# Dirac-like Hamiltonians associated to Schrödinger factorizations

D. Demir<EMAIL_ADDRESS>ORCID:
0000-0002-7381-8327, Ş<EMAIL_ADDRESS>ORCID:
0000-0001-6380-280X, J<EMAIL_ADDRESS>ORCID: 0000-0002-0847-6420

a Department of Medical Services and Techniques, Ankara Medipol University,
06050 Ankara, Turkey
b Department of Physics, Faculty of Science, Ankara University, 06100 Ankara,
Turkey
c Departamento de Física Teórica, Atómica y Óptica, and IMUVA,
Universidad de Valladolid, 47011 Valladolid, Spain

###### Abstract

In this work, we have extended the factorization method of scalar shape-
invariant Schrödinger Hamiltonians to a class of Dirac-like matrix
Hamiltonians. The intertwining operators of the Schrödinger equations have
been implemented in the Dirac-like shape invariant equations. We have
considered also another kind of anti-intertwining operators changing the sign
of energy. The Dirac-like Hamiltonians can be obtained from reduction of
higher dimensional spin systems. Two examples have been worked out, one
obtained from the sphere ${\cal S}^{2}$ and a second one, having a non-
Hermitian character, from the hyperbolic space ${\cal H}^{2}$.

Keywords: Dirac Hamiltonian; matrix Hamiltonian; factorization; symmetries;
pseudo-Hermiticity.

##  1 Introduction

In this work, our starting point will be the one-dimensional Schrödinger
equations that can be algebraically solved by the well known method of
factorization [1]. This method has a long history of applications to quantum
mechanics since the early times when it was already used by Dirac and
Schrödinger [2, 3]. The basis of the method can be described as the building
of a sequence (or hierarchy) of shape invariant Schrödinger Hamiltonians
connected by intertwining operators [4, 5, 6]. The knowledge of such
intertwining operators and their annihilated functions solves the eigenvalue
problem for all the eigenvalue equations in the Hamiltonian hierarchy. Our
objective here is to widen the factorization method from the Schrödinger
second order scalar equations to a kind of first order matrix Dirac-like
equations. We will associate to each of these hierarchies of Schrödinger
equations another hierarchy made up of matrix equations similar to Dirac
Hamiltonians in one variable [7]. This type of Dirac-like factorization
hierarchies will have also a kind of intertwining operators which correspond
to those of the scalar case in a natural way. Let us remark that some
particular cases of such Dirac-like Hamiltonian hierarchies have been found
previously when solving the Dirac equation for some external magnetic fields
[8, 9, 10, 11, 12, 13, 14, 15, 16]. In this work we plan to undertake a
systematic study of such a kind of matrix Hamiltonians.

Among the purposes of this work, we want to understand the origin of the
“Dirac factorizations” from symmetry considerations. In fact, we will show
that the class of Dirac-like Hamiltonians related to factorizable Schrödinger
equations can be explained by means of a reduction of higher dimensional
problems in flat and curved spaces. These initial systems must have
symmetries, including spin, which allow the reducing process. Our program
consist in the following steps. We will begin with a free Hamiltonian on a
flat or curved space. If this Hamiltonian is scalar it will lead, by
reduction, to some of the Schrödinger factorized Hamiltonians. If that
Hamiltonian is a kind of spin-orbit coupling then, after a reduction process,
it will lead us to a particular kind of the Dirac-like Hamiltonians proposed
for each factorization type. The intertwining operators of the reduced Dirac
Hamiltonians will be recovered from the symmetries of the initial system, but
we remark that there will appear new intertwining operators. We have
restricted in this paper to some specific two-dimensional spaces, the sphere
${\cal S}^{2}$ and the hyperboloid ${\cal H}^{2}$, for the sake of simplicity
and to keep it within a reasonable length, but this program can in principle
be applied to all the factorization cases [1].

In a similar way as the Klein-Gordon equation is obtained by “squaring” the
Dirac equation, we will have a similar property where the scalar second order
Schrödinger factorized Hamiltonian is obtained by taking the square (in an
appropriate way) of the associated Dirac-like Hamiltonian.

The organization of this work is as follows. In the second section, the
factorization method is introduced briefly in order to set the notation of
increasing and decreasing hierarchies used along the paper. The third section
is devoted to Dirac-like Hamiltonians related with increasing hierarchies.
Next, in the fourth section, an example on ${\cal S}^{2}$ is given and the
connection with symmetry group reduction is examined. In the fifth section the
same procedure is applied to find Dirac-like Hamiltonians for decreasing
hierarchies which in this case are non-Hermitian [21]. Here, the space ${\cal
H}^{2}$ will be used in order to get an example of this type of Dirac-like
Hamiltonians. Finally, this work will end with some conclusions and remarks.

##  2 Hierarchies of 1D Schrödinger Hamiltonians

In the following sections we will specify how is the correspondence of 1D
Schrödinger and Dirac Hamiltonian hierarchies of equations. But first, it is
necessary to distinguish three cases of hierarchies of shape invariant
Schrödinger equations (also referred as shape-invariant potentials). These are
well known [1, 4, 5, 6], but here we will briefly introduce them in order to
set the notation.

  * •

Increasing hierarchy (or I-hierarchy)

Consider the Schrödinger hierarchy of second order Hamiltonians $H_{n}$ which
can be factorized as follows:

$H_{n}=-\partial_{xx}+V_{n}(x)=a^{+}_{n}a^{-}_{n}+\mu_{n}^{2}=a^{-}_{n-1}a^{+}_{n-1}+\mu_{n-1}^{2},\qquad
n=0,1,\dots$ (2.1)

where $\partial_{xx}:=\frac{d^{2}}{dx^{2}}$. Let us explain the standard
notation here used. The first order operators $a^{\pm}_{n}$ are defined by

$a^{\pm}_{n}(x)=\mp\partial_{x}+w_{n}(x)$ (2.2)

Here, $\partial_{x}$ is the derivative with respect to $x$,
$\partial_{x}:=\frac{d}{dx}$; the functions $w_{n}(x)$ are called
superpotentials and, according to eq. (2.1), satisfy the following relations
with the potential $V_{n}(x)$,

$V_{n}(x)=w_{n}(x)^{2}-w_{n}^{\prime}(x)+\mu_{n}^{2}=w_{n-1}(x)^{2}+w_{n-1}^{\prime}(x)+\mu_{n-1}^{2}$
(2.3)

where $w_{n}^{\prime}(x):=\frac{dw_{n}(x)}{dx}$. The factorization energies
$\mu_{n}^{2}$ are assumed to be positive, but the following results can be
also adapted to other factorization energies, as we will see later.

In all the hierarchies, in particular in this I-hierarchy, we will also assume
that $a^{-}_{n}$ are annihilation operators, in other words the equation

$a^{-}_{n}\psi_{n}^{0}=0,\quad\implies\quad\psi_{n}^{0}(x)=K\,e^{-\int
w(x)dx}$ (2.4)

(where $K$ is an integration constant) determines a square-integrable function
$\psi_{n}^{0}(x)$. The function $\psi_{n}^{0}(x)$ corresponds to the ground
state of $H_{n}$, while $\mu_{n}^{2}$ is the ground state energy (this is
consistent with the factorization (2.1)):

$E_{n}^{0}=\mu_{n}^{2}\,$ (2.5)

This sequence of Hamiltonians $H_{n}$ has increasing ground energies. The key
factorization property (2.1) implies that the operators $a^{-}_{n}$ intertwine
the consecutive Hamiltonians $H_{n}$ and $H_{n+1}$:

$a^{-}_{n}H_{n}=H_{n+1}a^{-}_{n},\qquad H_{n}a^{+}_{n}=a^{+}_{n}H_{n+1}$ (2.6)

The annihilation operators $a^{-}_{n}$ go towards the Hamiltonians on the
right (index $n$) but decrease the energy level (index $k$). The creation
operators $a^{+}_{n}$ act in the opposite way:

$a^{-}_{n}:\psi_{n}^{k}\ \to\ \psi_{n+1}^{k-1},\qquad
a^{+}_{n}:\psi_{n+1}^{k-1}\ \to\ \psi_{n}^{k}$ (2.7)

In fact

$a^{-}_{n}\psi_{n}^{k}=\sqrt{\mu_{n+k}^{2}-\mu_{n}^{2}}\,\psi_{n+1}^{k-1},\qquad
a^{+}_{n}\psi_{n+1}^{k-1}=\sqrt{\mu_{n+k}^{2}-\mu_{n}^{2}}\,\psi_{n}^{k}$
(2.8)

The $k$-excited eigenfunction $\psi_{n}^{k}$ of $H_{n}$ can be obtained from
the ground eigenfunction $\psi_{n+k}^{0}$ of $H_{n+k}$, by means of creation
operators:

$\psi_{n}^{k}\ \propto\ a^{+}_{n}a^{+}_{n+1}\dots
a^{+}_{n+k-1}\psi_{n+k}^{0}\,,\qquad E_{n}^{k}=\mu_{n+k}^{2}$ (2.9)

An example of this increasing hierarchy type is the trigonometric Pöschl-
Teller (PT) Hamiltonians as we will see later.

  * •

Decreasing hierarchy (or D-hierarchy)

Next, we start from a Schrödinger hierarchy of second order Hamiltonians
$H_{n}$ satisfying the following factorization property:

$H_{n}=-\partial_{xx}+V_{n}(x)=a^{+}_{n}a^{-}_{n}-\mu_{n}^{2}=a^{-}_{n+1}a^{+}_{n+1}-\mu_{n+1}^{2},\qquad
n=0,1,\dots$ (2.10)

The first order operators $a^{\pm}_{n}$ are defined in the same way as (2.2).
The superpotentials $w_{n}(x)$ and the potentials $V_{n}(x)$ are here related
by

$V_{n}(x)=w_{n}(x)^{2}-w_{n}^{\prime}(x)-\mu_{n}^{2}=w_{n+1}(x)^{2}+w_{n+1}^{\prime}(x)-\mu_{n+1}^{2}$
(2.11)

The factorization energies are given by $-\mu_{n}^{2}$, where $\mu_{n}^{2}$
are real numbers which we will assume to be positive. The annihilation
operator $a^{-}_{n}$ determines the ground state wavefunction
$\psi_{n}^{0}(x)$ of $H_{n}$,

$a^{-}_{n}\psi_{n}^{0}=0,\qquad E_{n}^{0}=-\mu_{n}^{2}$

where $-\mu_{n}^{2}$ is the negative ground state energy, in a similar way as
in the previous positive case (2.4)-(2.5).

Here the sequence of Hamiltonians $H_{n}$ have decreasing ground energies. In
the D-hierarchies, the factorization property (2.10) implies that the
operators $a^{\pm}_{n}$ intertwine the consecutive Hamiltonians, $H_{n}$ and
$H_{n-1}$, as follows:

$a^{-}_{n}H_{n}=H_{n-1}a^{-}_{n},\qquad H_{n}a^{+}_{n}=a^{+}_{n}H_{n-1}$
(2.12)

The annihilation operators $a^{-}_{n}$ go towards the Hamiltonians to the left
(in index $n$) and decrease the energy level (index $k$). The creation
operators $a^{+}_{n}$ go in the opposite way:

$\begin{array}[]{ll}a^{-}_{n}:\psi_{n}^{k}\ \to\
\psi_{n-1}^{k-1},&a^{+}_{n}:\psi_{n-1}^{k-1}\ \to\ \psi_{n}^{k}\\\\[8.61108pt]
a^{-}_{n}\psi_{n}^{k}=\sqrt{\mu_{n}^{2}-\mu_{n-k}^{2}}\,\psi_{n-1}^{k-1},&a^{+}_{n}\psi_{n-1}^{k-1}=\sqrt{\mu_{n}^{2}-\mu_{n-k}^{2}}\,\psi_{n}^{k}\end{array}$
(2.13)

The $k$-excited eigenfunction $\psi_{n}^{k}$ of $H_{n}$ can be obtained from
the ground eigenfunction $\psi_{n-k}^{0}$ of Hamiltonians to the left
$H_{n-k}$, by means of creation operators:

$\psi_{n}^{k}\ \propto\ a^{+}_{n}a^{+}_{n-1}\dots
a^{+}_{n-k+1}\psi_{n-k}^{0},\qquad E_{n}^{k}=E_{n-k}^{0}=-\mu_{n-k}^{2},\qquad
k=1,2\dots$ (2.14)

An example of this decreasing hierarchy type is the sequence of hyperbolic
Pöschl-Teller Hamiltonians.

