard model after John Hubbard, who made a fundamental contribution to studying the statistical mechanics of that system, although the local form of Coulomb interaction was first introduced for an impurity model in a metal by Anderson [4]. We also recall that the Hubbard model is a particular case of the Shubin-Wonsowsky polaron model [5], which had appeared 30 years before [1] [2] [3]. In the Shubin-Wonsowsky model, along with the on-site Coulomb interaction, the interaction of electrons on neighboring sites is also taken into account. The Hubbard model is an approximation used in solid state physics to describe the transition between conducting and insulating states. It is the simplest model describing particle interaction on a lattice. Its Hamiltonian contains only two terms: the kinetic term corresponding to the tunneling (hopping) of particles between lattice sites and a term corresponding to the on-site interaction. Particles can be fermions, as in Hubbard’s original work, and also bosons. The simplicity and sufficiency of Hamiltonian (1) have made the Hubbard model very popular and effective for describing strongly correlated electron systems.
The Hubbard model well describes the behavior of particles in a periodic potential at sufficiently low temperatures such that all particles are in the lower Bloch band and long-range interactions can be neglected. If the interaction between particles on different sites is taken into account, then the model is often called the extended Hubbard model. It was proposed for describing electrons in solids, and it remains especially interesting since then for studying high-temperature superconductivity. Later, the extended Hubbard model also found applications in describing the behavior of ultracold atoms in optical lattices. In considering electrons in solids, the Hubbard model can be considered a sophisticated version of the model of strongly bound electrons, involving only the electron hopping term in the Hamiltonian. In the case of strong interactions, these two models can give essentially different results. The Hubbard model exactly predicts the existence of so-called Mott insulators, where conductance is absent due to strong repulsion between particles. The Hubbard model is based on the approximation of strongly coupled electrons. In the strong coupling approximation, electrons initially occupy orbitals in atoms (lattice sites) and then hop over to other atoms, thus conducting the current. Mathematically, this is represented by the so-called hopping integral. This process can be considered the physical phenomenon underlying the occurrence of electron bands in crystal materials. But the interaction between electrons is not considered in more general band theories. In addition to the hopping integral, which explains the conductance of the material, the Hubbard model contains the so-called on-site repulsion, corresponding to the Coulomb repulsion between electrons. This leads to a competition between the hopping integral, which depends on the mutual position of lattice sites, and the on-site repulsion, which is independent of the atom positions. As a result, the Hubbard model explains the metal-insulator transition in oxides of some transition metals. When such a material is heated, the distance between nearest-neighbor sites increases, the hopping integral decreases, and on-site repulsion becomes dominant.
The Hubbard model is currently one of the most extensively studied multielectron models of metals [6] [7] [8] [9] [10]. But little is known about exact results for the spectrum and wave functions of the crystal described by the Hubbard model, and obtaining the corresponding statements is therefore of great interest. The spectrum and wave functions of the system of two electrons in a crystal described by the Hubbard Hamiltonian were studied in [6]. It is known that two-electron systems can be in two states, triplet and singlet [6] [7] [8] [9] [10]. It was proved in [6] that the spectrum of the system Hamiltonian
in the triplet state is purely continuous and coincides with a segment
, and the operator
of the system in the singlet state, in addition to the continuous spectrum
, has a unique antibound state for some values of the quasimomentum. For the antibound state, correlated motion of the electrons is realized under which the contribution of binary states is large. Because the system is closed, the energy must remain constant and large. This prevents the electrons from being separated by long distances. Next, an essential point is that bound states (sometimes called scattering-type states) do not form below the continuous spectrum. This can be easily understood because the interaction is repulsive. We note that a converse situation is realized for
: below the continuous spectrum, there is a bound state (antibound states are absent) because the electrons are then attracted to one another.
