A schematic model of scattering in PT-symmetric Quantum Mechanics
arXiv:0704.0214v1  [quant-ph]  2 Apr 2007
A schematic model of scattering
in PT −symmetric Quantum Mechanics
Miloslav Znojil
Ústav jaderné fyziky AV ČR, 250 68 Řež, Czech Republic
e-mail: znojil@ujf.cas.cz
Abstract
One-dimensional scattering problem admitting a complex, PT −symmetric short-
range potential V (x) is considered. Using a Runge-Kutta-discretized version of
Schrödinger equation we derive the formulae for the reflection and transmission coef-
ficients and emphasize that the only innovation emerges in fact via a complexification
of one of the potential-characterizing parameters.
PACS 03.65.Ge
http://arxiv.org/abs/0704.0214v1
1 Introduction
Standard textbooks describe the stationary one-dimensional motion of a quantum
particle in a real potential well V (x) by the ordinary differential Schrödinger equation
+ V (x)
ψ(x) = E ψ(x) , x ∈ (−∞,∞) (1)
which may be considered and solved in the bound-state regime at E < V (∞) ≤ +∞
or in the scattering regime with, say, E = κ2 > V (∞) = 0. In this way one either
employs the boundary conditions ψ(±∞) = 0 and determines the spectrum of bound
states or, alternatively, switches to the different boundary conditions, say,
ψ(x) =
Aeiκx +B e−iκx , x≪ −1 ,
C eiκx , x≫ 1 .
Under the conventional choice of A = 1 the latter problem specifies the reflection
and transmission coefficients B and C, respectively [1].
The conventional approach to the quantum bound state problem has recently
been, fairly unexpectedly, generalized to many unconventional and manifestly non-
Hermitian Hamiltonians H 6= H† which are merely quasi-Hermitian, i.e., which are
Hermitian only in the sense of an identity H† = ΘH Θ−1 which contains a nontrivial
“metric” operator Θ = Θ† > 0 as introduced, e.g., in ref. [2]. The key ideas and
sources of the latter new development in Quantum Mechanics incorporate the so
called PT −symmetry of the Hamiltonians and have been summarized in the very
fresh review by Carl Bender [3]. This text may be complemented by a sample [4] of
the dedicated conference proceedings.
In this context we intend to pay attention to a very simple PT −symmetric scat-
tering model where
V (x) = Z(x) + i Y (x) , Z(−x) = Z(x) = real , Y (−x) = −Y (x) = real
and where the ordinary differential equation (1) is replaced by its Runge-Kutta-
discretized, difference-equation representation
ψ(xk−1)− 2ψ(xk) + ψ(xk+1)
+ V (xk)ψ(xk) = E ψ(xk) (3)
xk = k h , h > 0 , k = 0,±1, . . .
as employed, in the context of the bound-state problem, in refs. [5].
2 Runge-Kutta scattering
Once we assume, for the sake of simplicity, that the potential in eq. (3) vanishes
beyond certain distance from the origin,
V (x±j) = 0 j =M,M + 1, . . . ,
we may abbreviate ψk = ψ(xk), Vk = h
2V (xk) and 2 cosϕ = 2− h
2E in eq. (3),
−ψk−1 + (2 cosϕ+ Vk) ψk − ψk+1 = 0 . (4)
In the region of |k| ≥ M with vanishing potential Vk = 0 the two independent
solutions of our difference Schrödinger eq. (4) are easily found, via a suitable ansatz,
as elementary functions of the new “energy” variable ϕ,
ψk = const · ̺
k =⇒ ̺ = ̺± = exp(±i ϕ) .
This enables us to replace the standard boundary conditions (2) by their discrete
scattering version
ψ(xm) =
Aeimϕ +B e−imϕ , m ≤ −(M − 1) ,
C eimϕ , m ≥ M − 1
with a conventional choice of A = 1.
Two comments may be added here. Firstly, one notices that the condition of the
reality of the new energy variable ϕ imposes the constraint upon the original energy
itself, −2 ≤ 2 − h2E ≤ 2, i.e., E ∈ (0, 4/h2). At any finite choice of the lattice
step h > 0 this inequality is intuitively reminiscent of the spectra in relativistic
quantum systems. Via an explicit display of the higher O(h4) corrections in eq. (3),
this connection has been given a more quantitative interpretation in ref. [6].