  * •

Equal hierarchy (or E-hierarchy).

This is a hierarchy where all the factorizacion energies are equal:

$\mu^{2}=\mu_{n}^{2},\qquad\forall n$ (2.15)

In this case the spectrum of all the Hamiltonians $H_{n}$ are the same (they
are isospectral). The operators $a^{\pm}_{n}$ are neither annihilation nor
creation, they could be called iso-operators. This case will not be considered
in this work.

##  3 Hierarchies of 1D Dirac Hamiltonians: Increasing Case

Consider an increasing hierarchy of Schrödinger Hamiltonians $H_{n}$
characterized by the factorization (2.1). Then, we can form a $2\times 2$
matrix Hamiltonian hierarchy $h_{n}$ defined by

$h_{n}=\left(\begin{array}[]{cc}\mu_{n}&ia^{+}_{n}\\\\[4.30554pt]
-ia^{-}_{n}&-\mu_{n}\end{array}\right)$ (3.16)

We will refer to this first order matrix operator $h_{n}$ as a Dirac-like
Hamiltonian. This type of Dirac-like Hamiltonians have been obtained in a
number of problems related with Dirac-Weyl systems in planar graphene,
spherical surfaces or for other cases with cylindrical symmetry [8, 9, 10, 11,
12, 13, 14].

The matrix Hamiltonian $h_{n}$ in (3.16) is explicitly Hermitian from a formal
point of view. The square of $h_{n}$ gives a pair of Schrödinger Hamiltonians
of the scalar I-hierarchy in the diagonal, therefore in this sense it mimics
the Dirac Hamiltonians:

$h_{n}^{2}=\left(\begin{array}[]{cc}H_{n}&0\\\\[4.30554pt]
0&H_{n+1}\end{array}\right):={\boldsymbol{H}}_{n}$ (3.17)

Next, we will look for a pair of first order matrix differential
(anti)intertwining operators $A^{\pm}_{n}$ between the matrix Hamiltonians
$h_{n}$ and $h_{n+1}$ (similar to the operators $a^{\pm}_{n}$ between $H_{n}$
and $H_{n+1}$ in (2.6) for the I-hierarchy):

$A_{n}^{-}h_{n}=\gamma\,h_{n+1}A^{-}_{n},\qquad
h_{n}A_{n}^{+}=\gamma\,A^{+}_{n}h_{n+1}$ (3.18)

where $\gamma$ is for a sign. As a consequence of (3.18), the operators
$A^{\pm}_{n}$ will also intertwine $h_{n}^{2}$ and $h_{n+1}^{2}$:

$A_{n}^{-}h_{n}^{2}=h_{n+1}^{2}A^{-}_{n},\qquad
h_{n}^{2}A_{n}^{+}=A^{+}_{n}h_{n+1}^{2}$ (3.19)

where $A_{n}^{\pm}={(A_{n}^{\mp})}^{\dagger}$. There are two solutions to this
problem (up to a multiplicative constant):

a) True intertwining operators ($\gamma=1$)

$R^{-}_{n}=\left(\begin{array}[]{cc}a^{-}_{n}&i(\mu_{n+1}-\mu_{n})\\\\[4.30554pt]
0&a^{-}_{n+1}\end{array}\right),\qquad R_{n}^{-}h_{n}=h_{n+1}R^{-}_{n}$ (3.20)

b) Anti-intertwining operators ($\gamma=-1$)

$T^{-}_{n}=\left(\begin{array}[]{cc}a^{-}_{n}&-i(\mu_{n+1}+\mu_{n})\\\\[4.30554pt]
0&-a^{-}_{n+1}\end{array}\right),\qquad T_{n}^{-}h_{n}=-h_{n+1}T^{-}_{n}$
(3.21)

One couple, $R_{n}^{\pm}$, plays the role of the usual intertwining operators;
but there is a new pair $T_{n}^{\pm}$ carrying a minus sign that we call anti-
intertwining. This possibility is specific of Dirac-like Hamiltonians where
the spectrum includes a negative sector, as we will see in the next
subsection.

###  3.1 Spectrum of the Dirac hierarchy $\boldsymbol{h_{n}}$

We will compute the eigenfunctions and eigenvalues of each Dirac Hamiltonian
(3.16),

$h_{n}\Psi_{n}^{k}=\varepsilon_{n}^{k}\Psi_{n}^{k}$ (3.22)

where $\varepsilon_{n}^{k}$ designs the $k$th excited level of the energy
eigenvalues of Hamiltonian $h_{n}$ and $\Psi_{n}^{k}$ is a corresponding
spinor eigenfunction. According to the square of $h_{n}$ given in (3.17), we
will express the spectrum and eigen-spinors of $h_{n}$ ($\varepsilon_{n}^{k}$
and $\Psi_{n}^{k}$) in terms of the eigenfunctions of the scalar Schrödinger
Hamiltonians $H_{n}$ ($\mu_{n+k}^{2}$, $\psi_{n}^{k}$) and $H_{n+1}$
($\mu_{n+k+1}^{2}$, $\psi_{n+1}^{k}$), respectively. We make the natural
ansatz

$\Psi_{n}^{k}=\left(\begin{array}[]{c}\alpha\,\psi_{n}^{k}\\\\[6.45831pt]
i\beta\,\psi_{n+1}^{k-1}\end{array}\right),\qquad\alpha,\beta\in\mathbb{C}$
(3.23)

where $\alpha,\beta$ must be computed by substituting in the eigenvalue
equation for $h_{n}$ defined in (3.16). There are two signs for the energy.
Then, to specify the energy sign $\pm$ we will write $\varepsilon_{n}^{k\pm}$
and $\Psi_{n}^{k\pm}$ for the respective eigenfunctions. After some
computations we get the following solutions (they are not normalized).

  * •

Positive spectrum

$\begin{array}[]{ll}\varepsilon_{n}^{k+}=\mu_{n+k},&\Psi_{n}^{k+}=\left(\begin{array}[]{c}\sqrt{\mu_{n+k}+\mu_{n}}\,\psi_{n}^{k}\\\\[6.45831pt]
-i\sqrt{\mu_{n+k}-\mu_{n}}\,\psi_{n+1}^{k-1}\end{array}\right)\ ,\qquad
k=1,2,\dots\\\\[17.22217pt] {\rm Ground\ level}:\\\\[4.30554pt]
\varepsilon_{n}^{0+}=\mu_{n},&\Psi_{n}^{0+}=\left(\begin{array}[]{c}\psi_{n}^{0}\\\\[4.30554pt]
0\end{array}\right)\ ,\qquad k=0\end{array}$ (3.24)

  * •

Negative spectrum

$\varepsilon_{n}^{k-}=-\mu_{n+k},\qquad\Psi_{n}^{k-}=\left(\begin{array}[]{c}\sqrt{\mu_{n+k}-\mu_{n}}\,\psi_{n}^{k}\\\\[6.45831pt]
i\sqrt{\mu_{n+k}+\mu_{n}}\,\psi_{n+1}^{k-1}\end{array}\right)\ ,\qquad
k=1,2,\dots$ (3.25)

Notice some properties of the spectrum.

  1. 1.

The positive and negative solutions $\Psi_{n}^{k\pm}$ are solutions of the
square of $h_{n}$ with the same positive eigenvalue:

$(h_{n})^{2}\Psi_{n}^{k\pm}=\left(\begin{array}[]{cc}H_{n}&0\\\\[4.30554pt]
0&H_{n+1}\end{array}\right)\left(\begin{array}[]{c}\sqrt{\mu_{n+k}\pm\mu_{n}}\,\psi_{n}^{k}\\\\[6.45831pt]
\mp
i\sqrt{\mu_{n+k}\mp\mu_{n}}\,\psi_{n+1}^{k-1}\end{array}\right)=\mu_{n+k}^{2}\,\Psi_{n}^{k\pm}$
(3.26)

  2. 2.

The ground level $k=0$ exists only for positive energies, while the “ground
negative” energy level is reached for $k=1$.

  3. 3.

The eigenfunctions for positive energy have the upper component with bigger
norm than the lower one, while eigenfunctions of negative energy have oposite
character: the norm of the lower component is bigger than the upper.

  4. 4.

Eigenfunctions of positive and negative energies are orthogonal.

###  3.2 The spectrum for a null factorization energy (Dirac–Weyl like
equation)

Up to now we have assumed that each Dirac Hamiltonian $h_{n}$, as it is
defined in (3.16), has the factorization energy (it may play the role of mass)
given by $\mu_{n}$. However, we could be interested in the spectrum of a
particular $h_{n_{0}}$ with $n=n_{0}$, having a different value of the
factorizacion energy, for instance $\mu_{n_{0}}=0$, (it would correspond to a
null mass case means that the Dirac–Weyl like equation).

In principle, the Schrödinger Hamiltonian hierarchy (increasing (2.1) or
decreasing (2.10)) will have $\mu_{n_{0}}^{2}\neq 0$. Then, we will subtract
from all the Hamiltonian hierarchy
$\\{H_{n}=a_{n}^{+}a_{n}^{-}+\mu_{n}^{2}\\}$ the same value:
$\mu_{n_{0}}^{2}=E_{n_{0}}^{0}$. Hence, we will have the new sequence
$\\{\tilde{H}_{n}=a_{n}^{+}a_{n}^{-}+\tilde{\mu}_{n}^{2}\\}$ where

$\tilde{\mu}_{n}^{2}=\mu_{n}^{2}-\mu_{n_{0}}^{2}$

This new hierarchy has the same properties, intertwining, eigenfunctions, etc.
except the spectrum that now will be displaced the same amount:

$\tilde{E}_{n}^{k}=\tilde{E}^{0}_{n+k}=\tilde{\mu}_{n+k}^{2}=E^{0}_{n+k}-E^{0}_{n_{0}},\qquad\tilde{E}^{0}_{n_{0}}=0$

In other words, the ground energy of the displaced Hamiltonian
$\tilde{H}_{n_{0}}$ is zero and the corresponding Dirac Hamiltonian
$\tilde{h}_{n_{0}}$ will have zero mass:

$\tilde{h}_{n}=\left(\begin{array}[]{cc}\tilde{\mu}_{n}&ia^{+}_{n}\\\\[4.30554pt]
-ia^{-}_{n}&-\tilde{\mu}_{n}\end{array}\right)\,,\qquad\tilde{h}_{n_{0}}=\left(\begin{array}[]{cc}0&ia^{+}_{n_{0}}\\\\[4.30554pt]
-ia^{-}_{n_{0}}&0\end{array}\right)\,,\qquad\tilde{\mu}_{n_{0}}=0$ (3.27)

The formulas for the positive (3.24) or negative (3.25) spectrum
$\tilde{\varepsilon}_{n}^{k\pm}$ and wave functions $\tilde{\Psi}_{n}^{k\pm}$
will still be valid, with the replacement of the new values of
$\tilde{\mu}_{n}$ in their respective expressions.

###  3.3 Eigenfunctions annihilated by the intertwining operators

Next, we want to know what kind of eigenfunctions annihilate the intertwining
operators $R_{n}^{-}$ and $T_{n}^{-}$. Since they are first order linear
$2\times 2$-matrix differential operators, they must annihilate two linearly
independent spinor functions.