For the first band, the spectrum is independent of the parameter U of the on-site Coulomb interaction of two electrons and corresponds to the energy of two noninteracting electrons, being exactly equal to the triplet band. The second band is determined by Coulomb interaction to a much greater degree: both the amplitudes and the energy of two electrons depend on U, and the band itself disappears as
and increases without bound as
. The second band largely corresponds to a one-particle state, namely, the motion of the doublet, i.e., two-electron bound states.
The spectrum and wave functions of the system of three electrons in a crystal described by the Hubbard Hamiltonian were studied in [11]. In the three-electron systems are exists quartet state, and two type doublet states. In [11] proved the essential spectrum of the system in a quartet state consists of a single segment and the three-electron bound states or three-electron anti-bound states is absent. Furthermore, shown what the essential spectrum of the system in doublet states is the union of at most three segments. In [11] also proved that three-electron bound states exist in doublet states.
The spectrum of the energy operator of system of four electrons in a crystal described by the Hubbard Hamiltonian in the triplet state was studied in [12]. In the four-electron systems exist quintet state, and three type triplet states, and two type singlet states. In the work [12] proved that the essential spectrum of the system in a triplet state is the union of at most three segments. There are also proved that the four-electron bound states or a four-electron anti-bound states exists in triplet states. Furthermore, also proved the spectrum of these three triplet states are different. The spectrum of the energy operator of four-electron systems in the Hubbard model in the quintet, and singlet states were studied in [13]. In [13] proved the essential spectrum of the system in a quintet state consists of a single segment and the four-electron bound states or four-electron anti-bound states is absent. In the system exists two type four-electron singlet states, and they are different origins. In the singlet states the essential spectra of four-electron systems consist of the union of no more than three segments. Furthermore, in the system exists no more one four-electron anti-bound states or no more one bound states.
Here, we consider the energy operator of five-electron systems in the Hubbard model and describe the structure of the essential spectra and discrete spectrum of the system for third and fourth doublet states.
The Hamiltonian of the chosen model has the form
(2)
Here, A is the electron energy at a lattice site, B is the transfer integral between neighboring sites (we assume that
for convenience),
, where
are unit mutually orthogonal vectors, which means that summation is taken over the nearest neighbors, U is the parameter of the on-site Coulomb interaction of two electrons,
is the spin index,
or
,
and
denote the spin values
and
, and
and
are the respective electron creation and annihilation operators at a site
.
In the five-electron systems exists sextet state, four type quartet states, and five type doublet states.
The energy of the system depends on its total spin S. Along with the Hamiltonian, the
electron system is characterized by the total spin S,
.
Hamiltonian (2) commutes with all components of the total spin operator
, and the structure of eigenfunctions and eigenvalues of the system therefore depends on S. The Hamiltonian H acts in the antisymmetric Fo’ck space
.
Belove we will give constructions of the Fo’ck space.
Let
be a Hilbert space and denote by
the n-fold tensor product
. We set
and define
The
is called the Fock space over
; it will be separable, if
is. For example, if
, then an element
is a sequence of functions
, so that
Actually, it is not
itself, but two of its subspaces which are used most frequently in quantum field theory. These two subspaces are constructed as follows: Let
be the permutation group on n elements, and let
be a basis for space
. For each
, we define an operator (which we also denote by
) on basis elements
by
. The operator
extends by linearity to a bounded operator (of norm one) on space
, so we can define
. It is an easy exercise to show that, the operator
is the operator of orthogonal projection:
, and
. The range of
is called n-fold symmetric tensor product of
. In the case, where
and
,
is just the subspace of
, of all functions, left invariant under any permutation of the variables. We now define
. The space
is called the symmetric Fock space over
, or Boson Fock space over
.
Let
is function from
to
, which is one on even permutations and minus one on odd permutations. Define
, then
is an orthogonal projector on
.
is called the n-fold antisymmetric tensor product of
. In the case where
,
is just the subspace of
consisting of those functions odd under interchange of two coordinates. The subspace
is called the antisymmetric Fock space over
, or the Fermion Fock space over
.
