Acta Polytechnica


doi:10.14311/AP.2014.54.0122
Acta Polytechnica 54(2):122–123, 2014 © Czech Technical University in Prague, 2014

available online at http://ojs.cvut.cz/ojs/index.php/ap

THE FLOQUET METHOD FOR PT-SYMMETRIC PERIODIC
POTENTIALS

H. F. Jones

Physics Department, Imperial College, London SW7 2BZ, UK
correspondence: h.f.jones@imperial.ac.uk

Abstract. By the general theory of PT -symmetric quantum systems, their energy levels are either
real or occur in complex-conjugate pairs, which implies that the secular equation must be real. However,
for periodic potentials it is by no means clear that the secular equation arising in the Floquet method
is indeed real, since it involves two linearly independent solutions of the Schrödinger equation. In this
brief note we elucidate how that reality can be established.

Keywords: band structure, PT symmetry, Floquet method.

The study of systems governed by Hamiltonians
for which the standard requirement of Hermiticity
is replaced by that of PT -symmetry has undergone
significant development in recent years [1–6]. Pro-
vided that the symmetry is not broken, that is, that
the energy eigenfunctions respect the symmetry of the
Hamiltonian, the energy eigenvalues are guaranteed to
be real. In the case where the symmetry is broken en-
ergy levels may instead appear as complex-conjugate
pairs. This phenomenon is particularly interesting for
the case of periodic PT -symmetric potentials, where
unusual band structures may occur [7, 8].
An important physical realization of such systems

arises in classical optics, because of the formal simi-
larity of the time-dependent Schrödinger equation to
the paraxial equation for the propagation of electro-
magnetic waves. This equation takes the form [9]

i
∂ψ

∂z
= −

(
∂2

∂x2
+ V (x)

)
ψ, (1)

where ψ(x,z) represents the envelope function of the
amplitude of the electric field and z is a scaled prop-
agation distance. The optical potential V (x) is pro-
portional to the variation in the refractive index of
the material through which the wave is passing. In
optics this potential may well be complex, with its
imaginary part representing either loss or gain. If loss
and gain are balanced in a PT -symmetric way, so
that V ∗(x) = V (−x). we have the situation described
above. Optical systems of this type have a number of
very interesting properties [9–14], particularly when
they are periodic.

In such a case the potential V (x), whose period we
can take as π, without loss of generality, satisfies the
two conditions V ∗(−x) = V (x) = V (x + π). For peri-
odic potentials we are interested in finding the Bloch
solutions, which are solutions of the time-independent
Schrödinger equation

−
(
∂2

∂x2
+ V (x)

)
ψk(x) = Eψk(x) (2)

with the periodicity property ψk(x + π) = eikπψk(x).
The standard way of obtaining such solutions is

the Floquet method, whereby ψk(x) is expressed in
terms of two linearly-independent solutions, u1(x) and
u2(x), of Eq. (2), with initial conditions

u1(0) = 1, u′1(0) = 0,
u2(0) = 0, u′2(0) = 1. (3)

Then ψk(x) is written as the superposition

ψk(x) = cku1(x) + dku2(x). (4)

Imposing the conditions ψk(π) = eikπψ(0) and
ψ′k(π) = e

ikπψ′(0) and exploiting the invariance of
the Wronskian W(u1,u2) one arrives at the secular
equation

cos kπ = ∆ ≡
1
2

(u1(π) + u′2(π)) . (5)

In the Hermitian situation both u1(π) and u2(π) are
real, and the equation for E has real solutions (bands)
when |∆| ≤ 1. However, in the non-Hermitian, PT -
symmetric, situation it is not at all obvious that ∆
is real, since that implies a relation between u1(π)
and u′2(π), even though u1(x) and u2(x) are linearly
independent solutions of Eq. (2). It is that problem
that we wish to address in the present note. In fact
we will show that u′2(π) = u∗1(π).