  * •

Eigenfunctions annihilated by $R_{n}^{-}$

1) The ground state of positive energies:

$\Psi_{n}^{0+}=\left(\begin{array}[]{c}\psi_{n}^{0}\\\\[4.30554pt]
0\end{array}\right)\,,\qquad\varepsilon_{n}^{0+}=\mu_{n}$

2) The “ground state” of negative energies:

$\Psi_{n}^{1-}=\left(\begin{array}[]{c}\sqrt{\mu_{n+1}-\mu_{n}}\,\psi_{n}^{1}\\\\[6.45831pt]
i\sqrt{\mu_{n+1}+\mu_{n}}\,\psi_{n+1}^{0}\end{array}\right)\,,\qquad\varepsilon_{n}^{1-}=-\mu_{n+1}$

  * •

Eigenfunctions annihilated by $T_{n}^{-}$

1) The ground state of positive energies:

$\Psi_{n}^{0+}=\left(\begin{array}[]{c}\psi_{n}^{0}\\\\[4.30554pt]
0\end{array}\right)\,,\qquad\varepsilon_{n}^{0+}=\mu_{n}$

2) The first excited state of positive energies:

$\Psi_{n}^{1+}=\left(\begin{array}[]{c}\sqrt{\mu_{n+1}+\mu_{n}}\,\psi_{n}^{1}\\\\[6.45831pt]
-i\sqrt{\mu_{n+1}-\mu_{n}}\,\psi_{n+1}^{0}\end{array}\right)\,,\qquad\varepsilon_{n}^{1+}=\mu_{n+1}$

In conclusion, the matrix intertwining operators $R_{n}^{-}$ and $T_{n}^{-}$
annihilate some of lowest energy states of the Hamiltonians $h_{n}$, in a
similar way as the factor operators $a_{n}^{-}$ do for the ground states of
$H_{n}$. However, in the matrix case there are more options due to the two
sectors (positive and negative) of the spectrum and the fact that each
intertwining operator annihilates two independent states.

###  3.4 Symmetries from intertwining operators

As we have seen in a precedent subsection, the spectrum of this type of Dirac
Hamiltonians $h_{n}$ have simple degeneracy: the eigenspaces are one-
dimensional. This means that there are no additional independent symmetry, or
in other words, any symmetry must be a function of the Hamiltonian $h_{n}$.

A method to build symmetries is by means of the product of two intertwining
operators in opposite direction; for instance $S_{n}$ obtained by the product
$R_{n}^{+}R_{n}^{-}$ will be a symmetry of $h_{n}$:

$R_{n}^{-}h_{n}=h_{n+1}R_{n}^{-},\qquad h_{n}R_{n}^{+}=R_{n}^{+}h_{n+1}\\\
,\implies\,[S_{n},h_{n}]=0$

Therefore, this symmetry must be a function of $h_{n}$. After a computation we
find:

$S_{n}=R_{n}^{+}R_{n}^{-}=(h_{n}-\mu_{n})(h_{n}+\mu_{n+1})$

and the commutator is

$R_{n}^{+}R_{n}^{-}-R_{n-1}^{-}R_{n-1}^{+}=h_{n}(\mu_{n+1}-2\mu_{n}+\mu_{n-1})+\mu_{n}(\mu_{n-1}-\mu_{n+1})$

This result seems reasonable since we have seen before that the operator
$R_{n}^{-}$ annihilates eigenvectors with eigenvalues $\mu_{n}$ and
$-\mu_{n+1}$ and the same happens with the operators on the right hand side of
the last equality.

The symmetry obtained from the other intertwining (anti-intertwining)
operator:

$S^{\prime}_{n}=T_{n}^{+}T_{n}^{-}$

will have a similar expression:

$S^{\prime}_{n}=(h_{n}-\mu_{n})(h_{n}-\mu_{n+1})$

and in this case the commutator has the form

$T_{n}^{+}T_{n}^{-}-T_{n-1}^{-}T_{n-1}^{+}=-h_{n}(\mu_{n+1}+2\mu_{n}+\mu_{n-1})-\mu_{n}(\mu_{n-1}-\mu_{n+1})$

This is consistent with the eigenvectors (and their eigenvalues) annihilated
by $T_{n}^{-}$ (see the previous subsection).

###  3.5 Intertwining operators for $m_{0}$-mass Dirac-like Hamiltonians

In this subsection, we will mention briefly how our approach to Dirac-like
matrix Hamiltonians is compatible with a constant mass $m_{0}$.

Firstly, we define the $4\times 4$ Dirac-like Hamiltonian ${\cal H}_{n}$ with
mass $m_{0}$ from the previous $h_{n}$ as follows [8, 9, 16]:

${\cal H}_{n}=\left(\begin{array}[]{cc}m_{0}c^{2}&h_{n}\\\\[6.45831pt]
h_{n}&-m_{0}c^{2}\end{array}\right)$ (3.28)

Then, we can also introduce $4\times 4$ intertwining operators for the
hierarchy ${\cal H}_{n}$ , in terms of the previous $2\times 2$ operators,
$R_{n}^{\pm}=(R_{n}^{\mp})^{{\dagger}}$ and
$T_{n}^{\pm}=(T_{n}^{\mp})^{{\dagger}}$. In fact, we have two extensions for
each pair having the following form (together with their adjoint operators and
taking $c=1$):

${\cal R}_{n}^{-}=\left(\begin{array}[]{cc}R_{n}^{-}&0\\\\[6.45831pt]
0&R_{n}^{-}\end{array}\right),\qquad{\tilde{\cal
R}}_{n}^{-}=\left(\begin{array}[]{cc}-m_{0}M^{-}&R_{n}^{-}\\\\[6.45831pt]
R_{n}^{-}&m_{0}M^{-}\end{array}\right)$ (3.29)

${\cal T}_{n}^{-}=\left(\begin{array}[]{cc}-T_{n}^{-}&0\\\\[6.45831pt]
0&T_{n}^{-}\end{array}\right),\qquad{\tilde{\cal
T}}_{n}^{-}=\left(\begin{array}[]{cc}m_{0}M^{-}&-T_{n}^{-}\\\\[6.45831pt]
T_{n}^{-}&m_{0}M^{-}\end{array}\right)$ (3.30)

where

$M^{-}:=-i\sigma^{+}=-i(\sigma_{1}+i\sigma_{2})$

is a constant matrix which satisfies the following (anti)commutation relations
with the $2\times 2$ Hamiltonians $h_{n}$:

$M^{-}h_{n}+h_{n+1}M^{-}=-2R_{n}^{-},\qquad
M^{-}h_{n}-h_{n+1}M^{-}=-2T_{n}^{-}$

It can be checked that all these global intertwining operators fulfil the
usual type of intertwining relations, for example:

${\cal R}_{n}^{-}{\cal H}_{n}={\cal H}_{n+1}{\cal R}_{n}^{-},\qquad{\cal
T}_{n}^{-}{\cal H}_{n}={\cal H}_{n+1}{\cal T}_{n}^{-}$

The adjoint operators ${\cal R}_{n}^{+}=({\cal R}_{n}^{-})^{\dagger}$ and
${\cal T}_{n}^{+}=({\cal T}_{n}^{-})^{\dagger}$ will satisfy the opposite
intertwining relations. Finally, we can say that in fact we have obtained a
four dimensional vector space of global intertwining operators for the same
hierarchy of $4\times 4$ Hamiltonians ${\cal H}_{n}$. In principle, this is
surprising because in scalar Hamiltonian hierarchies there is only one set of
factor intertwining operators for the same hierarchy. In our case, the
degeneracy of the positive and negative energy levels of each Hamiltonian
${\cal H}_{n}$ allows that the intertwining operators be non unique.

Let us obtain the eigenfunctions and the spectrum of ${\cal H}_{n}$. The
eigenvalue equation is

${\cal H}_{n}\,\Xi_{n}^{ks}={\cal E}^{k}_{n\pm}\,\Xi_{n}^{ks}$ (3.31)

where the energy is ${\cal E}^{k}_{n\pm}$ and each state $\Xi_{n}^{ks}$ is
composed of two eigen-spinors of $h_{n}$ of same type:

$\Xi_{n}^{ks}=\left(\begin{array}[]{c}\Psi_{n}^{ks}\\\\[6.45831pt]
\Phi_{n}^{ks}\end{array}\right)\,$ (3.32)

For instance, $\Psi_{n}^{ks}$ corresponds to the eigenvalue
$\varepsilon_{n}^{ks}$ of the $n$-Hamiltonian $h_{n}$, $k$-energy level, and
energy sign $s=\pm$. Using ${\cal H}_{n}$ in the eigenvalue equation, we get
two coupled equations:

$m_{0}c^{2}\Psi_{n}^{ks}+h_{n}\Phi_{n}^{ks}={\cal
E}^{k}_{n\pm}\Psi_{n}^{ks},\qquad-
m_{0}c^{2}\Phi_{n}^{ks}+h_{n}\Psi_{n}^{ks}={\cal E}^{k}_{n\pm}\Phi_{n}^{ks}\,$
(3.33)

From these equations, we obtain the following relations

$\Phi_{n\pm}^{ks}=\frac{h_{n}\Psi_{n}^{ks}}{{\cal
E}^{k}_{n\pm}+m_{0}c^{2}},\qquad\Psi_{n\pm}^{ks}=\frac{h_{n}\Phi_{n}^{ks}}{{\cal
E}^{k}_{n\pm}-m_{0}c^{2}}\,$ (3.34)

Substituting these in (3.33), we get the equations for the spinors $\Psi$ and
$\Phi$

$h_{n}^{2}\Psi_{n}^{ks}=({({\cal
E}_{n}^{k})^{2}-m_{0}^{2}c^{4}})\Psi_{n}^{ks},\qquad
h_{n}^{2}\Phi_{n}^{ks}=({({\cal E}_{n}^{k})^{2}-m_{0}^{2}c^{4}})\Phi_{n}^{ks}$
(3.35)

taking into account
$h_{n}^{2}\Psi_{n}^{ks}=(\varepsilon_{n}^{ks})^{2}\Psi_{n}^{ks}$, we get the
eigenvalues

${\cal
E}^{k}_{n\pm}=\pm\sqrt{(\varepsilon_{n}^{ks})^{2}+m_{0}^{2}c^{4}}=\pm\sqrt{\mu_{n+k}^{2}+m_{0}^{2}c^{4}}$
(3.36)

where $E_{n}^{k}=(\varepsilon_{n}^{k})^{2}=\mu_{n+k}^{2}$, as shown in (3.26).
Then, we find the eigenfunctions:

$\Xi_{n+}^{ks}=\left(\begin{array}[]{c}\Psi_{n}^{ks}\\\\[6.45831pt]
\Phi_{n+}^{ks}\end{array}\right)\,,\qquad\Phi_{n+}^{ks}=\frac{s\,\mu_{n+k}\Psi_{n}^{ks}}{\sqrt{\mu_{n+k}^{2}+m_{0}^{2}c^{4}}+m_{0}c^{2}}$
(3.37)

or

$\Xi_{n-}^{ks}=\left(\begin{array}[]{c}\Psi_{n{-}}^{ks}\\\\[6.45831pt]
\Phi_{n}^{ks}\end{array}\right)\,,\qquad\Psi_{n-}^{ks}=-\frac{s\,\mu_{n+k}\Phi_{n}^{ks}}{\sqrt{\mu_{n+k}^{2}+m_{0}^{2}c^{4}}+m_{0}c^{2}}$
(3.38)

We notice that the spectrum (3.36) for a non-vanishing $m_{0}$ is symmetric in
the positive and negative eigenvalue sectors, contrary to $h_{n}$. The
eigenvalues have double degeneracy $s=\pm$, except for the ground energy of
each sector which have simple degeneracy.