The second eligible way of dealing with the uncertainty represented by the O(h4)
discrepancy between the difference- and differential-operator representation of the
Schrödinger’s kinetic energy is more standard and lies in its disappearance in the
limit h → 0. This is a purely numerical recipe known as the Runge-Kutta method
[7]. In the present context of scattering one has to keep in mind that the two “small”
parameters h and 1/M may and, in order to achieve the quickest convergence, should
be chosen and varied independently.
3 The matching method of solution
3.1 The simplest model of the scattering with M = 1
Once we are given the boundary conditions (5) the process of the construction of
the solutions is straightforward. Let us first illustrate its key technical ingredients
on the model with the first nontrivial choice of the cutoff M = 1. In this case our
difference Schrödinger eq. (4) degenerates to the mere three nontrivial relations,
−ψ−2 + 2 cosϕψ−1 − ψ
0 = 0
−ψ−1 + (2 cosϕ+ Z0) ψ0 − ψ1 = 0
0 + 2 cosϕψ1 − ψ2 = 0
where we may insert, from eq. (5),
ψ−1 = e
−i ϕ +B ei ϕ , ψ
0 = 1 +B , ψ
0 = C , ψ1 = C e
i ϕ (7)
and where we have to demand, subsequently,
0 = 1 +B = ψ
0 = C = ψ0 ,
−e−i ϕ −B ei ϕ + (2 cosϕ + Z0) C − C e
i ϕ = 0 .
Thus, at an arbitrary “energy” ϕ one identifies B = C − 1 and gets the solution
2i sinϕ
2i sinϕ− Z0
, B =
2i sinϕ− Z0
Of course, as long as we deal just with the real “interaction term” Z0, our M = 1
toy problem remains Hermitian since no PT −symmetry has entered the scene yet.
3.2 PT −symmetry and the scattering at M = 2
In the next, M = 2 version of our model we have to insert the four known quantities
ψ−2 = e
−2 i ϕ +B e2 i ϕ , ψ−1 = e
−i ϕ +B ei ϕ , ψ1 = C e
i ϕ , ψ2 = C e
2 i ϕ
in the triplet of relations
−ψ−2 + (2 cosϕ+ Z−1 − i Y−1) ψ−1 − ψ
0 = 0
−ψ−1 + (2 cosϕ+ Z0) ψ0 − ψ1 = 0
0 + (2 cosϕ+ Z−1 + i Y−1) ψ1 − ψ2 = 0
where the three symbols ψ0, ψ
0 and ψ
0 defined by these respective equations
should represent the same quantity and must be equal to each other, therefore.
Having this in mind we introduce ξ
0 = 1 +B and ξ
0 = C and decompose
0 = ξ
0 + χ
0 , ψ
0 = ξ
0 + χ
This enables us eliminate
0 = V−1 ψ−1 , χ
0 = V1 ψ1
and eq. (9) becomes reduced to the pair of conditions,
1 +B + V−1 ψ−1 = C + V1 ψ1 = ψ0 ,
−ψ−1 + (2 cosϕ+ Z0) ψ0 − ψ1 = 0
They lead to the two-dimensional linear algebraic problem which defines the reflection
and transmission coefficients B and C at any input energy ϕ. The same conclusion
applies to all the models with the larger M .
4 The matrix-inversion method of solution
Let us now re-write our difference Schrödinger eq. (4) as a doubly infinite system of
linear algebraic equations
. . .
. . .
. . .
. . . S−1 −1 0 . . .
. . . −1 S0 −1
. . .
. . . 0 −1 S1
. . .
. . .
. . .
. . .
= 0 , (11)
where
Sk ( ≡ S
−k) =
2 cosϕ+ Zk + i Yk sign k , |k| < M ,
2 cosϕ , |k| ≥M
and where the majority of the elements of the “eigenvector” are prescribed, in ad-
vance, by the boundary conditions (5). Once we denote all of them by a different
symbol,
ψ(xm) =
Aeimϕ +B e−imϕ ≡ ξ(−)m , m ≤ −(M − 1) ,
C eimϕ ≡ ξ(+)m , m ≥M − 1 ,
we may reduce eq. (11) to a finite-dimensional and tridiagonal non-square-matrix
problem
−1 S∗(M−1) −1
. . .
. . .
. . .
−1 S∗1 −1
−1 S0 −1
−1 S1 −1
. . .
. . .
. . .
. . .
. . .