2. Third Doublet State
In the system exists five type doublet states. The doublet state corresponds to the basis functions
and
and
and
and
.
The subspace
, corresponding to the third five-electron doublet state is the set of all vectors of the forms
, where
is the subspace of antisymmetric functions in the space
.
The restriction
of H to the subspace
, is called the five-electron third doublet state operator.
Theorem 1. The subspace
is invariant under the operator H, and the operator
is a bounded self-adjoint operator. It generates a bounded self-adjoint operator
, acting in the space
as
(3)
where
is the Kronecker symbol. The operator
acts on a vector
as
(4)
Proof. We act with the Hamiltonian H on vectors
using the standard anticommutation relations between electron creation and annihilation operators at lattice sites,
,
, and also take into account that
, where
is the zero element of
. This yields the statement of the theorem.
Lemma 1. The spectrum of the operators
and
coincide.
Proof. Because
and
are bounded self-adjoint operators, it follows from the Weyl criterion (see, for example, [14], Ch. VII, §14) that there exist a sequence of vectors
such that
,
,
and
(5)
where
. On the other hand,
. Here
and
. It follows that
. Consequently,
. Conversely, let
. Again by the Weyl criterion, there then exist a sequence
such that
and
(6)
as
.
Setting
, we have
and
. This, together with Formula (6) and the Weyl criterion, implies that
, and hence
. These two relations imply that
.
We let
denote the Fourier transform:
, where
is a
-dimensional torus with the normalized Lebesgue measure
.
We set
. In the quasimomentum representation, the operator
acts in the Hilbert space
as
(7)
where
is the subspace of antisymmetric functions in
.
Using tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces [15], we can verify that the operator
can be represented in the form
(8)
where
and I is the unit operator.
Taking into account that
, we can express the action of operators
and
in the form
The spectrum of the operator
, where A and B are densely defined bounded linear operators, was studied in [16] [17] [18]. Explicit formulas were given there that express the essential spectrum
and discrete spectrum
of operator
in terms of the spectrum
and the discrete spectrum
of A and in terms of the spectrum
and the discrete spectrum
of B:
(9)
(10)
It is clear that
.
Therefore, we must investigate the spectrum of the operators
,
and
.
Let
,
,
.
Let the total quasimomentum of the two-electron system
be fixed. We let
denote the space of functions that are square integrable on the manifold
. It is known ( [19], chapter II, pp. 63-84, and [15], chapter XIII, paragraph 16, pp. 303-341) that the operator
and the space
can be decomposed into a direct integral
,
of operators
and spaces
such that
are invariant under
and
act in
according to the formula
,
where
.
It is known that the continuous spectrum of
is independent of the parameter U and consists of the intervals
Definition 1. The eigenfunction
of the operator
corresponding to an eigenvalue
is called a bound state (BS) (antibound state (ABS)) of
with the quasi momentum
, and the quantity
is called the energy of this state.
We consider the operator
acting in the space
according to the formula
It is a completely continuous operator in
for
.
We set
.
Lemma 2. A number
is an eigenvalue of the operator
if and only if it is a zero of the function
, i.e.,
.
Proof. Let the number
be an eigenvalue of the operator
, and
be the corresponding eigenfunction, i.e.,
Let
. Then
i.e., the number
is an eigenvalue of the operator
. It then follows that
.
Now let
be a zero of the function
, i.e.,
. It follows from the Fredholm theorem than the homogeneous equation
has a nontrivial solution. This means that the number
is an eigenvalue of the operator
.
We consider the one-dimensional case.
Theorem 2. 1). At
and
, and all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of
, i.e.,
.
2). At
and
, and all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of
, i.e.,
.
Proof. If
, then in the one-dimensional case, the function
is a monotone decreasing function on
and
, i.e., outside the continuous spectrum domain of the operator
. For
the function
decreases from 1 to
,
as
,
as
. Therefore, below the value
, the function
has a single zero at the point
. For
, and
, the function
decreases from
to 1,
as
,
as
. Therefore, above the value
, the function
cannot vanish.