The clue to relating u1(π) and u2(π) comes from
considering a half-period shift, namely x = z + π/2.
We write ϕ(z) = ψ(z + π/2) and U(z) = V (z + π/2).
Then ϕ(z) satisfies the Schrödinger equation

−
(
∂2

∂z2
+ U(z)

)
ϕk(z) = Eϕk(z). (6)

The crucial point is that because of the periodicity
and PT -symmetry of V (x) the new potential U(z) is
also PT -symmetric. Thus U(−z) = V (−z + π/2) =
V (−z −π/2) = V ∗(z + π/2) = U∗(z).

122

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http://ojs.cvut.cz/ojs/index.php/ap


vol. 54 no. 2/2014 The Floquet Method for PT-symmetric Periodic Potentials

Now we can express the Floquet functions u1(x),
u2(x) in terms of Floquet functions v1(z), v2(z) of the
transformed equation (6), satisfying

v1(0) = 1, v′1(0) = 0,
v2(0) = 0, v′2(0) = 1. (7)

It is easily seen that the relation is

u1(x) = v′2(−π/2)v1(z) −v
′
1(−π/2)v2(z),

u2(x) = −v2(−π/2)v1(z) + v1(−π/2)v2(z), (8)

in order to satisfy the initial conditions on u1(x),
u2(x). So

u1(π) = v′2(−π/2)v1(π/2) −v
′
1(−π/2)v2(π/2),

u′1(π) = v
′
2(−π/2)v

′
1(π/2) −v

′
1(−π/2)v

′
2(π/2),

u2(π) = −v2(−π/2)v1(π/2) + v1(−π/2)v2(π/2),
u′2(π) = −v2(−π/2)v

′
1(π/2) + v1(−π/2)v

′
2(π/2). (9)

But, because of the PT -symmetry of Eq. (6) and the
initial conditions satisfied by v1(z), v2(z),

v1(−π/2) = (v1(π/2))∗,
v′1(−π/2) = −(v

′
1(π/2))

∗,

v2(−π/2) = −(v2(π/2))∗,
v′2(−π/2) = (v

′
2(π/2))

∗. (10)

Hence, indeed, u1(π) = (u′2(π))∗, so that ∆ in Eq. (5)
is real and the energy eigenvalues of the Bloch wave-
functions are either real or occur in complex conjugate
pairs. From Eq. (10) we also see that u′1(π) and u2(π)
are real. The statement u1(π) = (u′2(π))∗ is in fact
the PT -generalization of the relation u1(π) = u′2(π)
implied without proof by Eq. (20.3.10) of Ref. [16] for
the Hermitian case of the Mathieu equation, where
V (x) = cos(2x).
If we wish, we may express everything in terms of

u1, u2 because from Eq. (8)

u1(π/2) = v′2(−π/2),
u′1(π/2) = −v

′
1(−π/2),

u2(π/2) = −v2(−π/2),
u′2(π/2) = v1(−π/2). (11)

Hence

u1(π) = (u′2(π))
∗

= u1(π/2)(u′2(π/2))
∗ + u′1(π/2)(u2(π/2))

∗, (12)

which is the PT -generalization of a relation implied by
Eq. (20.3.11) of Ref. [16] after the use of the invariance
of the Wronskian.
Similarly

u′1(π) = 2Re (u
∗
1(π/2)u

′
1(π/2)) ,

u2(π) = 2Re (u∗2(π/2)u
′
2(π/2)) . (13)

To conclude, we have shown that the secular equa-
tion for the band structure of PT -symmetric periodic
potentials is indeed real, even though in the Floquet
method the discriminant involves the two ostensibly
independent functions u1(x) and u2(x). The crucial
point is that for such potentials there is also a PT sym-
metry about the midpoint of the Brillouin zone. The
proof involves expressing u1(x) and u2(x) in terms
of shifted functions v1(x) and v2(x), and shows that
u1(π) and u′2(π) are actually complex conjugates of
each other. The proof incidentally casts light on cer-
tain relations that hold for real symmetric potentials,
such as cos (2x).

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123

http://dx.doi.org/10.1103/PhysRevLett.80.5243
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	Acta Polytechnica 54(2):122–123, 2014
	References