##  4 Example: the Trigonometric Pöschl–Teller Potential

An example of a hierarchy of increasing type is the following sequence of PT
Schrödinger Hamiltonians

$H_{n}(x)=-\partial_{xx}+\frac{(n+1/2)(n-1/2)}{\sin^{2}x},\qquad
x\in(0,\pi),\qquad n\in\mathbb{N}$ (4.39)

$H_{n}(x)$ satisfy the factorization given by (2.1). The factor operators and
factor energies are

$a_{n}^{\pm}(x)=\mp\partial_{x}-(n+1/2)\cot
x,\qquad\mu_{n}^{2}=(n+1/2)^{2},\qquad n=0,1,2,\dots$ (4.40)

Therefore, the associated Dirac Hamiltonians $h_{n}$ are defined, according to
(3.16), by

$h_{n}=\left(\begin{array}[]{cc}n+1/2&i(-\partial_{x}-(n+1/2)\cot
x)\\\\[6.45831pt] -i(+\partial_{x}-(n+1/2)\cot x)&-n-1/2\end{array}\right)$
(4.41)

Their square reproduce the Hamiltonians $H_{n}$ and $H_{n+1}$, as it is given
in (3.17). The explicit intertwining operators, given by (3.20) and (3.21),
take here the expressions

$R^{-}_{n}=\left(\begin{array}[]{cc}\partial_{x}-(n+1/2)\cot
x&i\\\\[6.45831pt] 0&\partial_{x}-(n+3/2)\cot x\end{array}\right)$ (4.42)

and

$T^{-}_{n}=\left(\begin{array}[]{cc}\partial_{x}-(n+1/2)\cot
x&-i(2n+2)\\\\[6.45831pt] 0&-(\partial_{x}-(n+3/2)\cot x)\end{array}\right)$
(4.43)

In the next subsection, we will explain the origin of the scalar and matrix
hierarchies together with the (anti)intertwining operators.

###  4.1 Dirac-like equations for systems on the ${\cal S}^{2}$ sphere

Here we want to show how these two Hamiltonians, the scalar PT Schrödinger one
(4.39) and the matrix PT Dirac (4.41), can be obtained by reduction of a
scalar and a spinor particle systems respectively, defined on the ${\cal
S}^{2}$ sphere. We will start with the scalar case, which is more or less well
known, and then we will introduce the one of Dirac type, which is new up to
our knowledge.

(a) Scalar system

Let us consider a system on the two-dimensional sphere ${\cal S}^{2}$ defined
by the points $(x,y,z)\in\mathbb{R}^{3}$ satisfying

$x^{2}+y^{2}+z^{2}=1$

The $SO(3)$ group will act in the usual way on the scalar wavefunctions
$\tilde{\psi}({\bf x})$ defined on ${\cal S}^{2}$,

$U(g)\tilde{\psi}({\bf x})=\tilde{\psi}(g^{-1}{\bf x}),\qquad g\in SO(3)$
(4.44)

If we use Cartesian $(x,y,z)$ or spherical $(r,\theta,\phi)$ coordinates

$x=r\sin\theta\cos\phi,\qquad y=r\sin\theta\sin\phi,\qquad z=r\cos\theta$

the Hermitian generators $L_{i}$, $i=1,2,3$ (or $L_{x},L_{y},L_{z}$) of
$SO(3)$ represented by the (4.44) given by:

$\begin{array}[]{l}L_{x}=-i(y\partial_{z}-z\partial_{y})=i(\sin\phi\,\partial_{\theta}+\cos\phi\cot\theta\,\partial_{\phi})\\\\[8.61108pt]
L_{y}=-i(z\partial_{x}-x\partial_{z})=i(-\cos\phi\,\partial_{\theta}+\sin\phi\cot\theta\,\partial_{\phi})\\\\[8.61108pt]
L_{z}=-i(x\partial_{y}-y\partial_{x})=i(-\partial_{\phi})\end{array}$ (4.45)

We recall the notation of the lowering and raising angular momentum operators:

$L_{\pm}=L_{x}\pm iL_{y},\qquad L_{\pm}=e^{\pm
i\phi}(\pm\partial_{\theta}+i\cot\theta\partial_{\phi})$ (4.46)

such that

$[L_{z},L^{\pm}]=\pm L^{\pm},\qquad[L^{-},L^{+}]=-2L_{z}$

The quadratic Casimir operator $\tilde{H}:={\cal
C}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$ plays the role of a free two-dimensional
Hamiltonian in the space of wavefunctions defined on the sphere, with
eigenvalues $\ell(\ell+1)$ corresponding to the eigenfunctions of each
irreducible representation:

$\tilde{H}\tilde{\psi}_{\ell}=\left(-\partial_{\theta\theta}-\cot\theta\partial_{\theta}-\frac{\partial_{\phi\phi}}{\sin^{2}\theta}\right)\tilde{\psi}_{\ell}=\ell(\ell+1)\tilde{\psi}_{\ell},\qquad\ell=0,1,\dots$
(4.47)

We will eliminate the term with the first partial derivative by means of the
transformation

$\tilde{\psi}=\frac{1}{\sqrt{\sin\theta}}\,\psi\quad\implies\quad\partial_{\theta}\
\to\ \partial_{\theta}-\frac{1}{2}\cot\theta;\quad\tilde{H}\ \to\ H-1/4$
(4.48)

After applying this transformation, we get the following Hamiltonian equation
(with $H:=\tilde{H}+1/4$) in a explicitly Hermitian equivalent form

$H\psi_{\ell}=\left(-\partial_{\theta\theta}-\frac{1/4+\partial_{\phi\phi}}{\sin^{2}\theta}\right)\psi_{\ell}=(\ell+1/2)^{2}\,\psi_{\ell},\qquad\ell=0,1,\dots$
(4.49)

In summary, the above Schrödinger Hamiltonian can be identified, in the
realization (4.48), with the $SO(3)$ Casimir operator (plus the constant
$1/4$)

$H={\boldsymbol{L}}^{2}+1/4$ (4.50)

If we take separated solutions, which are also eigenfunctions of the generator
$L_{z}$ defined in (4.45),

$\psi_{\ell n}(\theta,\phi)=\psi_{\ell n}(\theta)e^{in\phi},\qquad
L_{z}\psi_{\ell n}(\theta,\phi)=n\,\psi_{\ell
n}(\phi,\theta),\qquad|n|\leq\ell$ (4.51)

the component $\psi_{\ell n}(\theta)$ will satisfy the reduced equation

$H_{n}\psi_{\ell
n}=\left(-\partial_{\theta\theta}+\frac{(n+1/2)(n-1/2)}{\sin^{2}\theta}\right)\psi_{\ell
n}=(\ell+1/2)^{2}\,\psi_{\ell n},\quad n=0,\pm 1,\dots\pm\ell$ (4.52)

If we fix the eigenvalue by means of the parameter $\ell$ then, the
Hamiltonians $H_{n}$ having this eigenvalue are those parametrised by $n=0,\pm
1,\dots,\pm\ell$. The states span the $2\ell+1$–dimensional representation
$``\ell"$ of $SO(3)$ with eigenvalue $\varepsilon_{\ell n}=(\ell+1/2)^{2}$.

On the other hand, once selected a value $n$, the spectrum of the reduced
Hamiltonian $H_{n}$ is given by the eigenvalues

$E_{n}^{k}=(n+k+1/2)^{2},\qquad k=0,1,\dots,\quad n+k:=\ell$

Equation (4.52) is the same as the initial one (4.39) of the PT potential
(with the variable $\theta$ instead of $x$) and the factorization energies
$\mu_{n}^{2}$ coincide with $E_{n}^{0}$. Therefore, we have the following
connection between the eigenvalue notation $(E_{n}^{k},\psi_{n}^{k}(\theta))$
and the notation of $SO(3)$ representations ($\varepsilon_{\ell
n}=(\ell+1/2)^{2},\psi_{\ell n})$:

$\left\\{\begin{array}[]{ll}H_{n}\psi_{n}^{k}=E_{n}^{k}\psi_{n}^{k},&E_{n}^{k}=(n+k+1/2)^{2}\\\\[8.61108pt]
H_{n}\psi_{\ell n}=\varepsilon_{\ell n}\psi_{\ell n},&\varepsilon_{\ell
n}=(\ell+1/2)^{2}\end{array}\right.\\!\\!\implies\ \ n+k=\ell\
\implies\left\\{\begin{array}[]{ll}\psi_{n}^{k}=\psi_{(n+k)n}\\\\[8.61108pt]
\psi_{\ell n}=\psi_{n}^{\ell-n}\end{array}\right.$

This relationship tell us that the eigenfunctions of this hierarchy $H_{n}$
are essentially spherical harmonics (subjected to the above transformation):

$\psi_{n}^{k}(\theta)e^{in\phi}\propto{\sqrt{\sin\theta}}\,Y_{\ell
n}(\theta,\phi),\quad\ell=n+k$ (4.53)

In particular, for $k=0$ have the form:

$\psi_{n}^{0}(\theta)e^{in\phi}\propto{\sqrt{\sin\theta}}\,Y_{nn}(\theta,\phi)$

If we apply the transformation (4.48) to the orbital angular operators (4.46),
they become

$L_{+}=e^{i\phi}\Big{(}\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)},\quad
L_{-}=\Big{(}-\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)}e^{-i\phi}$
(4.54)

where now it is explicit the adjoint relation $(L_{+})^{\dagger}=L_{-}$. If we
allow them to act on the separated solutions (4.51) we find the reduced
expressions

$L^{+}_{n}=\partial_{\theta}-\cot\theta(1/2+n),\quad
L^{-}_{n}=-\partial_{\theta}-\cot\theta(1/2+n)$ (4.55)

which coincide with the intertwining operators $a^{-}_{n}$ (corresponds to
$L^{+}_{n}$) and $a^{+}_{n}$ (to $L^{-}_{n}$), given in (4.40).

Next, we want to carry out a similar construction of the Dirac-like
Hamiltonian (4.41) by means of a spinor realization of $SO(3)$ (or better, its
universal covering $SU(2)$) on the sphere ${\cal S}^{2}$.

(b) Spinor system

We will consider an spinor system defined on the unit sphere by means of two-
component wavefunctions $\Psi({\bf x})$, ${\bf x}\in{\cal S}^{2}$. The action
of the group $SU(2)$ on a function of this space has the usual form

$U(g)\Psi({\bf x})=D(g)\Psi(g^{-1}{\bf x}),\qquad g\in SU(2)$

where $D$ is for the spin $1/2$ fundamental representation of $SU(2)$, and the
action of $g\in SU(2)$ on ${\cal S}^{2}$ is through the homomorphism of
$SU(2)$ onto $SO(3)$. The generators corresponding to this action are given by
the total rotations (orbital plus spin):

$J_{i}=L_{i}(\theta,\phi)+\frac{1}{2}\sigma_{i}:=L_{i}(\theta,\phi)+S_{i},\qquad
i=1,2,3$

with $\sigma_{i}$
($\sigma_{1}=\sigma_{x},\sigma_{2}=\sigma_{y},\sigma_{3}=\sigma_{z}$) being
the three $2\times 2$ Pauli spin matrices. These operators close the
$su(2)\approx so(3)$ Lie algebra. In this case, the Casimir operator is

${\cal C}_{\rm
s}={\boldsymbol{J}}^{2}={\boldsymbol{L}}^{2}+2{\boldsymbol{L}}{\boldsymbol{S}}+{\boldsymbol{S}}^{2}$
(4.56)

It is clear that ${\boldsymbol{L}}^{2}$ and ${\boldsymbol{S}}^{2}$ are also
Casimir operators for the spinor system (both commute with all the generators
$J_{i}$). Therefore, another Casimir operator commuting with the operators
${\boldsymbol{J}}$ is

$\tilde{h}:={\cal C}_{\rm
so}=2{\boldsymbol{L}}{\boldsymbol{S}}=2(\sum_{k}L_{k}S_{k})$ (4.57)

which has the form of a spin-orbit coupling term. Notice that this is a first
order Casimir in the momentum operators and it has a matrix character due to
$\boldsymbol{S}$; by these reasons we will identify this operator as a Dirac-
like Hamiltonian $\tilde{h}$.