If
, then the function
is an monotone increasing function on
and
, i.e., outside the continuous spectrum domain of the operator
. For
the function
increases from 1 to
,
as
,
as
. Therefore, below the value
, the function
cannot vanish.
For
, and
, the function
increases from
to 1,
as
,
as
. Therefore, above the value
, the function
has a single zero at the point
.
In the two-dimensional case, we have analogous results.
If
, then the function
is an increasing monotone function on
and
, i.e., outside the continuous spectrum domain of the operator
. For
the function
increases from 1 to
,
as
,
as
. Therefore, below the value
, the function
cannot vanish. For
and
, the function
increases from
to 1,
as
,
as
. Therefore, above the value
, the function
has a single zero at the point
.
If
, for
, then the function
decreases from
to 1,
as
,
as
. Therefore, above the value
, the function
cannot vanish. If
, for
, then the function
decreases from 1 to
,
as
,
as
. Therefore, below the value
, the function
has a single zero at the point
.
We consider three-dimensional case. Denote
.
For
, and
, below the continuous spectrum of the operator
, the function
has a single zero at the point
. For
, and
, below the continuous spectrum of the operator
, the function
cannot vanish.
Denote
.
For
, and
, above the continuous spectrum of the operator
, the function
has a single zero at the point
. For
, above the continuous spectrum of the operator
, the function
cannot vanish.
Now we investigated the spectrum of the operator
, i.e., the operator
It is known that the continuous spectrum of the operator
is independent of U and coincides with the segment
.
We let
denote the space of functions that are square integrable on the manifold
. That the operator
and the space
can be decomposed into a direct integral
,
. Each operator
acts in
as
, where
.
We set
.
The analogue of the Lemma 2 holds for the in this case. We consider the one-dimensional case.
Theorem 3. 1) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of operator
, i.e.,
.
2) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of operator
, i.e.,
.
The Theorem 3 is proved totally similarly to Theorem 2.
In the two-dimensional case, we have analogous results.
If
, then the equation
has a unique solution
, below the continuous spectrum of the operator
. If
, then the equation
has a unique solution
, above the continuous spectrum of the operator
.
We consider three-dimensional case. Denote
.
For
, and
, below of the continuous spectrum of the operator
, the function
has a single zero at the point
. For
, and
, below the continuous spectrum of the operator
, the function
cannot vanish.
Denote
.
For
, and
, above of the continuous spectrum of the operator
, the function
has a single zero at the point
. For
, and
, above the continuous spectrum of the operator
, the function
cannot vanish.
Now we investigated the spectrum of the operator
.
It is obvious that the spectrum of operator
is purely continuous and coincides with the value set of the function
, i.e.,
.
Now, using the obtained results and representation (8), we describe the structure of essential spectrum and the discrete spectrum of the operator
of third five-electron doublet state:
Theorem 4. At
and
the essential spectrum of the system of third five-electron doublet state operator
is exactly the union of four segments:
The discrete spectrum of the operator
is empty:
.
Here and hereafter
,
,
,
,
,
,
,
.
Theorem 5. At
and
the essential spectrum of the system of third five-electron doublet state operator
is exactly the union of four segments:
The discrete spectrum of the operator
is empty:
.
Here
,
.
Proof. It follows from representation (8) that
, and in the one-dimensional case, the continuous spectrum of operator
is
, and at
, the discrete spectrum of
consists of a single point
. The continuous spectrum of operator
is
, and at
, the discrete spectrum of
consists of a single point
. The spectra of the operator
is a purely continuous and consists of the segment
. Therefore, the essential spectrum of the system of third doublet-state operator
is the union of four segments, and the third doublet-state operator
has no eigenvalues.
The Theorem 5 is proved totally similarly to Theorem 4.
In the two-dimensional case to occur the analogous results.
We now consider the three-dimensional case.
