# PT-Rotations, PT-Spherical Harmonics and the PT-Hydrogen Atom

Juan M. Romero<EMAIL_ADDRESS>R. Bernal-Jaquez
<EMAIL_ADDRESS>O. González-Gaxiola<EMAIL_ADDRESS>Departamento de Matemáticas Aplicadas y Sistemas
Universidad Autónoma Metropolitana-Cuajimalpa, México 01120 DF, México

###### Abstract

We have constructed a set of non-Hermitian operators that satisfy the
commutation relations of the $SO(3)$-Lie algebra. It is shown that this
operators generate rotations in the configuration space and not in the
momentum space but in a modified non-Hermitian momentum space. This generators
are related with a new type of spherical harmonics that result to be
$\mathcal{PT}$-orthonormal. Additionally, we have shown that this operators
represent conserved quantities for a non-Hermitian Hamiltonian with an
additional complex term. As a particular case, the solutions of the
corresponding $\mathcal{PT}$-Hydrogen atom that includes a complex term are
obtained, and it is found that a non-Hermitian Runge-Lenz vector is a
conserved quantity. In this way, we obtain a set of non-Hermitian operators
that satisfy the $SO(4)$-Lie algebra.

###### pacs:

11.30.Er, 11.30.-j, 03.65.-w.

##  I Introduction

Quantum mechanics is considered as one the most solid and well established
theories in physics. Different experiments have corroborated its predictions.
However, at a theoretical level there are different facts that make us think
that it could be necessary to modify or extend this theory. For example, it
has not been possible to find a consistent quantum-mechanical formulation of
general relativity, then quantum theory maybe modified in order to make it
compatible with general relativity.

As it is well known, quantum mechanics has been formulated in terms of
Hermitian operators in order to obtain real spectra. However, it has become
clear that Hermiticity is not a necessary condition to obtain real spectra.
This opens the possibility for quantum mechanics to be extended using non-
Hermitian operators, this is the so called $\mathcal{PT}$-symmetry theory, see
review bender-0:gnus  and its references.

The $\mathcal{PT}$-version of quantum mechanics has strongly attracted
attention because it gives a way to deal with some problems that are out of
the scope of conventional quantum mechanics. For example, we can solve certain
kind of problems in which the potentials are given by complex-valued functions
and whose spectra results to be real Ben-1:gnus . In the same way, using this
formulation it has been possible to achieve a consistent quantization of a
system with high order derivatives: the so called the Pais-Uhlenbeck
oscillator model bender-uhlenbeck:gnus . This opens the possibility to
construct in a consistent way high order derivatives field theories. This fact
is important because different theories with high order derivatives have been
recently proposed, for example, in extensions of the standard model
pospelov:gnus , in the noncommutative spaces szabo:gnus  and gravity theories
gravedad:gnus . In this way, it becomes possible that an extension of
$\mathcal{PT}$-symmetry theory applied to field theory can give a consistent
description of these systems.

There is not a finished version of the theory, however, a growing number of
themes are under study in the $\mathcal{PT}$-framework, some of them can be
found das:gnus . An aspect that has been scarcely treated in the
$\mathcal{PT}$-context is the study of symmetries and conserved quantities. In
this work, we will study some aspects of this topic. We will obtain a non-
Hermitian set of operators that satisfy the commutation relations of the Lie
$SO(3)$ rotation group. It will be shown that these operators generate
rotations in the configuration space $x_{i}$, and not in the momentum space
$\vec{p}=-i\nabla$ but in a modified non-Hermitian momentum space
$\vec{p}_{f}=\vec{p}+i\vec{\nabla{f}}$, originally considered by Dirac in his
seminal book Dirac:gnus . Also, we will show that the Casimir of the algebra
has real spectra and that its eigenfunctions, under the $\mathcal{PT}$-inner
product, form a complete basis. This eigenfunctions will be called
$\mathcal{PT}$-spherical harmonics.

Additionally we will study a central potential Hamiltonian with an additional
complex term. It will be shown that the conventional angular momentum is not a
conserved quantity anymore and we will have a modified non-Hermitian angular
momentum operator. As a particular case, we obtain the solutions of the
corresponding $\mathcal{PT}$-Hydrogen atom that includes a complex term, and
it will be found that a non-Hermitian Runge-Lenz vector is a conserved
quantity. Then we will have the non-Hermitian generators of the $SO(4)$-Lie
algebra.

