Double General Point Interactions Symmetry and Tunneling Times

ORIGINAL RESEARCH
published: 31 March 2016
doi: 10.3389/fphy.2016.00010
Frontiers in Physics | www.frontiersin.org 1 March 2016 | Volume 4 | Article 10
Edited by:
Manuel Asorey,
Universidad de Zaragoza, Spain
Reviewed by:
Yutaka Shikano,
Institute for Molecular Science, Japan
Manuel Gadella,
University of Valladolid, Spain
*Correspondence:
Luiz A. Manzoni
manzoni@cord.edu
Specialty section:
This article was submitted to
Mathematical Physics,
a section of the journal
Frontiers in Physics
Received: 30 January 2016
Accepted: 14 March 2016
Published: 31 March 2016
Citation:
Lee MA, Lunardi JT, Manzoni LA and
Nyquist EA (2016) Double General
Point Interactions: Symmetry and
Tunneling Times. Front. Phys. 4:10.
doi: 10.3389/fphy.2016.00010
Double General Point Interactions:
Symmetry and Tunneling Times
Molly A. Lee1
, José T. Lunardi2, 3
, Luiz A. Manzoni1
* and Erik A. Nyquist1
1
Department of Physics, Concordia College, Moorhead, MN, USA, 2
School of Physics and Astronomy, University of
Glasgow, Glasgow, UK, 3
Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, Ponta Grossa,
Brazil
We consider the one dimensional problem of a non-relativistic quantum particle scattering
off a double barrier built from two generalized point interactions (each one characterized
as a member of the four parameter family of point interactions). The properties of the
double point barrier under parity transformations are investigated, using the distributional
approach, and the constraints on the parameters necessary for the interaction to have
a well-defined parity are obtained. We show that the limit of zero interbarrier distance
of a renormalized odd arrangement with two δ′ is either trivial or does not exist as
a generalized point interaction. Finally, we specialize to double barriers with defined
parity, calculate the phase and Salecker-Wigner-Peres clock times and argue that the
emergence of the generalized Hartman effect is an artifact of the extreme opaque limit.
Keywords: generalized point interactions, delta interaction, delta prime interaction, tunneling times, generalized
Hartman effect, Salecker-Wigner-Peres clock
1. INTRODUCTION
The search for a consistent definition of the time it takes for a particle to traverse a classically
forbidden region is an enduring problem in quantum mechanics (see, e.g., [1, 2] and references
therein). The difficulties involved in the problem are both experimental and theoretical. The
time scales involved, typically of the order of attoseconds, only recently became accessible
experimentally [3–5], and the interpretation of the experimental results is not yet free of
controversy [5, 6]. From the theoretical point of view, the difficulties are related to the impossibility
of defining a self-adjoint (observable) time operator in quantum mechanics. Thus, there arises the
need for operational definitions of quantum tunneling time.
Several competing definitions of tunneling time exist, among which the most well-known and
useful are the phase time [7], the dwell time [8], the Larmor time [9, 10], and the Salecker-Wigner-
Peres (SWP) clock time [11–14]. However, all of these tunneling times can lead to counter-intuitive
results in specific situations. In particular, all of them can result in superluminal speeds for particles
tunneling through an opaque barrier, the so-called Hartman effect (HE) [15], and thus their status
as traversal times is controversial.
The HE can be generalized for multiple potential barriers, in which case it has been established
that in the opaque limit (low probability of transmission) the phase time becomes independent of
the barrier spacing [16]; this so-called generalized Hartman effect (GHE) has been verified both in
non-relativistic [16–19]) and relativistic [20] quantum mechanics and it is usually assumed to hold
for other time scales, such as the dwell time [21]. At first sight the GHE seems to be even more
troublesome than the original version of the HE, since one would expect that for the free region in
between the barriers the traversal time would be well described by the phase time.

Lee et al. Double General Point Interactions
The vast literature discussing tunneling times and the HE, as
well as its generalized version, focuses mostly in one-dimensional
square potential barriers, with few other potentials such as the
sech-square potential [22] and Dirac delta barriers (see, e.g.,
[21, 23, 24] and references therein) having been considered. From
a practical point of view, the understanding of one-dimensional
tunneling may help to understand phenomena in nanoscale
systems such as, for example, conductance in nanowires (see [25]
and references therein). However, to reach a better understanding
of tunneling times and the associated questions, it is important
and useful to study a larger variety of potentials, such as the
generalized one-dimensional point interactions.