Theorem 6. The following statements hold:
1) Let
and
,
,
, or
,
,
. Then the essential spectrum of the system third five-electron doublet state operator
is the union of four segments:
The discrete spectrum of the operator
is empty:
.
Here,
,
,
,
,
,
and
are the eigenvalues of the operators
, correspondingly.
2) Let
and
,
, and
or
,
, and
. Then the essential spectrum of the system third five-electron doublet state operator
is the union of two segments:
or
. The discrete spectrum of the operator
is empty:
.
3) Let
and
,
, and
, or
,
, and
. Then the essential spectrum of the system third five-electron doublet state operator
is consists of single segment:
, and the discrete spectrum of the system third doublet-state operator
is empty:
.
Theorem 7. The following statements hold:
1) Let
and
,
,
, or
,
,
.
Then the essential spectrum of the system third five-electron doublet state operator
is the union of four segments:
The discrete spectrum of the operator
is empty:
.
Here,
,
,
,
,
,
,
and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
2) Let
and
,
, and
, or
,
, and
. Then the essential spectrum of the system third five-electron doublet state operator
is the union of two segments:
or
, and the discrete spectrum of the system third doublet-state operator
is empty:
.
3) Let
and
,
, and
, or
,
, and
. Then the essential spectrum of the system third five-electron doublet state operator
is consists of single segment:
, and the discrete spectrum of the system third five-electron doublet state operator
is empty:
.
We now consider the three-dimensional case. Let
, and
, and
.
Then the continuous spectrum of the operator
is consists of the segment
We consider the Watson integral
. (see. [20] ). Because the measure
is normalized,
.
Theorem 8. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is below of continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is below of continuous spectrum of operator
.
Theorem 9. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is above of continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above of continuous spectrum of operator
.
In this case the continuous spectrum of the operator
is consists of the segment
Theorem 10. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is above of continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above of continuous spectrum of operator
.
Theorem 11. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is below of continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is below of continuous spectrum of operator
.
We now using the obtaining results and representation (8), we can describe the structure of essential spectrum and discrete spectrum of the operator of third five-electron quartet state:
Let
and
, and
.
Theorem 12. The following statements hold:
1) Let
, and
,
, or
,
,
. Then the essential spectrum of the system third five-electron doublet state operator
is the union of four segments:
. The discrete spectrum of the operator
is empty:
.
Here and hereafter
,
,
,
,
,
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
2) Let
, and
,
, or
,
. Then the essential spectrum of the system third five-electron doublet state operator
is the union of two segments:
, or
. The discrete spectrum of the operator
is empty:
.
3) Let
,
,
or
,
. Then the essential spectrum of the system third five-electron doublet state operator
is consists of single segment:
, and discrete spectrum of the system third five-electron doublet state operator
is empty:
.
Theorem 13. The following statements hold:
1) Let
, and
,
, or
,
,
. Then the essential spectrum of the system third five-electron doublet state operator
is the union of four segments:
. The discrete spectrum of the operator
is empty:
.
Here and hereafter
,
,
,
,
,
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
2) Let
, and
,
, or
,
. Then the essential spectrum of the system third five-electron doublet state operator
is the union of two segments:
, or
. The discrete spectrum of the operator
is empty:
.
3) Let
, and
,
or
,
. Then the essential spectrum of the system third five-electron doublet state operator
is consists of single segment:
and the discrete spectrum of the system third five-electron doublet state operator
is empty:
.
Consequently, the essential spectrum of the system third five-electron doublet state operator
is the union of no more than four segments, and discrete spectrum of the operator
is empty.
3. Fourth Doublet State
The fourth doublet state corresponds to the basic functions
. The subspace
, corresponding to the fourth five-electron doublet state is the set of all vectors of the form
,
where
is the subspace of antisymmetric functions in the space
.
The restriction
of H to the subspace
, is called the five-electron fourth doublet state operator.