−1 S(M−1) −1
−(M−1)
ψ−(M−2)
= 0 (14)
or, better, to a non-homogeneous system of 2M − 1 equations
−(M−1)
ψ−(M−2)
where the (2M − 1)−dimensional square-matrix of the system can be partitioned as
follows,
S∗(M−1) −1
−1 S∗(M−2) −1
. . .
. . .
. . . S0
. . .
. . .
. . . −1
−1 S(M−2) −1
−1 S(M−1)
. (16)
Whenever this matrix proves non-singular, it may assigned the inverse matrixR=T−1,
the knowledge of which enables us to re-write eq. (15), in the same partitioning, as
follows,
−(M−1)
= R ·
, ~Ψ =
ψ−(M−2)
. (17)
In the next step we deduce that the matrix R has the following partitioned form
α∗ ~tT β
~u Q ~v
β ~wT α
We may summarize that in the light of the overall partitioned structure of eq. (17),
the knowledge of the (2M − 3)−dimensional submatrix Q as well as of the two
(2M − 3)−dimensional row vectors ~tT and ~wT (where T denotes transposition) is
entirely redundant. Moreover, the knowledge of the other two column vectors ~u and
~v only helps us to eliminate the “wavefunction” components ψ−(M−2), ψ−(M−3), . . . ,
ψM−3, ψM−2. In this sense, equation (15) degenerates to the mere two scalar relations
−(M−1) − α
−(M) − β ξ
M = 0 ,
M−1 − β ξ
−M − α ξ
M = 0 .
Once we insert the explicit definitions from eq. (13) we get the final pair of linear
equations
e−i (M−1)ϕ +B ei (M−1)ϕ − α∗
e−iM ϕ +B eiM ϕ
− C β eiM ϕ = 0 ,
C ei (M−1)ϕ − β
e−iM ϕ +B eiM ϕ
− C α eiM ϕ = 0
which are solved by the elimination of
B = −e−2iMϕ +
e−iϕ − α
and, subsequently, of
2iβe−2iMϕ sinϕ
β2 − (e−iϕ − α∗) (e−iϕ − α)
. (21)
This is our present main result.
5 Coefficients α and β
Our final scattering-determining formulae (20) and (21) indicate that the complex
coefficient α and the real coefficient β carry all the “dynamical input” information.
At any given energy parameter ϕ these matrix elements are, by construction, rational
functions of our 2M − 1 real coupling constants Z0, Z1, . . . , ZM−1 and Y1, . . . , YM−1.
In particular, β is equal to 1/ detT and α has the same denominator of course. An
explicit algebraic determination of the determinant detT and of the numerator (say,
γ) of α is less easy. Let us illustrate this assertion on a few examples.
5.1 M = 2 once more
detT = Z0 Z1
2 − 2Z1 + Y1
Re γ = Z0 Z1 − 1
Im γ = −Z0 Y1
5.2 M = 3
detT = Z0 Z1
2 − 2Z0 Z1 Z2 − 2Z1 Z2
Z0 Z2
2 + 2Z2 + Y2
Z0 Z1
2 − 2Y2
Z1 + Y2
Z0 + 2Z0 Y1 Y2 + Z0
Re γ = Z0 Z1
Z2 − 2Z1 Z2 + Y1
Z0 Z2 − Z1 Z0 + 1
Im γ = −Z0 Z1
Y2 + 2Z1 Y2 − Y1
Z0 Y2 −Y1 Z0
5.3 M = 4
The growth of complexity of the formulae occurs already at the dimension as low
as M = 4. The determinant detT and the real and imaginary parts of γ are them
represented by the sums of 15 and 14 and 32 products of couplings, respectively.
A simplification is only encountered in the weak coupling regime where one finds
just two terms in the determinant which are linear in the couplings,
detT = −2Z3 − 2Z1 + . . .
being followed by the 10 triple-product terms,
. . .+ Y1
Z0 + 4Z1 Z2 Z3 − 4Z1 Y2 Y3 + Z1
Z0 + 2Y3
Z2 + 2Y1 Z0 Y3+
+2Z1 Z0 Z3 + Z3
Z0 + Y3
Z0 + 2Z3
Z2 + . . .
etc. Similarly, we may decompose, in the even-number products,
Re γ = −1 + 2Z2 Z3 + Z0 Z1 + Z0 Z3 + 2Z2 Z1 + . . .