In order to simplify some calculations, we will make use of the lowering and
raising spin operators:

$S_{\pm}=S_{x}\pm iS_{y},\qquad
S_{+}=\left(\begin{array}[]{ll}0&1\\\\[4.30554pt]
0&0\end{array}\right)\,,\qquad
S_{-}=\left(\begin{array}[]{ll}0&0\\\\[4.30554pt] 1&0\end{array}\right)$

Then, we have another useful expression of $\tilde{h}$ defined by (4.57) in
terms of $L_{\pm}$ (given in (4.54)) and $S_{\pm}$:

$\tilde{h}=L_{-}S_{+}+L_{+}S_{-}+2L_{z}S_{z}$ (4.58)

After replacement of the operators in (4.58), the final explicit expression
for $\tilde{h}$ is

$\tilde{h}=\left(\begin{array}[]{cc}-i\partial_{\phi}&\Big{(}-\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)}e^{-i\phi}\\\\[8.61108pt]
e^{i\phi}\Big{(}\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)}&i\partial_{\phi}\end{array}\right)$
(4.59)

Since $\tilde{h}$ commutes with the generators ${\boldsymbol{J}}$, the
eigenfunctions $\Psi_{jm}$ can be labeled by the parameters $j$ and $m$
corresponding to the eigenvalues of ${\boldsymbol{J}}^{2}$ and $J_{z}$:

${\boldsymbol{J}}^{2}\Psi_{jm}=j(j+1)\Psi_{jm},\qquad
J_{z}\Psi_{jm}=m\Psi_{jm}$ (4.60)

If we apply the matrix Hamiltonian (4.59) on the eigenfunctions of the
generator $J_{z}$, with undetermined value of ${\boldsymbol{J}}^{2}$,

$J_{z}\Psi_{m}=m\Psi_{m}\ \implies\
\Psi_{m}(\theta,\phi)=\left(\begin{array}[]{l}\varphi_{m}(\theta)e^{i(m-1/2)\phi}\\\\[4.30554pt]
\chi_{m}(\theta)e^{i(m+1/2)\phi}\end{array}\right)$ (4.61)

then, we get the reduced Hamiltonian in the $\theta$ variable

$\tilde{h}_{m}=\left(\begin{array}[]{cc}m-1/2&-(\partial_{\theta}+m\cot\theta)\\\\[8.61108pt]
-(-\partial_{\theta}+m\cot\theta)&-m-1/2\end{array}\right)$ (4.62)

which is similar to $h_{n}$ proposed at the beginning of this section in
(4.41); in fact

$\tilde{h}_{m}+1/2=h_{n},\quad\quad m=n+1/2$ (4.63)

We must have in mind that, according to its definition $m$ is a pure half-
integer, i.e. $m-1/2=n$ and $m+1/2=n+1$, are integer values of the orbital
angular momentum $L_{z}$ corresponding to the spinor components in (4.61). We
will also use the notation of (4.63) for non reduced Hamiltonians:
$\tilde{h}+1/2=h$.

We can also compute the square of $\tilde{h}$ with the aim of recovering the
Schrödinger equation (4.49):

$(2{\boldsymbol{L}}{\boldsymbol{S}})^{2}={\boldsymbol{L}}^{2}-2{\boldsymbol{L}}{\boldsymbol{S}}\
\implies\
\Big{(}2{\boldsymbol{L}}{\boldsymbol{S}}+\frac{1}{2}\Big{)}^{2}={\boldsymbol{L}}^{2}+\frac{1}{4}:={\boldsymbol{H}}$
(4.64)

In other words, we must choose $h=\tilde{h}+1/2$, where $\tilde{h}$ is defined
in (4.59) and square it as in (4.64) to get

$h^{2}={\boldsymbol{H}}$ (4.65)

Therefore, by taking the square (in this modified way) of the Dirac operator
$h$, we get the scalar Hamiltonian (4.49), just in a similar manner as from
the Dirac equation we can get the Klein-Gordon equation.

In particular, if we apply this formula to the Hamiltonian $\tilde{h}_{m}$ in
(4.62), with $m=n+1/2$, we get the scalar reduced Hamiltonians $H_{n}$ and
$H_{n+1}$ in a diagonal matrix, as it was shown in (4.52),

$\Big{(}\tilde{h}_{m}+\frac{1}{2}\Big{)}^{2}=\left(\begin{array}[]{cc}H_{n}&0\\\\[4.30554pt]
0&H_{n+1}\end{array}\right)\,,\qquad m=n+1/2$ (4.66)

We are always making use of the relation $n=m-1/2$ and $n+1=m+1/2$ for the
spinor components.

In this way, we have explained the origin of the first order Dirac-like matrix
Hamiltonians $h_{n}$ corresponding to the scalar Hamiltonians $H_{n}$ from the
factorization method [1]. Next, we will determine the intertwining operators
of the sequence $h_{n}$ by means of the symmetries of $h$.

(c) Symmetries and “anti-symmetries”

  * 1)

Symmetries $J_{\pm},J_{z}$.

In the following we will compute the symmetries corresponding to the lowering
and raising operators $J_{\pm}$ for the eigenfunctions $\Psi_{jm}$ of $J_{z}$
and ${\boldsymbol{J}}^{2}$. Taking into account the expressions of (4.54), we
get

$J_{+}=L_{+}+S_{+}=\left(\begin{array}[]{cc}e^{i\phi}\Big{(}\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)}&1\\\\[6.45831pt]
0&e^{i\phi}\Big{(}\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)}\end{array}\right)$

$J_{-}=L_{-}+S_{-}=\left(\begin{array}[]{cc}\Big{(}-\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)}e^{-i\phi}&0\\\\[6.45831pt]
1&\Big{(}-\partial_{\theta}+\cot\theta(-1/2+i\partial_{\phi})\Big{)}e^{-i\phi}\end{array}\right)$

When $J_{+}$ act on $\Psi_{jm}$ and $J_{-}$ on $\Psi_{jm+1}$ their effect on
the components $\varphi_{m}(\theta)$ and $\chi_{m}(\theta)$ of (4.61) turns
into

$J^{+}_{m}=\left(\begin{array}[]{cc}\partial_{\theta}-m\cot\theta&1\\\\[6.45831pt]
0&\partial_{\theta}-(m+1)\cot\theta\end{array}\right)\,;\ \
J^{-}_{m}=\left(\begin{array}[]{cc}-\partial_{\theta}-m\cot\theta&0\\\\[6.45831pt]
1&-\partial_{\theta}-(m+1)\cot\theta\end{array}\right)$

These expressions are related with the previous ones shown in (4.42) by means
of a simple equivalence ($R^{\pm}_{n}\approx J^{\mp}_{m}$, $m=n+1/2$). We
should also include the symmetry $J_{z}=L_{z}+S_{z}$ which is used to define
the basis of eigenvectors $\Psi_{jm}$ (4.60).

  * 2)

Anti-symmetries $T^{\pm},T_{3}$.

We will also find the anti-symmetries $T^{\pm}$ which anti-commute with the
Hamiltonian $h$ and behave like shift operators of $J_{z}$:

$T^{\pm}(\tilde{h}+1/2)=-(\tilde{h}+1/2)\,T^{\pm},\qquad\quad[J_{z},T^{\pm}]=\pm
T^{\pm}$

Due to their anti-commutation with $\tilde{h}+1/2$, the operators $T^{\pm}$
change a positive energy state of $\tilde{h}+1/2$ into another one with the
same absolute but opposite sign energy. It can be checked that such anti-
symmetries are given by

$T^{+}=L_{+}S_{3}-L_{3}S_{+},\qquad(T^{+})^{\dagger}=T^{-}=L_{-}S_{3}-L_{3}S_{-}$

where $S_{3}=S_{z}$ and $L_{3}=L_{z}$. When they act on the basis $\Psi_{jm}$,
$T^{\pm}$ acquire an equivalent expression to that given in (4.43). Besides
the above anti-symmetry operators, we must include a third one, called $T_{3}$
defined by:

$T_{3}=-\frac{1}{2}(L_{+}S_{-}-L_{-}S_{+})$

We can also check that the set of anti-symmetries $T_{\pm},T_{3}$ constitute a
vector operator under the action of ${\boldsymbol{J}}$; in other words, they
satisfy the commutation rules

$[J_{+},T^{+}]=0,\quad[J_{+},T^{-}]=2T_{3},\quad[J_{\pm},T_{3}]=\mp
T^{\pm},\quad[J_{3},T^{\pm}]=\pm T^{\pm},\quad[J_{3},T_{3}]=0$

The operators $T^{\pm},T_{3}$ are “odd” operators in the sense that connect
the spaces of positive and negative energies states. The pair $T^{\pm}$
besides changing the energy sign, they also change the $J_{3}=J_{z}$
eigenvalue $m$, but $T_{3}$ keeps $m$ invariant. All of them change the
eigenvalue $j$ of ${\boldsymbol{J}}^{2}$ to $j-1$ or to $j+1$ if the initial
space was of positive or negative energies, respectively. This is consistent
with the following commutation rule,

$[{\boldsymbol{J}}^{2},T^{+}]=2(\tilde{h}+1/2)T^{+}$

The existence of anti-symmetry operators does not imply that the positive and
negative eigenspaces are completely symmetric, as we will see later. Notice
that this type of nontrivial anti-symmetry operators is a remarkable property
of this kind of Dirac-like Hamiltonians.

(d) Spectrum and eigenfunctions

In this subsection, we will compute the discrete spectrum and eigenfunctions
in the same way as we have seen above. Later on we will identify the
eigenfunctions in relation with the symmetries.

First of all we choose the eigenfunctions with a definite value of the
momentum $J_{z}$ given by (4.61). As we have seen in (4.66), the square of
$\tilde{h}_{m}+1/2$ is a diagonal matrix with $H_{n}$ and $H_{n+1}$ in the
diagonal. These two scalar Hamiltonians have the same eigenvalues
$E_{n}^{k}=(n+k+1/2)^{2}$ for the respective eigenfunctions $\psi_{n}^{k}$ and
$\psi_{n+1}^{k-1}$. Therefore, this suggest the following form of the spinor
eigenfunctions of $\tilde{h}_{m}+1/2$:

$\Psi_{n}^{k}=\left(\begin{array}[]{c}\alpha\,\psi_{n}^{k}(\theta)e^{i(m-1/2)\phi}\\\\[6.45831pt]
\beta\,\psi_{n+1}^{k-1}(\theta)e^{i(m+1/2)\phi}\end{array}\right),\qquad
m=n+1/2,\quad k=0,1,2\dots$

where the constants $\alpha,\beta\in\mathbb{C}$ must be found from the
eigenvalue equation:

$(\tilde{h}_{m}+1/2)\Psi_{n}^{k}=\tilde{\varepsilon}\,\Psi_{n}^{k},\qquad
m=n+1/2$

We have positive energy solutions

$\tilde{\varepsilon}_{n}^{k+}=+(n+k+1/2),\quad\Psi_{n}^{k+}=\left(\begin{array}[]{c}\alpha_{n}^{k+}\,\psi_{n}^{k}(\theta)e^{i(m-1/2)\phi}\\\\[6.45831pt]
\beta_{n}^{k+}\,\psi_{n+1}^{k-1}(\theta)e^{i(m+1/2)\phi}\end{array}\right),\qquad
m=n+1/2,\quad k=0,1,2\dots$

and negative energy solutions

$\tilde{\varepsilon}_{n}^{k-}=-(n+k+1/2),\quad\Psi_{n}^{k+}=\left(\begin{array}[]{c}\alpha_{n}^{k-}\,\psi_{n}^{k}(\theta)e^{i(m-1/2)\phi}\\\\[6.45831pt]
\beta_{n}^{k-}\,\psi_{n+1}^{k-1}(\theta)e^{i(m+1/2)\phi}\end{array}\right),\qquad
m=n+1/2,\quad k=1,2\dots$

with eigenfunctions determined by the coefficients

$\alpha_{n}^{k+}=\sqrt{2n+k+1},\
\beta_{n}^{k+}=-\sqrt{k},\qquad\alpha_{n}^{k-}=\sqrt{k},\
\beta_{n}^{k-}=\sqrt{2n+k+1}$

The zero ground energy level is positive:

$\tilde{\varepsilon}_{n}^{0+}=+(n+1/2)$

and its wave function is

$\Psi_{n}^{0+}(\theta,\phi)=\left(\begin{array}[]{c}\psi_{n}^{0}(\theta)e^{i(m-1/2)\phi}\\\\[6.45831pt]
0\end{array}\right)$

Next, we will interpret the eigenfunctions $\Psi_{n}^{k\pm}$ and their spinor
components in terms of the commuting symmetries of $\tilde{h}$: $J_{z}$,
${\boldsymbol{L}}^{2}$ and ${\boldsymbol{J}}^{2}$.