This work is organized as follows: section II make a brief review of
$\mathcal{PT}$-theory and of conventional spherical harmonics, in section III
we study the $\mathcal{PT}$-rotations, in section IV we study the completeness
relation and some examples, section V is devoted to the study of symmetry
transformations, in section VI we deal with the central potential problem, in
section VII we will study the Hydrogen atom and at last we conclude with a
summary of our results.

##  II $\mathcal{PT}$-Symmetry and Spherical Harmonics

In this section, a review of some well known facts of $\mathcal{PT}$-symmetry
theory and spherical harmonics is made before consider the
$\mathcal{PT}$-transformed version of this functions

###  II.1 $\mathcal{PT}$-Inner product

$\mathcal{PT}$-theory considers the transformations under the parity operator
$\mathcal{P}$ and the time reversal operator $\mathcal{T}$. Under the
$\mathcal{P}$-operator we have the transformation

$\displaystyle x,y,z\to-x,-y,-z$ (1)

and under $\mathcal{T}$

$\displaystyle i\to-i$ (2)

In this way, any function $f(\vec{x})$ can be transform as

$\displaystyle\mathcal{PT}\left(f(\vec{x})\right)=f^{*}(-\vec{x}).$ (3)

Note that, in spherical coordinates $\mathcal{P}$ produces the transformation

$\displaystyle(r,\theta,\varphi)\to(r,\pi-\theta,\varphi+\pi),$ (4)

under $\mathcal{P}$ a $f$ function transforms as

$\displaystyle\mathcal{P}\left(f(r,\theta,\varphi)\right)=f(r,\pi-\theta,\varphi+\pi),$
(5)

therefore

$\displaystyle\mathcal{PT}\left(f(r,\theta,\varphi)\right)=f^{*}(r,\pi-\theta,\varphi+\pi).$
(6)

Now, the $\mathcal{PT}$-inner product is defined as

$\displaystyle\langle f|g\rangle=\int d\vec{x}[\mathcal{PT}f(x)]g(x).$ (7)

This expressions will be used in sections bellow. An exhaustive study of the
$\mathcal{PT}$-theory can be found in bender-0:gnus .

###  II.2 Spherical Harmonics

The angular momentum components are given by the Hermitian operators
landau:gnus

$\displaystyle L_{x}=-i\left(y\frac{\partial}{\partial
z}-z\frac{\partial}{\partial
y}\right),\;L_{y}=-i\left(z\frac{\partial}{\partial
x}-x\frac{\partial}{\partial
z}\right),\;L_{z}=-i\left(x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}\right).$

Its algebra is given by

$\displaystyle\left[L_{x},L_{y}\right]=iL_{z},\qquad\left[L_{z},L_{x}\right]=iL_{y},\qquad\left[L_{y},L_{z}\right]=iL_{x}.$
(8)

Considering $L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$ and Eq. (8), we have

$\displaystyle\left[L^{2},L_{x}\right]=\left[L^{2},L_{y}\right]=\left[L^{2},L_{z}\right]=0.$
(9)

An important equation in mathematical-physics is the eigenvalue equation
$L^{2}Y_{lm}=l(l+1)Y_{lm},l=0,1,2\cdots,$ that in spherical coordinates is
written

$\displaystyle L^{2}Y_{lm}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle-\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial
Y_{lm}(\theta,\varphi)}{\partial\theta}\right)+\frac{1}{\sin\theta^{2}}\frac{\partial^{2}Y_{lm}(\theta,\varphi)}{\partial\varphi^{2}}\right]$
(10) $\displaystyle=$ $\displaystyle l(l+1)Y_{lm}(\theta,\varphi).$

If $\varphi\in(0,2\pi)$ and $\theta\in(0,\pi),$ the solutions of this equation
are given by the spherical harmonics fesbach:gnus