and continue
. . .+−2Z0 Z1 Z2 Z3 + 2Z0 Y1 Y2 Z3−
−Z2 Y1
Z0 − 2Z2
Z1 Z3 − Z2 Z1
Z0 − 2Y2
Z1 Z3 + . . .
etc, plus
Im γ = −2Z2 Y3 + 2Y2 Z1 − Z0 Y1 − Z0 Y3 − . . .
with a continuation
. . .− Y2 Y1
Z0 −Y2 Z1
Z1 Y3 + 2Z0 Z1 Z2 Y3 + 2Z2
Z1 Y3 − 2Z0 Y1 Y2 Y3 + . . .
etc. Symbolic manipulations on a computer should be employed at all the higher
dimensions M ≥ 4 in general.
6 Discussion
The main inspiration of the activities and attention paid to the PT −symmetry
originates from the pioneering 1998 letter by Bender and Boettcher [8] where the
operator P meant parity and where the (antilinear) T represented time reversal. Its
authors argued that the complex model V (x) = x2 (ix)δ seems to possess the purely
real bound-state spectrum at all the exponents δ ≥ 0. After a rigorous mathematical
proof of this conjecture by Dorey, Duncan, Tateo and Shin [9] and after the (crucial)
clarification of the existence of a nontrivial, “physical” Hilbert space H where the
Hamiltonian remains self-adjoint [2, 10, 11, 12], the bound-state version of eq. (1)
may be considered more or less well understood, especially after it has been clarified
that the physics-inspired concept of PT −symmetry of a Hamiltonian H should in
fact be understood, in the language of mathematics, as a P−pseudo-Hermiticity of
H specified by the property H† = P H P−1 [10, 11, 13].
In the spirit of the latter generalization, current literature abounds in the studies
of the potentials which are analytically continued [14], singular and multisheeted
[15], multidimensional [16], manybody [17], relativistic [18], supersymmetric [19] and
channel-coupling [20]. Among all these developments, a comparatively small number
of papers has been devoted to the problem of the scattering. For a sample one might
recollect the key reviews [21] and various Kleefeld’s conceptual conjectures [22] as well
as a very explicit study of the scattering by the separable PT −symmetric potentials
of rank one [23] or by the rectangular or reflectionless barriers [24], or the motion
considered along the so called tobogganic (i.e., complex and topologically nontrivial)
integration contours [25]. In this context our present difference-equation-based study
may be understood just as another attempt to fill the gap.
Technically we felt inspired by our old Runge-Kutta-type discretization of the
PT −symmetric Schrödinger equations [5] as well as by our recent chain-model ap-
proximations of bound states in a finite-dimensional Hilbert space [26]. In a certain
unification of these two approaches we succeeded here in showing that there exists
a close formal parallelism between the description of the (one-dimensional, Runge-
Kutta-approximated) scattering by a real (i.e., Hermitian) potentials and by their
complex, PT −symmetric generalizations. We showed that in both these contexts,
the definition of the transmission and reflection coefficients has the same form [cf.,
e.g., eq. (21)], with all the differences represented by the differences in the form of
the “dynamical input” information. It has been shown to be encoded, in both the
Hermitian and non-Hermitian cases, in the two functions α and β of the lattice po-
tentials, with the vanishing or non-vanishing coefficients Yk, respectively (cf. a few
samples of the concrete form of α and β in section 5).
On the level of physics we would like to emphasize that one of the main dis-
tinguishing features of the scattering problem in PT −symmetric quantum mechan-
ics lies in the manifest asymmetry between the “in” and “out” states [22]. In its
present solvable exemplification we showed that such an asymmetry is merely formal
and that the problem remains tractable by the standard, non-matching and non-
recurrent techniques of linear algebra. A key to the success proved to lie in the
partitioning of the Schrödinger equation which enabled us to separate its essential
and inessential components and to reduce the construction of the amplitudes to the
mere two-dimensional matrix inversion [cf. eq. (18)] where all the dynamical input
is represented by the four corners of the inverse matrix R = T−1 [cf. the definition
(16)].
We believe that the merits of the present discrete model were not exhausted by
its present short analysis and that its further study might throw new light, e.g., on
the non-Hermitian versions of the inverse problem of scattering.
Acknowledgement
Work supported by the MŠMT “Doppler Institute” project Nr. LC06002, by the In-
stitutional Research Plan AV0Z10480505 and by the GAČR grant Nr. 202/07/1307.
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