  * $J_{z}$:

The label $n$ of the eigenfunctions $\Psi_{n}^{k\pm}$ means that, in fact they
are eigenfunctions of $J_{z}$ with eigenvalue $m=n+1/2$:

$J_{z}\Psi_{n}^{k\pm}=m\Psi_{n}^{k\pm},\qquad m=n+1/2$

  * ${\boldsymbol{L}}^{2}$:

We know from the previous scalar case that the scalar functions
$\psi_{n}^{k}(\theta)e^{i(m-1/2)}$ and $\psi_{n+1}^{k-1}(\theta)e^{i(m+1/2)}$
(which are de components of the spinors $\Psi_{n}^{k\pm}$) are eigenfunctions
of ${\boldsymbol{L}}^{2}$ with eigenvalue “$\ell(\ell+1)$”.

Therefore, the spinor eigenfunctions of $\tilde{h}$ are also eigenfunctions of
${\boldsymbol{L}}^{2}$ with the same eigenvalue

$({\boldsymbol{L}}^{2}+1/4)\Psi_{n}^{k\pm}=(\ell+1/2)^{2}\Psi_{n}^{k\pm},\qquad\ell=n+k,\quad
m=n+1/2$

  * ${\boldsymbol{J}}^{2}$:

The above eigenfunctions of $\tilde{h}$ are also eigenfunctions of
${\boldsymbol{J}}^{2}$. We will make use of expression (4.56) in order to see
this:

${\boldsymbol{J}}^{2}\Psi_{n}^{k+}=\big{(}{\boldsymbol{L}}^{2}+\tilde{h}+{\boldsymbol{S}}^{2}\big{)}\Psi_{n}^{k+}=(\ell(\ell+1)+\ell+3/4)\Psi_{n}^{k+}=(\ell+1/2)(\ell+3/2)\Psi_{n}^{k+}$

In other words, it can be written as

${\boldsymbol{J}}^{2}\Psi_{n}^{k+}=(j^{+})(j^{+}+1)\Psi_{n}^{k+},\qquad
j^{+}=\ell+1/2$

In the same way, we check that

${\boldsymbol{J}}^{2}\Psi_{n}^{k-}=(j^{-})(j^{-}+1)\Psi_{n}^{k+},\qquad
j^{-}=\ell-1/2$

We could explain these results from the fact that the spinor wave functions
$\Psi$ belong to tensor products of the representations “$\ell$” from the
orbital part and $s=1/2$ from spin: $\Psi\in\ell\otimes 1/2$. As we know (from
the composition of angular momenta):

$\ell\otimes 1/2=j^{+}\oplus j^{-},\qquad j^{\pm}=\ell\pm 1/2$

Then, the positive energy spinors $\Psi_{n}^{k+}$ belong to the representation
of the total angular momentum $j^{+}$, while the negative spinors
$\Psi_{n}^{k-}$ to $j^{-}$.

In summary, the eigenfunctions of $\tilde{h}+1/2$ could be labeled in two
equivalent ways with the notations $\Psi_{n}^{k\pm}$ and $\Psi_{j^{\pm}\ell
m}$:

$\begin{array}[]{lll}\Psi_{n}^{k+}=\Psi_{j^{+}\ell
m};&\tilde{\varepsilon}_{n}^{k+}=\ell+1/2,&\ell=n+k,\quad m=n+1/2,\quad
j^{+}=\ell+1/2\\\\[8.61108pt] \Psi_{n}^{k-}=\Psi_{j^{-}\ell
m};&\tilde{\varepsilon}_{n}^{k-}=-(\ell+1/2),&\ell=n+k,\quad m=n+1/2,\quad
j^{-}=\ell-1/2\end{array}$

As a consequence, this result confirms us that the degeneracy of positive
energy $\tilde{\varepsilon}_{n}^{k+}$ levels is greater than the corresponding
negative level $\tilde{\varepsilon}_{n}^{k-}$. This is due to the fact that
the positive eigenspace is the support of the representation $j^{+}=\ell+1/2$,
whose dimension is $2\ell+2$, while the negative eigenspace supports $j^{-}$
of dimension $2\ell$.

##  5 Hierarchies of 1D Dirac Hamiltonians: Decreasing Case

Next, we will carry out this program for a decreasing Hamiltonian hierarchy
$H_{n}$ characterized by the factorization

$H_{n-1}=-\partial_{xx}+V_{n}(x)=a^{+}_{n-1}a^{-}_{n-1}-\mu_{n-1}^{2}=a^{-}_{n}a^{+}_{n}-\mu_{n}^{2},\qquad
n=1,2,\dots$ (5.67)

which was given in (2.10). Then, we can form a Dirac-like Hamiltonian
hierarchy $h_{n}$ defined by

$h_{n}=\left(\begin{array}[]{cc}\mu_{n}&ia^{-}_{n}\\\\[4.30554pt]
ia^{+}_{n}&-\mu_{n}\end{array}\right)$ (5.68)

The square of $h_{n}$ gives a pair of Schrödinger Hamiltonians of the
D-hierarchy, with an extra minus sign

$h_{n}^{2}=-\left(\begin{array}[]{cc}H_{n-1}&0\\\\[4.30554pt]
0&H_{n}\end{array}\right):=-{\boldsymbol{H}}_{n}$ (5.69)

Thus, we see that the Dirac-like Hamiltonians (5.68) in this case have a
relevant difference with the increasing hierarchy: they are non-Hermitian. In
fact, we could say that they are $\sigma_{3}$-Hermitian [21, 22] in the sense
that

$h_{n}=\sigma_{3}\,h_{n}^{\dagger}\,\sigma_{3}:=h_{n}^{\sharp}$ (5.70)

However, since $\sigma_{3}$ commutes with $h_{n}^{2}$ given in (5.69) we find
that this is Hermitian: $(h_{n}^{2})^{\dagger}=h_{n}^{2}$.

Next, we are looking for a pair of intertwining operators $A^{\pm}_{n}$
between the matrix Hamiltonians $h_{n}$ and $h_{n-1}$ in the same way as
(2.12) for the scalar D-hierarchy:

$A_{n}^{-}h_{n}=\gamma h_{n-1}A^{-}_{n},\qquad h_{n}A_{n}^{+}=\gamma
A^{+}_{n}h_{n-1}$ (5.71)

where $A^{\pm}_{n}=\sigma_{3}(A_{n}^{\mp})^{\dagger}\sigma_{3}$ and $\gamma$
is for a sign. As a consequence, the operators $A^{\pm}_{n}$ will intertwine
$h_{n}^{2}$ and $h_{n-1}^{2}$:

$A_{n}^{-}h_{n}^{2}=h_{n-1}^{2}A^{-}_{n},\qquad
h_{n}^{2}A_{n}^{+}=A^{+}_{n}h_{n-1}^{2}$ (5.72)

The solution for the case of intertwining operators ($\gamma=1$) is

$R^{-}_{n}=\left(\begin{array}[]{cc}a^{-}_{n-1}&0\\\\[4.30554pt]
i(\mu_{n-1}-\mu_{n})&a^{-}_{n}\end{array}\right),\qquad
R_{n}^{-}h_{n}=h_{n-1}R^{-}_{n}$ (5.73)

While the anti-intertwining operators ($\gamma=-1$) are $T^{\pm}_{n}$:

$T^{-}_{n}=\left(\begin{array}[]{cc}a^{-}_{n-1}&0\\\\[4.30554pt]
i(\mu_{n-1}+\mu_{n})&-a^{-}_{n}\end{array}\right),\qquad
T_{n}^{-}h_{n}=-h_{n-1}T^{-}_{n}$ (5.74)

###  5.1 Spectrum of the Dirac hierarchy for decreasing case

The spectrum and spinor eigenfunctions of the Hamiltonian $h_{n}$ (5.68) can
also be found following the same procedure as before. We get the solutions:

  * •

Positive spectrum

$\varepsilon_{n}^{k+}=\mu_{n-k},\quad\Psi_{n}^{k+}=\left(\begin{array}[]{c}\sqrt{\mu_{n}+\mu_{n-k}}\,\psi_{n-1}^{k-1}\\\\[6.45831pt]
i\sqrt{\mu_{n}-\mu_{n-k}}\,\psi_{n}^{k}\end{array}\right),\qquad k=1,2,\dots$
(5.75)

  * •

Negative spectrum

$\begin{array}[]{ll}\varepsilon_{n}^{k-}=-\mu_{n-k},&\Psi_{n}^{k-}=\left(\begin{array}[]{c}\sqrt{\mu_{n}-\mu_{n-k}}\,\psi_{n-1}^{k-1}\\\\[6.45831pt]
i\sqrt{\mu_{n}+\mu_{n-k}}\,\psi_{n}^{k}\end{array}\right),\qquad
k=1,2,\dots\\\\[17.22217pt] {\rm Ground\ level}:\\\\[4.30554pt]
\varepsilon_{n}^{0-}=-\mu_{n},&\Psi_{n}^{0-}=\left(\begin{array}[]{c}0\\\\[4.30554pt]
\psi_{n}^{0}\end{array}\right),\qquad k=0\end{array}$ (5.76)

Remark that in this Dirac-like version of the scalar D-hierarchies the
“absolute” ground state has negative energy. Now, the role of antiparticle
states are played by the positive energy spectrum. This is an example of non-
Hermitian matrix Hamiltonian, as it is explicit in (5.68), where the discrete
energy levels are real and the eigenfunctions square integrable.

###  5.2 Eigenfunctions annihilated by the intertwining operators

Next, we want to know what kind of eigenfunctions annihilate the intertwining
operators $R_{n}^{-}$ and $T_{n}^{-}$.

  * •

Eigenfunctions annihilated by $R_{n}^{-}$

1) The ground state of negative energies:

$\Psi_{n}^{0-}=\left(\begin{array}[]{c}0\\\\[4.30554pt]
\psi_{n}^{0}\end{array}\right)\,,\qquad\varepsilon_{n}^{0-}=-\mu_{n}$

2) The “ground state” of positive energies:

$\Psi_{n}^{1+}=\left(\begin{array}[]{c}\sqrt{\mu_{n}+\mu_{n-1}}\,\psi_{n-1}^{0}\\\\[6.45831pt]
i\sqrt{\mu_{n}-\mu_{n-1}}\,\psi_{n}^{1}\end{array}\right)\,,\qquad\varepsilon_{n}^{1+}=\mu_{n-1}$

  * •

Eigenfunctions annihilated by $T_{n}^{-}$

1) The ground state of negative energies:

$\Psi_{n}^{0-}=\left(\begin{array}[]{c}0\\\\[4.30554pt]
\psi_{n}^{0}\end{array}\right)\,,\qquad\varepsilon_{n}^{0-}=-\mu_{n}$

2) The first excited state of negative energies:

$\Psi_{n}^{1-}=\left(\begin{array}[]{c}\sqrt{\mu_{n}-\mu_{n-1}}\,\psi_{n-1}^{0}\\\\[6.45831pt]
i\sqrt{\mu_{n}+\mu_{n-1}}\,\psi_{n}^{1}\end{array}\right)\,,\qquad\varepsilon_{n}^{1-}=-\mu_{n-1}$

###  5.3 Example: The hyperbolic Pöschl–Teller potential

The following sequence of hyperbolic PT Schrödinger Hamiltonians

$H_{n}(x)=-\partial_{xx}-\frac{(n+1/2)(n-1/2)}{\cosh^{2}x},\qquad
x\in(-\infty,\infty),\qquad n\in\mathbb{N}$ (5.77)

are of decreasing type and satisfy the factorization (2.10). The factor
operators and factor energies are