$\displaystyle Y_{lm}(\theta,\varphi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)!}}e^{im\varphi}P_{l}^{m}(\cos\theta),\qquad-l\leq
m\leq l,$ (11)

with

$\displaystyle l=0,1,2,3\cdots,\qquad P_{l}^{m}(u)$ $\displaystyle=$
$\displaystyle(-)^{m}(1-u^{2})^{\frac{m}{2}}\frac{d^{m}}{du^{m}}P_{l}(u),\qquad
P_{l}(u)=\frac{1}{2^{l}l!}\frac{d^{l}}{du^{l}}\left(u^{2}-1\right)^{l}$ (12)

where $P_{l}^{m}(u)$ denoted the associated Legendre polynomials and
$P_{l}(u)$ the Legendre polynomials respectively. The spherical harmonics
satisfy the orthonormality relation

$\displaystyle<Y_{l^{\prime}m^{\prime}}(\theta,\varphi)|Y_{lm}(\theta,\varphi)>=\int
d\Omega
Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi)=\delta_{mm^{\prime}}\delta_{l^{\prime}l}.$
(13)

Given that the spherical harmonics constitute an orthonormal basis, we can
write any function $F(\theta,\varphi)$ as a linear combination of them, that
is

$\displaystyle F(\theta,\varphi)=\sum_{l\geq
0}\sum_{m=-l}^{l}C_{lm}Y_{lm}(\theta,\varphi).$ (14)

Using Eq. (13), we find

$\displaystyle C_{lm}=\int d\Omega
Y^{*}_{lm}(\theta,\varphi)F(\theta,\varphi).$ (15)

Substituting $C_{lm}$ in Eq. (14) and making the change of variables
$u^{\prime}=\cos\theta^{\prime},u=\cos\theta$ we obtain

$\displaystyle
F(\theta,\varphi)=\int_{0}^{2\pi}d\varphi\int_{-1}^{1}du^{\prime}F(u^{\prime},\varphi^{\prime})\left(\sum_{l\geq
0}\sum_{m=-l}^{l}Y^{*}_{lm}(u^{\prime},\varphi^{\prime})Y_{lm}(u,\varphi)\right).$

Therefore, the expression inside the parenthesis must be equal to
$\delta(\varphi-\varphi^{\prime})\delta(u-u^{\prime}),$ that is

$\displaystyle\sum_{l\geq
0}\sum_{m=-l}^{l}Y^{*}_{lm}(\theta^{\prime},\phi^{\prime})Y_{lm}(\theta,\phi)=\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime}),$
(16)

this expression is called completeness relation.

Under the parity operator the spherical harmonics transform as

$\displaystyle\mathcal{P}\left(Y_{lm}(\theta,\varphi)\right)=Y_{lm}(\pi-\theta,\varphi+\pi)=(-)^{l}Y_{lm}(\theta,\varphi),$
(17)

and under the $\mathcal{PT}$-operator, we have

$\displaystyle\mathcal{PT}\left(Y_{lm}(\theta,\varphi)\right)=Y^{*}_{lm}(\pi-\theta,\varphi+\pi)=(-)^{l}Y^{*}_{lm}(\theta,\varphi).$
(18)

This results will be used below.

##  III $\mathcal{PT}$ and Rotations

Given any $f=f(r,\theta,\varphi)$, we can define

$\displaystyle L_{fi}=e^{f}L_{i}e^{-f}.$ (19)

In general $L_{fi}$ is a non-Hermitian operator, however

$\displaystyle\left[L_{fx},L_{fy}\right]=iL_{fz},\qquad\left[L_{fz},L_{fx}\right]=iL_{fy},\qquad\left[L_{fy},L_{fz}\right]=iL_{fx},$
(20)

that we can identify as the $SO(3)$-Lie algebra commutation relations.
Considering $L_{f}^{2}=L_{fx}^{2}+L_{fy}^{2}+L_{fz}^{2}$ and Eq. (20), we have

$\displaystyle\left[L_{f}^{2},L_{fi}\right]=0,$ (21)

therefore, we have the same algebra as the one satisfied by $L_{i}.$ Besides

$\displaystyle
L_{f}^{2}Y_{flm}(\theta,\varphi)=l(l+1)Y_{flm}(\theta,\varphi),\qquad
Y_{flm}(\theta,\varphi)=e^{f}Y_{lm}(\theta,\varphi),$ (22)

that will be called $\mathcal{PT}$-spherical harmonics. In this case, the
$\mathcal{PT}$-inner product is given by

$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$ $\displaystyle\int
d\Omega\mathcal{PT}\left(Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)\right)Y_{flm}(\theta,\varphi).$
(23)