Point interactions in one-dimensional quantum mechanics
have been the source of significant theoretical and mathematical
interest for providing the opportunity to investigate
regularization and renormalization schemes in the framework
of quantum mechanics (see, e.g., [26–29]). There have also been
controversies in the literature due to conflicting regularization-
dependent results obtained in the treatment of the so-called
δ′-interaction [30–33]. These controversies, however, were
clarified by using mathematically rigorous methods such as self-
adjoint extensions (SAE) [34–37] and Schwartz’s distribution
theory [38, 39] (there are also alternative approaches using
non-standard distribution theory, e.g., [40, 41]), both of which
unequivocally demonstrate the existence of a four-parameter
family of point interactions in one dimensional quantum
mechanics—these include, but are not restricted to, the δ
interaction as well as other interactions commonly associated
with the δ′ in the literature. It should be noticed that the
distributional approach [39] makes it unambiguously clear,
from parity considerations, that a “genuine” δ′ interaction does
not exist (for a particular regularization of the δ′ interaction
displaying even behavior under parity transformations see [42]).
From the point of view of applications, point-like interactions
have long provided models of significant experimental relevance
for lower dimensional systems [43, 44], such as the Girardeau-
Tonks gas [45, 46] and the Lieb-Liniger model [47, 48] and, more
recently, as a zero-range model of ultra-thin heterostructures
[49, 50]. The theoretical study of tunneling across point barriers
is another important application of these interactions, since
it has the important feature that one does not need to be
concerned with above-the-barrier components when considering
tunneling of wave packets, while for barriers of finite height
such components dominate the transmitted packet in the case
of opaque barriers and lead to difficulties in the interpretation
of tunneling times—tunneling across point barriers is, therefore,
free of such difficulties.
The general four-parameter family of interactions in one
dimension has been used to investigate tunneling times for a
non-relativistic particle incident on a single point scatterer [51].
However, in order to generate a richer tunneling structure and
to investigate some of the more counterintuitive phenomena
associated with tunneling times it is necessary to consider (at
least) two point scatterers. For example, it is not difficult to
convince oneself that for a single point barrier the SWP clock
time (and the dwell time) must be trivially zero, while for two
point barriers it is non-trivial. Futhermore, in addition to its
relevance to the study of resonant tunneling in heterostructures
in the zero-range limit [49, 50], the presence of two point barriers
allows us to investigate the GHE—in contradistinction, the HE
cannot be addressed by considering only one point scatterer,
since the distance “traveled” in that case vanishes (the support of
a group of point interactions is a set of zero measure). Therefore,
in this work we investigate the scattering of a non-relativistic
particle by two point scatterers given by the most general
potential [34–36, 39]. Given the relevance of parity-symmetric
considerations in the controversial discussion of regularized
implementations of point interactions, and the related discussion
of the δ′-interaction, we start by introducing the singular
interaction following the distributional approach introduced
in Lunardi and Manzoni [39], and analyzing the behavior of
such potential under parity transformations, thus obtaining the
necessary conditions to have even and odd potentials under
such transformations. We also analyze the limits in which the
separation between the barriers shrinks to zero and extend the
work of Šeba [35] to include the limit for a renormalized1
odd-arrangement of two δ′-interactions2. Then, we proceed to
calculate the phase and SWP clock times for barriers with well
defined parity symmetry, analyze the behavior of these times in
the opaque regime and discuss whether GHE occurs or not for
these potentials (a preliminary report on tunneling times, not
accounting for the symmetry of the interactions, appeared in Lee
et al. [52]).
2. DOUBLE BARRIER OF GENERALIZED
POINT INTERACTIONS: SYMMETRY
UNDER PARITY
Let us consider the scattering of a non-relativistic particle of
mass m and energy E = ¯h2
k2
2m by two general point interactions
localized at x = x1 and x = x2, with x1 < x2. According to
Calçada et al. [39] the distributional Schrödinger equation can
be written as (throughout this paper we will adopt the atomic
Rydberg units such that ¯h = 1 and 2m = 1, as usual when dealing
with point interactions)
ψ′′
(x) + k2
ψ = s1 [ψ] (x) + s2 [ψ] (x) ≡ s [ψ] (x) , (1)
where the prime indicates a space derivative, and the interaction
distributions are given by
sj [ψ] (x) = cj eiθj ψ(x−
j ) + dj eiθj − 1 ψ′
(x−
j ) δ x − xj
+ aj eiθj − 1 ψ(x−
j ) + bj eiθj ψ′
(x−
j ) δ′
x − xj (2)
or
sj [ψ] (x) = cj e−iθj ψ(x+
j ) − aj e−iθj − 1 ψ′
(x+
j ) δ x − xj
− dj e−iθj − 1 ψ(x+
j ) − bj e−iθj ψ′
(x+
j ) δ′
x − xj ,
(3)
1Following Šeba’s nomenclature [35].