Theorem 14. The subspace
is invariant under the operator H, and the operator
is a bounded self-adjoint operator. It generates a bounded self-adjoint operator
acting in the space
as
(11)
where
is the Kronecker symbol. The operator
acts on a vector
as
(12)
Proof. The proof of the theorem can be obtained from the explicit form of the action of H on vectors
using the standard anticommutation relations between electron creation and annihilation operators at lattice sites,
,
, and also take into account that
, where
is the zero element of
. This yields the statement of the theorem.
We set
. In the quasimomentum representation, the operator
acts in the Hilbert space
as
(13)
where
is the subspace of antisymmetric functions in
.
Taking into account that the function
is antisymmetric, we can rewrite Formula (13) as
(14)
where
and I is the unit operator.
Taking into account that
, we can express the action of operators
in the form
We must therefore investigate the spectrum and bound states (antibound states) of the operators
,
and
. Let the total quasimomentum of the two-electron systems by fixed:
,
,
. We let
(correspondingly,
and
) denote the space of functions that are square integrable on the manifold
(correspondingly,
and
). That the operator
,
and
and the space
can be decomposed into a direct integral
(correspondingly,
and
),
(correspondingly,
and
). Each operator
(correspondingly,
and
) acts in
(correspondingly,
and
) as
correspondingly,
where
(correspondingly,
and
).
The continuous spectrum of operator
(correspondingly,
and
) does not depend on the parameter U and consists of the intervals
(correspondingly,
and
).
We set
.
The analogue of the Lemma 2 holds for the in this case. We consider the one-dimensional case.
Theorem 15. 1) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
that is below the continuous spectrum of
, i.e.,
.
2) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
that is above the continuous spectrum of
. i.e.,
.
Proof. If
, then in the one-dimensional case, the function
is a monotone decreasing function on
and
, i.e., outside the continuous spectrum domain of the operator
. For
the function
decreases from 1 to
,
as
,
as
. Therefore, below the value
, the function
has a single zero at the point
. For
, and
, the function
decreases from
to 1,
as
,
as
. Therefore, above the value
, the function
cannot vanish. If
, then the function
is a monotone increasing function on
and
, i.e., outside the continuous spectrum domain of the operator
. For
the function
increases from 1 to
,
as
,
as
. Therefore, below the value
, the function
cannot vanish. For
, and
, the function
increases from
to 1,
as
,
as
. Therefore, above the value
,
has a single zero at the point
.
In the two-dimensional case, we have analogous results.
We consider three-dimensional case. Denote
.
For
, and
, below of the continuous spectrum of the operator
the function
has a single zero at the point
. For
, and
, below of the continuous spectrum of the operator
the function
cannot vanish.
Denote
.
For
and
, above the continuous spectrum of the operator
the function
has a single zero at the point
. For
, above the continuous spectrum of the operator
the function
cannot vanish.
We now investigate the spectrum of the operator
:
We set
.
The analogue of the Lemma 2 holds for the in this case. We consider the one-dimensional case.
Theorem 16. 1) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
that is above the continuous spectrum of
, i.e.,
.
2) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
that is below the continuous spectrum of
, i.e.,
.
Proof. If
, then in the one-dimensional case, the function
increases monotonically outside the continuous spectrum domain of the operator
. For
the function
increases from 1 to
,
as
,
as
. Therefore, below the value
, the function
cannot vanish.
For
, and
, the function
increases from
to 1,
as
,
as
. Therefore, above the value
, the function
has a single zero at the point
. If
, then the function
decreases monotonically outside the continuous spectrum domain of the operator
. For
the function
decreases from 1 to
,
as
,
as
. Therefore, below the value
, the function
has a single zero at the point
. For
, and
, the function
decreases from
to 1,
as
,
as
. Therefore, above the value
, the function
cannot vanish.
In the two-dimensional case, we have analogous results.
We consider three-dimensional case. Denote
.
For
, and
, below of the continuous spectrum of the operator
the function
has a single zero at the point
. For
, and
, below of the continuous spectrum of the operator
the function
cannot vanish.