Under a $\mathcal{PT}$-transformation, we have

$\displaystyle\mathcal{PT}(Y_{flm})$ $\displaystyle=$ $\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)}(-)^{l}Y^{*}_{lm}(\theta,\varphi).$ (24)

therefore we can write

$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$ $\displaystyle(-)^{l}\int d\Omega
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi).$

It becomes clear that, under this inner product not any function $f$ allows
the set $Y_{flm}$ to be an orthogonal set. However, if the following condition
is fulfilled

$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=\lambda,\qquad\lambda={\rm
const}$ (25)

then we have

$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}\lambda\delta_{l^{\prime}l}\delta_{m^{\prime}m}.$ (26)

In this way, the spherical harmonics $Y_{flm}(\theta,\varphi)$ are orthogonal
under the $\mathcal{PT}$-inner product only if Eq. (25) is satisfy. It is
worthy to mention that due to the parity of the wave functions, in some
$\mathcal{PT}$-symmetry systems the following orthogonality relations

$\displaystyle\langle\phi_{m}|\phi_{n}\rangle=(-1)^{n}\delta_{m,n},$ (27)

may be obtained bender-0:gnus .

##  IV Completeness Relation

Using the $\mathcal{PT}$-spherical harmonics $Y_{flm}(\theta,\varphi)$, we can
have the expansion

$\displaystyle F(\theta,\varphi)=\sum_{l\geq
0}\sum_{m=-l}^{l}a_{lm}Y_{flm}(\theta,\varphi).$ (28)

Appeling to the orthonormality relations Eq. (26), we find

$\displaystyle
a_{lm}=\frac{(-)^{l}}{\lambda}<Y_{flm}(\theta,\varphi)|F(\theta,\varphi)>_{f}=\frac{(-)^{l}}{\lambda}\int
d\Omega\mathcal{PT}\left(Y_{flm}(\theta,\varphi)\right)F(\theta,\varphi).$
(29)

substituting this result into Eq.(28), we obtain

$\displaystyle F(\theta,\varphi)$ $\displaystyle=$ $\displaystyle\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{(-)^{l}}{\lambda}\int
d\Omega^{\prime}\mathcal{PT}\left(Y_{flm}(\theta^{\prime},\varphi^{\prime})\right)F(\theta^{\prime},\varphi^{\prime})Y_{flm}(\theta,\varphi)$
$\displaystyle=$ $\displaystyle\int
d\Omega^{\prime}F(\theta^{\prime},\varphi^{\prime})\left[\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{(-)^{l}}{\lambda}\mathcal{PT}\left(Y_{flm}(\theta^{\prime},\varphi^{\prime})\right)Y_{flm}(\theta,\varphi)\right]$
$\displaystyle=$ $\displaystyle\int
d\Omega^{\prime}F(\theta^{\prime},\varphi^{\prime})\left[\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{e^{f^{*}(r,\pi-\theta^{\prime},\varphi^{\prime}+\pi)+f(r,\theta,\varphi)}}{\lambda}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi)\right],$

therefore

$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}\frac{e^{f^{*}(r,\pi-\theta^{\prime},\varphi^{\prime}+\pi)+f(r,\theta,\varphi)}}{\lambda}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(30)

This is the completeness relation for the $\mathcal{PT}$-spherical harmonics.
A similar completeness relation is found for different systems in
$\mathcal{PT}$-quantum mechanics bender-0:gnus .

###  IV.1 Examples on Completeness

In this subsection we will see some examples of functions that satisfy Eq.
(25).

Let us suppose that $a$ is a real constant, then we can define

$\displaystyle f(r,\theta,\varphi)=a\theta$ (31)

therefore

$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{a(\pi-\theta)+a\theta}=e^{a\pi},$
(32)

where

$\displaystyle\lambda=e^{a\pi}.$ (33)

In this case, the orthonormality relations are given by

$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}e^{a\pi}\delta_{l^{\prime}l}\delta_{m^{\prime}m}$ (34)

and the completeness relation is given by

$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{a(\theta-\theta^{\prime})}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(35)

