Acta Polytechnica


Acta Polytechnica 53(3):280–282, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

GENERALIZED CONTINUITY EQUATION FOR
QUASI-HERMITIAN HAMILTONIANS

Amine B. Hammou∗

USTO–Mohamed Boudiaf, Département de Physique, BP 1505 El M’naouer, Oran, Algeria
∗ corresponding author: hbamine@gmail.com

Abstract. The continuity relation is generalized to quasi-Hermitian one-dimensional Hamiltonians. As
an application we show that the reflection and transmission coefficients computed with the generalized
current obey the conventional unitarity relation for the continuous double delta function potential.

Keywords: complex scattering potential, metric operator, quasi-Hermitian, PT -symmetry.

1. Introduction
In [1], scattering from a discrete quasi-Hermitian delta
function potential was studied and a generalized con-
tinuity relation was obtained in the physical Hilbert
space Hphys. A generalized probability current density
was defined. Using a quasi-Hermitian toy model of
discrete delta function potential, it was shown that
the reflection and transmission coefficients computed
using this current obey the unitarity relation.

In this paper we will review the construction of the
generalized continuity relation of [1] and then apply
it to the continuous double delta function potential
where a metric was obtained to first order in the imag-
inary part of the potential strength [2]. We will then
compute the reflection and transmission coefficients in
the physical Hilbert space Hphys and show that they
indeed obey the unitarity relation. This result comes
as a surprise since, although the potential is local,
the metric is highly non local and so the existence
of asymptotic states is questionable [3–5]. This is as
far as I know the first example of a quasi-Hermitian
Hamiltonian with a non-diagonal metric that possesses
a conserved probability current density. Recently it
has been found [6] that, for the non-Hermitian com-
plex PT-symmetric version of the Scarf II potential,
unitarity is obeyed for a particular choice of the pa-
rameters.

2. The continuity relation in Hphys
A Hamiltonian H that obeys

H† = ηHη−1 (2.1)

is said to be quasi-Hermitian [7]. Where η is a posi-
tive definite metric operator. By an appropriate mod-
ification of the inner product on the Hilbert space,
quasi-Hermitian operators can be made Hermitian [8].
The inner product of the new Hilbert space is given by
〈·|·〉η := 〈·|η·〉, where 〈·|·〉 is the inner product which
defines the original Hilbert space H. The new Hilbert
space endowed with the inner product 〈.|.〉η is recog-
nized as the physical Hilbert space Hphys in which H

acts as a Hermitian operator [8]. A generalized con-
tinuity equation in the physical Hilbert space Hphys
was defined in [1] as

∂

∂t
ρη(x) +

∂

∂x
jη(x) = 0 (2.2)

in the x-representation, where

ρη(x) =
∫
dy ψ∗(y)η(y,x)ψ(x) (2.3)

is the probability density. The probability current
density is given by

jη(x) = i
∫
dy
(
ψ∗(y)η(y,x)

∂ψ(x)
∂x

−
∂
(
ψ∗(y)η(y,x)

)
∂x

ψ(x)
)
. (2.4)

3. The continuous double delta
function potential

In this section, we will test the continuity relation
obtained from the generalized current (2.4) in the
continuous case. There are very few cases in the
literature where the metric was computed [2, 9–11].
Let us consider the double delta function potential
given by

V (x) = iλ
(
δ(x + a) −δ(x−a)

)
(3.1)

with the scattering wave function, in the situation
where the plane wave comes in from the left and is
either reflected or transmitted at the delta function
potential, is of the form{

ψ<(x) = eikx + re−ikx if x �−a,
ψ>(x) = teikx if x � a,

(3.2)

where ψ<(x) and ψ>(x) are respectively the incoming
and the outgoing wave functions. The metric was
given in [2] as a perturbative series in λ, to first order
it is given by

η(x,y) = δ(x−y) + η(1)(x,y) + O(λ2) (3.3)

280

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vol. 53 no. 3/2013 Generalized Continuity Equation for Quasi-Hermitian Hamiltonians

where

η(1)(x,y) = i
λ

2
sign(x−y)

(
θ(x + y + 2a)
−θ(x + y − 2a)

)
. (3.4)

Let us apply our formula for the generalized continu-
ity relation introduced in (2.4) to this case. Therefore
we should compute the functions

χ(x) = ψ(x) +
∫
η(1)(x,y)ψ(y) dy. (3.5)

A close inspection of the metric shows that for both
x �−a and x � a with −(x + 2a) < y < −(x− 2a)
we have η(1)(x,y) 6= 0; otherwise it is zero.