Denote
.
For
and
, above the continuous spectrum of the operator
the function
has a single zero at the point
. For
, above the continuous spectrum of the operator
the function
cannot vanish.
We now investigate the spectrum of the operator
It is known that the continuous spectrum of
is independent of U and coincides with the segment
.
We set
.
The analogue of the Lemma 2 holds for the in this case. We consider the one-dimensional case.
It is clear that the at
(
) exists only one solution of the equation
, the above (the below) the continuous spectrum of the operator
.
Theorem 17. 1) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of operator
, i.e.,
.
2) At
and
and for all values of the parameter of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of operator
, i.e.,
.
In the two-dimensional case to occur the analogous situation. For
, the function
above the continuous spectrum of the operator
has a single zero at the point
. For
, the function
below the continuous spectrum of the operator
has a single zero at the point
.
We now consider three-dimensional case. For
, and
, the function
below the continuous spectrum of the operator
has a single zero at the point
. For
, the function
below the continuous spectrum of the operator
cannot vanish. For
, and
, the function
above the continuous spectrum of the operator
has a single zero at the point
. For
, the function
above the continuous spectrum of the operator
cannot vanish.
Now, using the obtained results and representation (14), we describe the structure of the essential spectrum and the discrete spectrum of the operator
fourth five-electron doublet state:
Theorem 18. At
and
the essential spectrum of the system of fourth five-electron doublet state operator
is exactly the union of seven segments:
The discrete spectrum of the operator
consists of no more than one point:
, or
.
Here and hereafter
,
,
,
,
,
,
,
,
.
Theorem 19. At
and
the essential spectrum of the system of fourth five-electron doublet state operator
is exactly the union of seven segments:
The discrete spectrum of the operator
consists of no more than one point:
, or
.
Here
,
,
.
In the two-dimensional case to occur the analogous results.
We now consider the three-dimensional case. Let
.
Theorem 20. The following statements hold:
1) Let
, and
,
, or
,
,
or
,
,
. Then the essential spectrum of the system fourth five-electron doublet state operator
is the union of seven segments:
The discrete spectrum of the operator
consists of no morethanone point:
, or
.
Here,
,
,
,
,
,
,
and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
2) Let
,
, and
, or
,
, and
, or
,
, and
, or
,
, and
, or
,
, and
, or
,
, and
, Then the essential spectrum of the system fourth five-electron doublet state operator
is the union of four segments:
or
or
The discrete spectrum of the operator
is empty:
.
3) Let
,
, and
, or
,
, and
, or
,
, and
. Then the essential spectrum of the system fourth five-electron doublet state operator
is the union of two segments:
or
,
or
The discrete spectrum of the operator
is empty:
.
4) Let
,
and
, or
, and
, and
,
, or
,
,
, and
. Then the essential spectrum of the system fourth five-electron doublet state operator
is consists of single segment:
, and the discrete spectrum of the operator
is empty:
.
Let
.
Theorem 21. The following statements hold:
1) Let
, and
,
,
, or
, and
,
or
, and
,
,
. Then the essential
spectrum of the system fourth five-electron doublet state operator
is the union of seven segments:
The discrete spectrum of the operator
consists of no more than one point:
, or
.
Here,
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
2) Let
,
, and
, or
,
, and
,
, or
,
, and
,
. Then
the essential spectrum of the system fourth five-electron doublet state operator
is the union of four segments:
or
or
The discrete spectrum of the operator
is empty:
.
3) Let
,
, and
,
, or
,
, and
,
, or
,
, and
,
, or
,
, and
,
. Then the essential spectrum of the system fourth five-electron doublet state operator
is the union of two segments:
or
or
The discrete spectrum of the operator
is empty:
.
4) Let
,
, and
, or
,
, and
, or
,
, and
,
. Then the essential spectrum of the system fourth five-electron doublet state operator
is consists of single segment:
, and the discrete spectrum of the operator
is empty:
.