Now consider

$\displaystyle f(r,\theta,\varphi)=ai\sin\theta,$ (36)

then we obtain

$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{-ai\sin\theta+ia\sin\theta}=1=\lambda$
(37)

with the orthonormality relations

$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}\delta_{l^{\prime}l}\delta_{m^{\prime}m}.$ (38)

In this case the completeness relation is given by

$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{ai(\sin\theta-\sin\theta^{\prime})}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(39)

Using

$\displaystyle f(r,\theta,\varphi)=a\cos\theta,$ (40)

we have

$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{-a\cos\theta+a\cos\theta}=\lambda=1$
(41)

in this case the orthogonality relations are given by

$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}\delta_{l^{\prime}l}\delta_{m^{\prime}m},$ (42)

and the completeness relation is given by

$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{a(\cos\theta-\cos\theta^{\prime})}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(43)

With the function

$\displaystyle f(r,\theta,\varphi)=ai\varphi$ (44)

we obtain

$\displaystyle
e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}=e^{-ai(\varphi+\pi)+ia\varphi}=e^{-ia\pi}=\lambda,$
(45)

with the orthogonality relation given by

$\displaystyle<Y_{fl^{\prime}m^{\prime}}(\theta,\varphi)|Y_{flm}(\theta,\varphi)>_{f}$
$\displaystyle=$
$\displaystyle(-)^{l}e^{-ia\pi}\delta_{l^{\prime}l}\delta_{m^{\prime}m},$ (46)

and the completeness relation

$\displaystyle\delta(\phi-\phi^{\prime})\delta(\cos\theta-\cos\theta^{\prime})=\sum_{l\geq
0}\sum_{m=-l}^{l}e^{ai(\varphi-\varphi^{\prime})}Y^{*}_{lm}(\theta^{\prime}.\varphi^{\prime})Y_{lm}(\theta,\varphi).$
(47)

Considering the above examples, it is clear that many functions satisfy Eq.
(25).

##  V Symmetry Transformations

Consider $A,B,C$ that satisfy the following commutation relations

$\displaystyle[A,B]=C.$ (48)

Transforming the $A,B,C$ operators, we obtain
$A_{f}=e^{f}Ae^{-f},B_{f}=e^{f}Be^{-f}$ y $C_{f}=e^{f}Ce^{-f},$ then

$\displaystyle[A_{f},B_{f}]=C_{f}.$ (49)

We know that $[L_{i},x_{j}]=i\epsilon_{ijk}x_{k}$. If we consider the
transformation $x_{fi}=e^{f}x_{i}e^{-f}=x_{i}$, we arrive to

$\displaystyle[L_{fi},x_{j}]=i\epsilon_{ijk}x_{k},$ (50)

therefore the operators $L_{fi}$ generate infinitesimal rotations in the space
$x_{i}$. However, as $p_{i}=-i\frac{\partial}{\partial x^{i}},$ then

$\displaystyle[L_{fi},p_{j}]\not=i\epsilon_{ijk}p_{k},$ (51)

and we say that the operators $L_{fi}$ does not generate infinitesimal
rotations in the space $p_{i}$. Now, if $p_{fi}$ is given by
$p_{fi}=e^{f}p_{i}e^{-f}$, then we have

$\displaystyle[L_{fi},p_{fj}]=i\epsilon_{ijk}p_{fk}.$ (52)

Note that

$\displaystyle\vec{p}_{f}=e^{f}\vec{p}e^{-f}=\vec{p}+i\vec{\nabla}f.$ (53)

this operator was studied by Dirac in his seminal book Dirac:gnus . If the
Hamiltonian operator is given by

$\displaystyle H=\frac{\vec{p}^{\;2}}{2m}+V(r),$ (54)

then

$\displaystyle[L_{i},H]=0.$ (55)

Defining $H_{f}$ by

$\displaystyle H_{f}=e^{f}He^{-f},$ (56)

we have that, in general

$\displaystyle[L_{i},H_{f}]\not=0,$ (57)

therefore the angular momentum $L_{i}$ is not a conserved quantity for
Hamiltonians of the form $H_{f}.$ However

$\displaystyle[L_{fi},H_{f}]=0,$ (58)

then the modified angular momentum $L_{fi}$ is conserved. Note that the
Hamiltonian $H_{f}$ is given by

$\displaystyle
H_{f}=\frac{\vec{p}_{f}^{\;2}}{2m}+V(r)=\frac{m}{2}\left(\vec{p}+i\vec{\nabla}f\right)^{2}+V(r)$
(59)

In the next section we will consider one important example.