We therefore obtain for x �−a

χ<(x) = ψ<(x) − i
λ

2

∫ −(x−2a)
−(x+2a)

ψ<(y) dy

= eikx +
(
r − i

λ

k
t sin 2ak

)
e−ikx (3.6)

and for x � a we find

χ>(x) = ψ>(x) + i
λ

2

∫ −(x−2a)
−(x+2a)

ψ<(y) dy

=
(
t + i

λ

k
r sin 2ak

)
eikx + i

λ

k
sin 2ak e−ikx. (3.7)

The second term in this function can be interpreted
as a reflected wave function traveling from right to
left and therefore it has no physical meaning [4, 5].
This causes no problem in our approach since χ(x) has
no physical interpretation. The generalized reflection
coefficient is easily computed using its definition in
terms of reflected and incident generalized currents
that can be worked out using (2.4), and is given by

R = |r|2 +
iλ

4
t∗r sin 2ak. (3.8)

Similarly the generalized transmission coefficient can
be computed using the transmitted and incident cur-
rents, and is given by

T = |t|2 −
iλ

4
r∗t sin 2ak (3.9)

leading to

R + T = |r|2 + |t|2 +
iλ

4
(t∗r −r∗t) sin 2ak. (3.10)

Note that if we use the complex conjugate of rela-
tion (2.4) to compute the generalized reflection and
transmission coefficient we obtain the same result.
This is easily seen from the fact that the reflection
and transmission coefficients are real and hence the
currents are real too. The first term in both R and T
are obviously real and the second terms are also real
because of the relation t∗r = −r∗t, which is a general
feature of PT -symmetric models [12].

For the model at hand (3.1), the reflection and
transmission amplitudes can be easily worked out,
leading to

r = it
λ

k

(
1 −

λ

2k

)
sin 2ak (3.11)

and
t =

1
1 + i λ22k2 e

2iak sin 2ak
. (3.12)

It is worth noting that if we do usual quantum me-
chanical scattering (η(x,y) = δ(x,y)) by using the
PT-symmetric double delta function potential we will
not have unitarity

|r|2 + |t|2 =
1 + λ

2

k2
(1 − λ2k )

2 sin2 2ak
1 − λ2

k2
(1 − λ24k2 ) sin

2 2ak
. (3.13)

By using the expressions (3.11) and (3.12) we obtain

t∗r = −r∗t = i |t|2
λ

k

(
1 −

λ

2k

)
sin 2ak (3.14)

and after a straightforward computation we can show
that

R + T = 1. (3.15)

We have obtained a unitarity relation in the Hilbert
space Hphys. Note that this result is exact in λ, as
we have used the full expressions of the transmission
(3.12) and reflection amplitudes (3.11) in terms of λ
(to all orders). This is a puzzling result that we have
yet to understand. It is clear that if one uses the
expressions for t and r to first order in λ, the relation
(3.15) holds trivially. A possible explanation is that
the higher orders in λ of the metric somehow will not
change the result of the continuity relation. For this
to be true, the higher order corrections should not
contribute to the functions χ<(x) and χ>(x).

4. Conclusion
We have been able to show that the generalized current
defined with respect to the metric η is conserved

(Jη)in − (Jη)out = 0 (4.1)

and the corresponding generalized reflection and
transmission coefficients obey the unitarity relation for
the quasi-Hermitian continuous double delta function
potential.

Although the metric used in the continuous case is
perturbative in the strength of the imaginary part of
the potential λ, the results of the unitarity relation
were obtained using the expression for the amplitudes
to all orders in λ. And so this arises a question that
deserves further investigation. To this end, one has
to work out the metric to a higher order in λ. Finally
an interesting and urgent question is to understand
the physical implementation of the metric η in these
scattering processes.

281



Amine B. Hammou Acta Polytechnica

Acknowledgements
I would like to thank the Abdus Salam International Center
for Theoretical Physics ICTP, Trieste, Italy for Hospitality
and support. I would also like to thank M. Znojil for help-
ful discussions and for organizing the conference “Analytic
and Algebraic methods in Physics X”, Prague.

References
[1] Amine B. Hammou, J. Phys. A: Math. Theor. 45
(2012) 215310.

[2] H. Mehri-Dehnavi, A. Mostafazadeh, A. Batal, J.
Phys. A43, 145301 (2010).

[3] M. Znojil, Phys. Rev. D78, 025026 (2008).
[4] H. F. Jones, Phys. Rev. D76, 125003 (2007).

[5] H. F. Jones, Phys. Rev. D78, 065032 (2008).
[6] Z. Ahmed, arXiv:1207.6896.
[7] F. G. Scholtz, H. B. Geyer and F. J. W. Hahne, Ann.
Phys. 213, 74 (1992).

[8] A, Mostafazadeh, Int. J. Geom. Meth. Mod. Phys. 7,
1191 (2010).

[9] A. Mostafazadeh, J. Math. Phys. 46, 102108, (2005).
[10] A. Mostafazadeh, J. Phys. A: Math. Gen. 39, 13495
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[11] A. Mostafazadeh, J. Phys. A39, 10171 (2006).
[12] F. Cannata, J.-P. Dedonder and A. Ventura, Ann.
Phys. 322, 397 (2007).

282

http://arxiv.org/abs/1207.6896

	Acta Polytechnica 53(3):280–282, 2013
	1 Introduction
	2 The continuity relation in H_phys
	3 The continuous double delta function potential
	4 Conclusion
	Acknowledgements
	References