Let
and
, and
.
It is known that the continuous spectrum of
is independent of U and coincides with the segment
.
Theorem 22. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is below the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is below the continuous spectrum of operator
.
Theorem 23. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is above the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above the continuous spectrum of operator
.
It is known that the continuous spectrum of
is independent of U and coincides with the segment
.
Theorem 24. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is above the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above the continuous spectrum of operator
.
Theorem 25. At
and
and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
if
, that is below the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is below the continuous spectrum of operator
.
Now, using the obtained results and representation (14), we describe the structure of the essential spectrum and the discrete spectrum of the system fourth five-electron doublet state operator
:
Let
and
, and
.
Theorem 26. The following statements hold:
1) Let
,
,
, and
, or
,
,
and
, or
,
, and
. Then the essential spectrum of the system fourth five-electron doublet state operator
is consists of the union of seven segments:
.
The discrete spectrum of the operator
consists of no more than one point:
, or
.
Here and hereafter
,
,
,
,
,
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
2) Let
,
,
, and
, or
,
, and
, or
,
,
, and
, or
,
,
, and
. Then the essential spectrum of the system fourth five-electron doublet state operator
is consists of the union of four segments:
,
or
,
or
.
The discrete spectrum of the operator
is empty:
.
3) Let
,
,
, and
, or
,
, and
, or
,
,
, and
, or
,
,
, and
. Then the essential spectrum of the system fourth five-electron doublet state operator
is consists of the union of two segments:
,
or
,
or
,
The discrete spectrum of the operator
is empty:
.
4) Let
,
,
, and
, or
,
,
, and
, or
,
,
and
, or
,
and
. Then the essential
spectrum of the system fourth five-electron doublet state operator
is consists of single segments:
, and discrete spectrum of the operator
is empty:
.
Theorem 27. The following statements hold:
1) Let
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
. Then the essential spectrum of the system
fourth five-electron doublet state operator
is consists of the union of seven segments:
The discrete spectrum of the operator
consists of no more than one point:
, or
.
Here and hereafter,
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
2) Let
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
, and
. Then the
essential spectrum of the system fourth five-electron doublet state operator
is consists of the union of four segments:
,
or
,
or
.
The discrete spectrum of the operator
is empty:
.
3) Let
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
, and
. Then the essential spectrum of the system fourth five-electron doublet state operator
is consists of the union of two segments:
,
or
,
or
.
The discrete spectrum of the operator
is empty:
.
4) Let
,
,
, (
), and
, or
,
,
and
, or
,
, and
. Then the essential
spectrum of the system fourth five-electron doublet state operator
is consists of single segments:
, and discrete spectrum of the operator
is empty:
.
4. Conclusions
In this paper, we consider the energy operator of five electron systems in the Hubbard model and describe the structure of the essential spectra and discrete spectrum of the system in the third and fourth doublet states. The obtained results show that in the one-dimensional and two-dimensional case, the essential spectrum of the system in the third doublet state is exactly the union of four segments, and the discrete spectrum is empty. In the three-dimensional case, the essential spectrum of the system in the third doublet state consists of the union of four segments, or the union of two segments, or consists of a single segment. The discrete spectrum of the system in the third doublet state always is empty. Consequently, the essential spectrum of the system in the third doublet state consists of the union of no more than four segments, and the discrete spectrum is empty.
In the case of the fourth doublet state, the essential spectrum of the system in the one-dimensional and two-dimensional case consists of union of seven segments, and the discrete spectrum of the system consists of most one point. In the three-dimensional case, the essential spectrum of the system in the fourth doublet state consists of the union of seven segments, or the union of four segments, or the union of two segments, or consists of a single segment. The discrete spectrum of the system in the fourth doublet state consists of no more than one point. Consequently, the essential spectrum of the system in the fourth doublet state consists of the union of no more than seven segments, and the discrete spectrum consists of only one point, or is empty.