$a_{n}^{\pm}(x)=\mp\partial_{x}+(n-1/2)\tanh
x,\qquad\mu_{n}^{2}=(n-1/2)^{2},\qquad n=1,2,\dots$ (5.78)

Therefore, the associated Dirac Hamiltonians $h_{n}$ are defined by

$h_{n}=\left(\begin{array}[]{cc}(n-1/2)&i(\partial_{x}+(n-1/2)\tanh
x)\\\\[4.30554pt] i(-\partial_{x}+(n-1/2)\tanh x)&-(n-1/2)\end{array}\right)$
(5.79)

and the intertwining operators by

$R^{+}_{n}=\left(\begin{array}[]{cc}-\partial_{x}+(n-3/2)\tanh
x&-i\\\\[4.30554pt] 0&-\partial_{x}+(n-1/2)\tanh x\end{array}\right)$ (5.80)
$T^{+}_{n}=\left(\begin{array}[]{cc}-\partial_{x}+(n-3/2)\tanh
x&i\,(2n-2)\\\\[4.30554pt] 0&-(-\partial_{x}+(n-1/2)\tanh
x)\end{array}\right)$ (5.81)

###  5.4 Dirac-like equations for systems on the ${\cal H}^{2}$ hyperboloid

We can get the hyperbolic PT scalar Hamiltonian (5.77) and the matrix PT
Hamiltonian corresponding to (5.79) from separation of variables of a system
defined on a hyperboloid in a similar way as that employed in the
trigonometric case in the space of functions on the sphere. In order to get
the potential in (5.77) we start with one sheet hyperboloid (this choice is
different from the two sheets hyperboloid and it has some peculiarities). We
have made this election in order to work with representations of
eigenfunctions in the angular $\phi$ variable and to obtain the specific form
of the hyperbolic potential (5.77).

(a) The scalar system

Consider the hyperbolic surface ${\cal H}^{2}$,

$x^{2}+y^{2}-z^{2}=1$ (5.82)

parametrised with pseudo-spherical coordinates:

$x=\cosh\chi\cos\phi,\qquad y=\cosh\chi\sin\phi,\qquad z=\sinh\chi$ (5.83)

The surface (5.82) is invariant under the group $SO(2,1)$ and the scalar
wavefunctions defined on ${\cal H}^{2}$ constitute the support space of a
quasi-regular representation $U$,

$U(g)\tilde{\psi}({\bf x})=\tilde{\psi}(g^{-1}{\bf x}),\qquad g\in SO(2,1)$
(5.84)

The generators of the group $SO(2,1)$ are given by: one of them, $L_{z}$,
generating rotations around the $z$-axis and two, $L_{x}$ and $L_{y}$, of
hyperbolic transformations around the $x$ and $y$ axes. A differential
realization in Cartesian coordinates is

$L_{x}=y\partial_{z}+z\partial_{y},\qquad
L_{y}=-x\partial_{z}-z\partial_{x},\qquad L_{z}=-x\partial_{y}+y\partial_{x}$

If we pass to the pseudo-spherical coordinates (5.83) and multiply by the
imaginary unit $i$, they take the form

$L_{x}=i(\sin\phi\,\partial_{\chi}+\tanh\chi\cos\phi\,\partial_{\phi}),\quad
L_{y}=i(-\cos\phi\,\partial_{\chi}+\tanh\chi\sin\phi\,\partial_{\phi}),\quad
L_{z}=i(-\partial_{\phi})$ (5.85)

Their commutation rules close the $so(2,1)\approx su(1,1)$ Lie algebra

$[L_{z},L_{x}]=i\,L_{y},\qquad[L_{z},L_{y}]=-i\,L_{x},\qquad[L_{x},L_{y}]=-i\,L_{z}$

We build the lowering and raising operators from the realization (5.85),

$L_{\pm}=L_{x}\pm iL_{y}=e^{\pm
i\phi}\Big{(}\pm\partial_{\chi}+i\,\tanh\chi\partial_{\phi}\big{)}$ (5.86)

In this basis the $so(2,1)\approx su(1,1)$ Lie algebra is

$[L_{z},L_{\pm}]=\pm L_{\pm},\qquad[L_{-},L_{+}]=2L_{z}$

The Casimir operator, which we will identify with the scalar Schrödinger
Hamiltonian, has the form

${\cal C}:=\tilde{H}=L_{x}^{2}+L_{y}^{2}-L_{z}^{2}=L_{+}L_{-}-L_{z}(L_{z}-1)$
(5.87)

The discrete series of unitary irreducible representations (uir’s) of
$SO(2,1)$ are characterized by the Casimir eigenvalues [18, 19, 20]

$\tilde{H}\tilde{\psi}=-\lambda(\lambda-1)\tilde{\psi},\qquad\lambda=1,2,3,\dots$
(5.88)

In terms of the pseudo-spherical coordinates ${\cal C}$ ($=\tilde{H}$) takes
the expression

$\tilde{H}=-\partial_{\chi\chi}-\tanh\chi\,\partial_{\chi}+\frac{\partial_{\phi\phi}}{\cosh^{2}\chi}$

There is a first order term in $\partial_{\chi}$ that can be eliminated by
means of the following transformation:

$\tilde{\psi}=\frac{1}{\sqrt{\cosh\chi}}\psi\ \implies\
\partial_{\chi}\to\partial_{\chi}-\frac{1}{2}\,\tanh\chi$ (5.89)

Then, the displaced Hamiltonian $H:=\tilde{H}-\frac{1}{4}$ becomes

$H=\tilde{H}-\frac{1}{4}=-\partial_{\chi\chi}+\frac{1/4+\partial_{\phi\phi}}{\cosh^{2}\chi}$
(5.90)

The eigenvalues of this Hamiltonian corresponding to the uir’s of $SO(2,1)$
are, according to (5.88),

$H\psi=-(\lambda-1/2)^{2}\psi,\qquad\lambda=1,2,\dots$

If we consider a basis in this representation of common eigenfunctions of
${\cal C}$ and $L_{z}$,

$\psi(\chi,\phi)=\psi_{n}(\chi)e^{in\phi},\qquad n\in{\mathbb{N}}$

we find

$H_{n}\psi_{n}=\left(-\partial_{\chi\chi}-\frac{(n+1/2)(n-1/2)}{\cosh^{2}\chi}\right)\psi_{n}=-(\lambda-1/2)^{2}\psi_{n},\qquad\lambda=1,2,3,\dots,n$
(5.91)

where $n\geq\lambda\geq 1$. Therefore, once fixed $n$ then the number of bound
states is finite: $k=0,1,\dots,n-1$. In case we fixed the eigenvalue
$\lambda$, the potentials that may support such a negative level are labeled
by $n\geq\lambda$.

We will use the notation $\psi_{n}^{k}$ for the eigenfunction corresponding
to:

$\psi_{n}^{k}=\left\\{\begin{array}[]{ll}{\rm
Potential}&V_{n}=-\frac{(n+1/2)(n-1/2)}{\cosh^{2}\chi}\\\\[8.61108pt] {\rm
Energy\ level}&E_{n}^{k}=-(n-k-1/2)^{2},\qquad k=0,1\dots n-1\\\\[8.61108pt]
{\rm Representation}&\lambda=n-k,\qquad
L_{z}\psi_{n}^{k}=n\psi_{n}^{k},\qquad\lambda=n,n-1,\dots,1\end{array}\right.$

We can check that indeed this equation coincides with the initial hyperbolic
PT equation (5.77) as well as the eigenvalues (5.78).

On the other hand, the lowering and raising operators, when they act between
the Hamiltonians $H_{n}$ and $H_{n-1}$, after the transformation (5.89),
become

$\begin{array}[]{l}L_{+}=L_{x}+iL_{y}=\big{(}+\partial_{\chi}-(n-1/2)\tanh\chi\big{)}\\\\[8.61108pt]
L_{-}=L_{x}-iL_{y}=\big{(}-\partial_{\chi}-(n-1/2)\tanh\chi\big{)}\end{array}$
(5.92)

These operators coincide with the intertwining operators $a_{n}^{\pm}$ given
before in (5.78).

(b) The spinor system

Next, we will consider the spinor representation of $so(2,1)\approx su(1,1)$
defined in the space of spinor wavefunctions $\Psi({\bf x})$ on the
hyperboloid (5.82). In this space, the generators of $SU(1,1)$, which will be
called $K_{i}$, are defined by

$K_{i}=L_{i}+S_{i},\qquad i=1,2,3$ (5.93)

where the orbital generators ($L_{1}=L_{x},L_{2}=L_{y},L_{3}=L_{z}$) have been
given in (5.85), while the spinor generators
($S_{1}=S_{x},S_{2}=S_{y},S_{3}=S_{z}$) have been chosen as

$\textstyle S_{x}=i\frac{1}{2}\sigma_{x},\qquad
S_{y}=i\frac{1}{2}\sigma_{y},\qquad S_{z}=\frac{1}{2}\sigma_{z}$

and therefore, $S_{\pm}=S_{x}\pm iS_{y}$ are

$S_{+}=\left(\begin{array}[]{cc}0&i\\\\[4.30554pt]
0&0\end{array}\right)\,,\qquad
S_{-}=\left(\begin{array}[]{cc}0&0\\\\[4.30554pt] i&0\end{array}\right)$
(5.94)

The matrices $S_{x},S_{y}$ are anti-Hermitian, while $S_{z}$ is Hermitian.
They satisfy the $su(1,1)$ Lie algebra:

$[S_{z},S_{x}]=i\,S_{y},\qquad[S_{z},S_{y}]=-i\,S_{x},\qquad[S_{x},S_{y}]=-i\,S_{z}$

or

$[S_{z},S_{\pm}]=\pm S_{\pm},\qquad[S_{-},S_{+}]=2S_{z}$

Then, the Casimir operator of this realization is

${\cal
C}_{s}={\boldsymbol{K}}^{2}=(L_{x}+S_{x})^{2}+(L_{y}+S_{y})^{2}-(L_{z}+S_{z})^{2}={\boldsymbol{L}}^{2}+2{\boldsymbol{L}}{\boldsymbol{S}}+{\boldsymbol{S}}^{2}=-\nu(\nu-1)$
(5.95)

where the cross terms constitute the matrix Hamiltonian $h$,

$h=2{\boldsymbol{L}}{\boldsymbol{S}}=2(L_{x}S_{x}+L_{y}S_{y}-L_{z}S_{z})=L_{+}S_{-}+L_{-}S_{+}-2L_{z}S_{z}$
(5.96)

and ${\boldsymbol{S}}^{2}=-3/4$. We will use the spinor expressions (5.94) and
the orbital operators $L_{\pm}$ of (5.86) after the transformation (5.89),

$L_{\pm}=e^{\pm i\phi}\big{(}\pm\partial_{\chi}+(i\partial_{\phi}\mp
1/2)\tanh\chi\big{)}$

to be replaced in the expression (5.96) of $h$ in order to find the explicit
matrix Hamiltonian

$h=\left(\begin{array}[]{cc}i\partial_{\phi}&ie^{-i\phi}(-\partial_{\chi}+(i\partial_{\phi}+1/2)\tanh\chi)\\\\[8.61108pt]
i(+\partial_{\chi}+(i\partial_{\phi}+1/2)\tanh\chi)e^{i\phi}&-i\partial_{\phi}\end{array}\right)$
(5.97)

As we know from the beginning this Hamiltonian is not explicitly Hermitian
since the matrix generators $S_{x}$ and $S_{y}$ were not Hermitian. Let us
write the form of $h$ when it acts on simultaneous eigenfunctions $\Psi_{\nu
m}$ of the Casimir ${\cal C}$ (5.95) and $K_{z}$:

$K_{z}\Psi_{\nu m}=m\Psi_{\nu m}\ \implies\ \Psi_{\nu
m}(\chi,\phi)=\left(\begin{array}[]{l}\varphi_{m}(\chi)e^{i(m-1/2)\phi}\\\\[4.30554pt]
\xi_{m}(\chi)e^{i(m+1/2)\phi}\end{array}\right)$ (5.98)

Then, the reduced Hamiltonian in the variable $\chi$, using this basis, is

$h_{m}=\left(\begin{array}[]{cc}-m+1/2&i(-\partial_{\chi}-m\tanh\chi)\\\\[8.61108pt]
i(\partial_{\chi}-m\tanh\chi)&m+1/2\end{array}\right)$ (5.99)

The next step is to compute the square of $h$:

$h^{2}=-{\boldsymbol{L}}^{2}+h$ (5.100)

According to (5.90) $H={\boldsymbol{L}}^{2}-1/4$ so that finally we get

$(h-1/2)^{2}=-H=-\left(-\partial_{\chi\chi}+\frac{1/4+\partial_{\phi\phi}}{\cosh^{2}\chi}\right)$
(5.101)

If we take the square $(h_{m}-1/2)^{2}$ of the reduced Hamiltonian (5.99), we
find

$(h_{m}-1/2)^{2}=-\left(\begin{array}[]{cc}H_{n-1}&0\\\\[4.30554pt]
0&H_{n}\end{array}\right)$ (5.102)

where $H_{m}$ coincides with $H_{n}$ of (5.91), provided $m=n-1/2$. Let us
make some remarks.