2Throughout this work the notation δ′ refers to the most widely used version of
such interaction, which will be presented in Section 2 together with an alternative
version denoted by δ(1).

with aj, bj, cj, dj ∈ R, ajdj − bjcj = 1 and θj ∈ [0, π),
j = 1, 2. In these expressions we used the notation ψ(x±
j ) =
limε→0+ ψ xj ± ε , and similarly for ψ′(x±
j ). Such interaction
terms correspond to non-separated solutions, and will be the
only type of interactions considered in this work. The above
distributional Schrödinger equation implies the same boundary
conditions at the point singularities as those obtained from the
SAE method [39, 53] (also see [54] for a rigorous study of the
boundary conditions for point interactions):
ψ(x+
j )
ψ′(x+
j )
= j
ψ(x−
j )
ψ′(x−
j )
, with j = eiθj
aj bj
cj dj
j = 1, 2. (4)
The general solution for the Schrödinger equation in the region
outside the double potential is
ψ(x) =
A1eikx + A2e−ikx, x < x1
C1eikx + C2e−ikx, x > x2
, (5)
where
C1
C2
= M 1, 2, k, q, x1, x2
A1
A2
, (6)
with k and q being the wave numbers in the region outside and
inside the double barrier, respectively, and3
M 1, 2, k, q, x1, x2 = M−1
k, x2 Ŵ 1, 2, q, x1, x2
M k, x1 , (7)
with the transfer matrix given by:
Ŵ 1, 2, q, x1, x2 = 2M q, x2 M−1
q, x1 1 (8)
and
M (ρ, x) =
eiρx e−iρx
iρeiρx −iρe−iρx . (9)
From Equations (5) and (6) we can write
ψ x+
2
ψ′ x+
2
= Ŵ 1, 2, q, x1, x2
ψ x−
1
ψ′ x−
1
. (10)
From the above expressions we observe that when x2 → x1 the
double point interaction converges to a single point interaction,
with the transfer matrix given by Ŵ = 2 1, i.e., in this limit the
composition law for the parameters has the U(1)×SL(2, R) group
structure (see [55] for the group structure of a restricted class of
double point interactions). The explicit form of the interaction
term s[ψ] in Equation (1) is suitable to study its symmetry
properties [39]. To this end, and without losing generality, let us
assume x2 = −x1 ≡ L > 0. Then, under a parity transformation
this term changes as
s [ψ] (x) → ˜s[ψ](x) = ˜s1 [ψ] (x) + ˜s2 [ψ] (x) , (11)
where
3Even though in this Section q = k, such a notation will be useful in the next
Section, where we will consider a small constant potential V0 = 0 in the region
between the barriers.
˜sj [ψ] (x) = cjeiθj ψ (−xj)+
− djeiθj − 1 ψ′
(−xj)+
δ x + xj + − aj eiθj − 1 ψ (−xj)+
+ bj eiθj ψ′
(−xj)+
δ′
x + xj , j = 1, 2. (12)
The interaction is even if ˜s [ψ] = s [ψ] and it is odd if ˜s [ψ] =
−s [ψ] (for details see [39]). These conditions, taking Equation
(3) into account, imply that the double general interactions with
well defined parity symmetry are characterized by the following
constraints:
– Even Interactions
θ2 = −θ1, a2 = d1, b2 = b1, c2 = c1 and d2 = a1,
(13)
which is equivalent to the two b.c. matrices (redefining the
parameters as θ1 ≡ θ, d1 ≡ d, b1 ≡ b, c1 ≡ c and a1 ≡ a)
(e)
1 = eiθ a b
c d
, (e)
2 = e−iθ d b
c a
,
with ad − bc = 1 . (14)
From the above, the even arrangements of two general point
interactions constitute a four parameter family of interactions.