##  VI The Central Problem

Consider the Hamiltonian

$\displaystyle H=\frac{m}{2}\vec{p}^{\;2}+V(x,y,z),$ (60)

then

$\displaystyle H_{f}$ $\displaystyle=$ $\displaystyle
e^{f}He^{-f}=\frac{m}{2}\left(\vec{p}+i\vec{\nabla}f\right)^{2}+V(x,y,z)$ (61)
$\displaystyle=$
$\displaystyle\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}+(\nabla^{2}f)-\left(\vec{\nabla}f\right)^{2}\right)+V(x,y,z),$

that is a non-Hermitian Hamiltonian. If the potential is given by

$\displaystyle
V(x,y,z)=-\frac{m}{2}\left(\nabla^{2}f-\left(\vec{\nabla}f\right)^{2}\right)+U(x,y,z),$
(62)

then we can write

$\displaystyle H_{f}$ $\displaystyle=$
$\displaystyle\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}\right)+U(x,y,z).$
(63)

This kind of Hamiltonians naturally arise in some statistical models
reichl:gnus .

Note that if $\psi$ is an eigenfunction in the equation

$\displaystyle H\psi=E\psi$ (64)

then we can define the $f$-states $\psi_{f}=e^{f}\psi$ that satisfy

$\displaystyle H_{f}\psi_{f}=E\psi_{f}.$ (65)

It follows that although $H_{f}$ is a non-Hermitian operator it does has a
real spectrum.

As an example, let us consider the central potential problem $V(r)$ whose
Schrodinger equation is given by landau:gnus

$\displaystyle H\psi=\left(\frac{m}{2}\vec{p}^{\;2}+V(r)\right)\psi=E\psi$
(66)

and whose solutions

$\displaystyle\psi_{E}(r,\theta,\varphi)=\phi_{E}(r)Y_{lm}(\theta,\varphi)$
(67)

satisfy the orthogonality relations

$\displaystyle<\psi_{E^{\prime}}(r,\theta,\varphi)|\psi_{E}(r,\theta,\varphi)>=\int
drr^{2}d\Omega\psi^{*}_{E^{\prime}}(r,\theta,\varphi)\psi_{E}(r,\theta,\varphi)=\delta_{EE^{\prime}}.$
(68)

Then the solutions of the equation

$\displaystyle H_{f}\psi_{f}$ $\displaystyle=$
$\displaystyle\left[\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}+\nabla^{2}f-\left(\vec{\nabla}f\right)^{2}\right)+V(r)\right]\psi_{f}$
(69) $\displaystyle=$ $\displaystyle E\psi_{f}$

are given by

$\displaystyle\psi_{Ef}(r,\theta,\varphi)=e^{f(r,\theta,\varphi)}\phi_{E}(r)Y_{lm}(\theta,\varphi).$
(70)

The $\mathcal{PT}$-inner product for the $\psi_{f}$-states is given by

$\displaystyle<\psi_{E^{\prime}f}(r,\theta,\varphi)|\psi_{Ef}(r,\theta,\varphi)>_{f}=\int
drd\Omega\mathcal{PT}\left(\psi_{E^{\prime}f}(r,\theta,\varphi)\right)\psi_{Ef}(r,\theta,\varphi)$
$\displaystyle=$ $\displaystyle\int
drd\Omega(-)^{l}e^{f^{*}(r,\pi-\theta,\varphi+\pi)+f(r,\theta,\varphi)}\phi_{E^{\prime}}^{*}(r)\phi_{E}(r)Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi).$

Besides, if $f(r,\theta,\varphi)$ satisfies Eq. (25) and considering Eq. (68),
we have

$\displaystyle<\psi_{E^{\prime}f}(r,\theta,\varphi)|\psi_{Ef}(r,\theta,\varphi)>_{f}$
$\displaystyle=$ $\displaystyle(-)^{l}\lambda\int
drd\Omega\phi_{E^{\prime}}^{*}(r)\phi_{E}(r)Y^{*}_{l^{\prime}m^{\prime}}(\theta,\varphi)Y_{lm}(\theta,\varphi)$
(71) $\displaystyle=$ $\displaystyle\lambda(-)^{l}\delta_{EE^{\prime}}.$

Given that $[L_{i},H_{f}]\not=0$, it follows that $L_{i}$ is not a conserved
quantity. However, as

$\displaystyle[L_{fi},H_{f}]=0,$ (72)

then $L_{fi}$ is conserved.