(a) These two formulas for the square of $h$ are reasonable since the square
of a real spectrum of $h$ will always produce a positive spectrum, that is,
this square must have oposite sign than the scalar Hamiltonians $H$, which
have a negative spectrum as shown in (5.91).

(b) The matrix hyperbolic Hamiltonian $h$ given in (5.97), or its reduced form
(5.99), are not explicitly Hermitian, therefore this seems an inconsistency.
However, $h$ although non-Hermitian it has a real discrete spectrum with
square integrable solutions. We will show this property in the following
subsection by computing the eigenvalues and eigenvectors.

(c) Let us notice that, as a consequence of the previous remark, this spinor
representation of $SU(1,1)$ is not unitary. However, there is a Hermitian
invariant product given by

$\langle\Phi,\Psi\rangle=\int\Phi^{\dagger}(\chi,\phi)\sigma_{3}\Psi(\chi,\phi)\,{\rm
d}\chi{\rm d}\phi$

where the invariant measure on the hyperboloid is hidden in the wavefunctions
by the transformation (5.89). One can check that indeed, the reduced
Hamiltonian (5.97) is Hermitian with respect to $\sigma_{3}$:
$h^{\dagger}\sigma_{3}=\sigma_{3}h$. This is a non positive definite product
which is consistent with the non-unitary character.

(c) Spectrum and eigenfunctions

If we have in mind the diagonal form of the square $(h-1/2)^{2}$, and at the
same time we write the operators in the entries of the matrix $h_{m}$ given in
(5.99) we guess that the eigenfunctions will have the following form, making
use of the factor operators $a_{m}^{\pm}$ of (5.78):

$\left(\begin{array}[]{cc}-m&-ia_{m}^{-}\\\\[6.45831pt]
-ia_{m}^{+}&m\end{array}\right)\left(\begin{array}[]{l}\alpha\psi_{m-1}^{k-1}\\\\[6.45831pt]
i\,\beta\psi_{m}^{k}\end{array}\right)=\varepsilon_{n}^{k}\left(\begin{array}[]{l}\alpha\psi_{m-1}^{k-1}\\\\[6.45831pt]
i\,\beta\psi_{m}^{k}\end{array}\right)$ (5.103)

Taking into account that the action of these factor operators is

$a_{m}^{-}\psi_{m}^{k}=\sqrt{\mu_{m}^{2}-\mu_{m-k}^{2}}\
\psi_{m-1}^{k-1},\qquad
a_{m}^{+}\psi_{m-1}^{k-1}=\sqrt{\mu_{m}^{2}-\mu_{m-k}^{2}}\
\psi_{m}^{k},\qquad\mu_{m-k}^{2}=(m-k)^{2}$

Then, we get the eigenvalues of $h$:

$\varepsilon_{n}^{k\pm}=\pm(m-k)\,,\qquad m=n-1/2$ (5.104)

The corresponding eigenspinors $\Psi_{n}^{k\pm}$ are given by their
coefficients $\alpha^{\pm}$ and $\beta^{\pm}$:

$\alpha^{-}=\sqrt{k}\,,\ \
\beta^{-}=\sqrt{2m-k}\,,\qquad\alpha^{+}=\sqrt{2m-k}\,,\ \ \beta^{+}=\sqrt{k}$

The ground state of these eigenfunctions is negative:

$\varepsilon_{n}^{0-}=-m,\qquad\Psi_{n}^{0-}=\left(\begin{array}[]{l}0\\\\[6.45831pt]
\psi_{m}^{0}\end{array}\right)$

We notice that the greater spinor component of the negative spectrum
eigenfunctions is the lower one with spin $-1/2$. However, the greater
component of the positive spectrum spinor eigenfunctions is the upper
corresponding to spin $1/2$. The degeneracy is infinite for each positive or
negative eigenvalue. In the following we will determine the $SU(1,1)$
representations corresponding to the eigenvalues of the same absolute value
but opposite sign: $\varepsilon_{n}^{k\pm}=\pm(n-k-1/2)$. In order to find
these representations, we will characterize the eigenfunctions
$\Psi_{n}^{k\pm}$ and their spinor components in terms of the commuting
symmetries of $h$: $K_{z}$, ${\boldsymbol{L}}^{2}$ and ${\boldsymbol{K}}^{2}$.

  * $K_{z}$:

The label $n$ of the spinor functions $\Psi_{n}^{k\pm}$ means that they are
eigenfunctions of $K_{z}$ with eigenvalue $m=n-1/2$:

$K_{z}\Psi_{n}^{k\pm}=m\Psi_{n}^{k\pm},\qquad m=n-1/2$

  * ${\boldsymbol{L}}^{2}$:

We know that the scalar functions $\psi_{n}^{k}(\theta)e^{i(m+1/2)\phi}$ and
$\psi_{n-1}^{k-1}(\theta)e^{i(m-1/2)\phi}$ (which are de components of the
spinors $\Psi_{n}^{k\pm}$) are eigenfunctions of ${\boldsymbol{L}}^{2}$ with
eigenvalue $\lambda=n-k$. Therefore, the spinor eigenfunctions of $h$ are also
eigenfunctions of ${\boldsymbol{L}}^{2}$ with the same eigenvalue

$({\boldsymbol{L}}^{2}-1/4)\Psi_{n}^{k\pm}=-(\lambda-1/2)^{2}\Psi_{n}^{k\pm},\qquad\lambda=n-k,\qquad
m=n-1/2$

  * ${\boldsymbol{K}}^{2}$:

The above eigenfunctions of $h$ are also eigenfunctions of
${\boldsymbol{K}}^{2}$. We will make use of expression (4.56) in order to see
this:

${\boldsymbol{K}}^{2}\Psi_{n}^{k-}=\big{(}{\boldsymbol{L}}^{2}+\tilde{h}+{\boldsymbol{S}}^{2}\big{)}\Psi_{n}^{k-}=(-\lambda(\lambda-1)-\lambda+1/4)\Psi_{n}^{k-}=-(\lambda+1/2)(\lambda-1/2)\Psi_{n}^{k-}$

In other words, we can write as

${\boldsymbol{K}}^{2}\Psi_{n}^{k-}=(\nu^{+})(\nu^{+}-1)\Psi_{n}^{k-},\qquad\nu^{+}=\lambda+1/2$

and in a similar way

${\boldsymbol{K}}^{2}\Psi_{n}^{k+}=(\nu^{-})(\nu^{-}-1)\Psi_{n}^{k+},\qquad\nu^{-}=\lambda-1/2$

We could explain these results from the fact that the spinor wave functions
$\Psi$ belong to tensor products of the representations $\lambda$ from the
orbital part and $s=1/2$ from spin: $\Psi\in\lambda\otimes 1/2$. As we know
from the composition of “angular momenta”:

$\lambda\otimes 1/2=\nu^{+}\oplus\nu^{-},\qquad\nu^{\pm}=\lambda\pm 1/2$

Then, the negative energy spinors $\Psi_{n}^{k-}$ belong to the representation
of the total “angular momentum” $\nu^{+}$, while the positive spinors
$\Psi_{n}^{k+}$ to $\nu^{-}$.

In conclusion, we have obtained a real spectrum for a non-Hermitian matrix
Hamiltonian.

(d) Symmetries and “anti-symmetries”

  * 1)

Symmetries $K_{\pm},K_{3}$.

In the same way as for $su(2)$ the spinor symmetries in the case of $su(1,1)$
are given by

$K_{+}=L_{+}+S_{+},\qquad K_{-}=L_{-}+S_{-},\qquad K_{3}=L_{3}+S_{3}$

Their restriction to the eigenfunctions of $K_{3}=K_{z}$ give rise to the
intertwining operators $R^{\pm}_{n}$.

  * 2)

Anti-symmetries

We will also find the anti-symmetries $T^{\pm}$ which anti-commute with the
Hamiltonian $h_{n}-1/2$,

$T^{\pm}(\tilde{h}-1/2)=-(\tilde{h}-1/2)\,T^{\pm},\qquad\quad[K_{3},T^{\pm}]=\pm
T^{\pm}$

and have the same form as in the trigonometric case:

$T^{+}=L_{+}S_{3}-L_{3}S_{+},\qquad T^{-}=L_{-}S_{3}-L_{3}S_{-},\qquad
T_{3}=\frac{1}{2}(L_{+}S_{-}-L_{-}S_{+})$

The spectrum of Dirac-Weyl like equations, symmetries in terms of intertwining
operators and global intertwining operators for $4\times 4$ Dirac-like
Hamiltonians can also be obtained following the same procedure as above.
However, in order to shorten the length of the paper we have not discussed
them here.

##  6 Conclusions and remarks

We have introduced a kind of matrix Dirac-like Hamiltonians which are in close
correspondence with factorizable Schrödinger Hamiltonians. The scalar
intertwining operators, $a^{\pm}_{n}$ and the factorization energies
$\pm\mu_{n}^{2}$, are used as ingredients to construct the matrix Hamiltonian
and the intertwining matrix operators. This type of Dirac Hamiltonians have
appeared in many previous references taking part in different problems. Here,
we have tried to explain them by a reduction process of spinor systems defined
on curved spaces. We have called $h_{n}$ (the reduced cases in one variable)
or $h$ (in two coordinates of the surface) to these Dirac-like Hamiltonians.
In the case of spherical symmetry they are called Dirac operators [23].

We have worked out two simple examples. One of them defined on a sphere ${\cal
S}^{2}$, giving rise to a matrix version of a trigonometric Pöschl-Teller
potential. The symmetries and anti-symmetries were characterized as well as
the degeneracy of the energy levels. In particular the anti-symmetries
interchanged the sign of energy. The second example which was defined in a two
dimensional hyperboloid ${\cal H}^{2}$ gave rise to a matrix non-Hermitian
Hamiltonian with a real spectrum, which is related to a scalar hyperbolic
Pöschl-Teller potential. The scalar systems were obtained as the square of the
Dirac-like systems, just in a similar way as the Dirac and Klein-Gordon
equations are related. We expect to complete this picture in the near future
by finding the Dirac-like cases of all the scalar factorizable potentials with
or without the mass $m_{0}$. In particular they will need higher space
dimensions and higher gamma matrices. Another point that we pursue is to
identify the role of this kind of Dirac-like Hamiltonians; it is reasonable
that they be related with the symmetries of different classes of Dirac
systems, since our building relies only on symmetry considerations. We hope
that this kind of Dirac-like Hamiltonians be also consistent with some
symmetric interactions with external fields [24, 25].

## Acknowledgments

This work was partially supported by Junta de Castilla y León (BU229P18) and
by Ankara University BAP No. 20L0430005. D. Demir Kızılırmak acknowledges
Ankara Medipol University.

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