##  VII The Hydrogen Atom

In the case of the Hydrogen atom, we have the potential

$\displaystyle V(r)=-\frac{Ze^{2}}{r},$ (73)

where the solutions are given by

$\displaystyle\psi_{Nlm}(\rho,\theta,\varphi)$ $\displaystyle=$
$\displaystyle\frac{2}{N^{2}}\sqrt{\frac{Z^{3}}{a_{RB}^{3}}\frac{(N-l-1)!}{(N+l)!}}\rho^{l}L_{N-(l+1)}^{2l+1}(\rho)e^{-\frac{\rho}{2}}Y_{lm}(\theta,\varphi),$
$\displaystyle E_{N}$ $\displaystyle=$
$\displaystyle-\frac{Ze^{2}}{a_{RB}N^{2}},\qquad N=n+l+1,\qquad
n=0,1,2\cdots,$

where $a_{RB}$ is the Bohr radius and

$\displaystyle\rho=\alpha
r,\qquad\qquad\alpha=2\sqrt{\frac{-2mE}{\hbar^{2}}}.$ (74)

Taking into account Eq. (69), we have the equation

$\displaystyle H_{f}\psi_{f}$ $\displaystyle=$
$\displaystyle\left[\frac{m}{2}\left(\vec{p}^{\;2}+2i\vec{\nabla}f\cdot\vec{p}+\nabla^{2}f-\left(\vec{\nabla}f\right)^{2}\right)-\frac{Ze^{2}}{r}\right]\psi_{f}=E\psi_{f}$
(75)

whose solutions are given by

$\displaystyle\psi_{fNlm}(\rho,\theta,\varphi)$ $\displaystyle=$
$\displaystyle e^{f(r,\theta,\varphi)}\psi_{Nlm}(\rho,\theta,\varphi)$ (76)

this are orthogonal functions if equation (25) is satisfied. A remarkable fact
is that $L_{fi}$ is a conserved quantity. In the conventional Hydrogen atom,
the Runge-Lenz vector is also conserved landau:gnus

$\displaystyle
R_{i}=\frac{1}{2}\left(\vec{L}\times\vec{p}-\vec{p}\times\vec{L}+\frac{Ze^{2}\vec{r}}{r}\right)_{i}.$
(77)

In the case of the Hamiltonian $H_{f}$, we have

$\displaystyle[R_{fi},H_{f}]=0,$ (78)

and we can say that the transformed non-Hermitian Runge-Lenz vector is
conserved.
Note that in this case, we have obtained a set of conserved quantities
$L_{fi},L_{f}^{2},R_{fi}$, that are the non-Hermitian generators of the
$SO(4)$ algebra.

##  VIII Summary

In this work we have constructed a set of non-Hermitian operators $L_{fi}$
that satisfy the commutation relations of the $SO(3)$-Lie algebra. We have
shown that this operators generate rotations in the configuration space and
not in the conventional momentum space but in a modified non-Hermitian
momentum space $\vec{p}_{f}=\vec{p}+i\vec{\nabla{f}}.$ It is worthy to mention
that this operator was originally considered by Dirac in his seminal book.
Besides, the $L_{if}$ generators are related with a new type of spherical
harmonics that result to be $\mathcal{PT}$-orthonormal. Additionally, we have
shown that this quantities are conserved for mechanical systems described by a
central potential Hamiltonian with an additional complex term. As a particular
case, we have obtained the solutions of the corresponding
$\mathcal{PT}$-Hydrogen atom that includes a complex term, and we have found
that a non-Hermitian Runge-Lenz vector is a conserved quantity. Considering
this case, one remarkable result is that, as we have obtained the non-
Hermitian generators of the $SO(3)$-Lie algebra and also a non-Hermitian
Runge-Lenz vector, then we have the non-Hermitian generators of the
$SO(4)$-Lie algebra. In a future work we will study the non-Hermitian
generators corresponding to others symmetry groups.

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