– Odd Interactions
θ1 = θ2 = 0, a2 = 2 − d1, b2 = −b1, c2 = −c1,
d2 = 2 − a1 , (15)
and the conditions aidi − bici = 1 (i = 1, 2) further imply
a2 = a1 (thus d2 = d1), with b1c1 ≤ 0.
Redefining b1 ≡ b and c1 ≡ c, odd interactions can be
conveniently characterized by the b.c. matrices
(o)
1 =
1 + s
√
−bc b
c 1 − s
√
−bc
,
(o)
2 =
1 + s
√
−bc −b
−c 1 − s
√
−bc
, (16)
with bc ≤ 0 and s = ±1. Thus, the odd arrangements of two
general point interactions constitute a three parameter family of
interactions.
From Equation (14) one can see that, given an arbitrary point
interaction, it is always possible to find a suitable second point
interaction resulting on an even arrangement4; on the other
hand, odd double point interactions are more restrictive and
can only be built by combining very specific kinds of point
interactions, as revealed by the constraints on the parameters
of the two b.c. matrices indicated in Equation (16). Particularly
interesting among the even arrangements of double-interactions
are those built from two delta (δ) interactions (θ = 0, a =
4The parity symmetric behavior of the even arrangements of δ and δ′ could
be inferred from the symmetries of quantum graphs considered in Boman and
Kurasov [56] and also from the investigation of PT -Hermiticity in Albeverio
et al. [57]. However, our treatment applies to a more general class of interactions
which include the parity symmetric properties of arrangements of single point
interactions without well defined symmetry (e.g., the even arrangement of two δ(1))
and, in particular, it allows for odd arrangements of two point interactions.

d = 1, b = 0, c arbitrary), two “delta prime” (δ′) interactions
(θ = 0, a = d = 1, c = 0, b arbitrary) and two δ(1)
interactions (θ = 0, b = c = 0, a = d−1 arbitrary). Among
the odd arrangements we are interested in those built from two
δ or two δ′ (it is not possible to built an odd arrangement
from two δ(1) interactions). As already mentioned in Section
1, neither δ′ nor δ(1) stand for a “genuine” derivative of a δ
interaction—they simply refer to particular choices of parameters
in the general point interaction [34–36] (a recent proposal for
physically interpreting the boundary conditions given by single
point interactions is given in Kulinskii and Panchenko [58]).
It is well known that single point interactions are either
even under parity or have no defined symmetry, that is, there
exists no single point interaction which is odd under parity
[39] (for instance, both a single δ and a single δ′ are even
interactions, whereas a single δ(1) has no defined parity). Thus,
there naturally arises the question: what is the resulting symmetry
of an interaction obtained by shrinking the inter-barrier spacing
to zero for an initial double point barrier with well defined odd
symmetry? It is simple to show that in this limit, characterized
by taking x2 → x1 in Equation (8), the resulting interaction
either does not have a defined symmetry or vanishes (when bc =
0), a result in accordance with the conclusion that there is no
possible regularization capable of producing an odd single point
interaction [39]. The equivalent limit for an even double point
barrier results, unsurprisingly, in an even point interaction.
Šeba [35] investigated the limit L → 0 of the renormalized
dipole interaction (1/2Lν) δ(x + L) − δ(x − L) , in the context
of SAE, and found that for ν < 1
2 this renormalized odd
interaction converges to the free case, for ν = 1
2 it converges to
the (even) delta interaction and for ν > 1
2 such a limit results
in separated solutions (i.e., free Hamiltonians on the two half-
lines defined from the origin). Šeba’s results can be immediately
reproduced in the formalism developed above by assuming the
parameters for an odd arrangement of delta interactions in
Equation (16), b = 0 and c ≡ λL−ν, and taking the limit L → 0 of
Ŵν
δ ( (o)
1 , (o)
2 , k, − L, L), where we introduced the indices δ and
ν to indicate the odd arrangement of deltas and the dependence
on the parameter ν, respectively.
Let us now extend the above analysis to the case of a
renormalized odd arrangement of “delta prime” interactions.
Thus, choosing b ≡ λL−ν and c = 0 in Equation (16) we obtain
Ŵν
δ′ ( (o)
1 , (o)
2 , k, − L, L) ≃
1 + 2λk2L1−ν 2λ2k2L1−2ν
−2k2L 1 − 2λk2L1−ν ;
ν > 0 , (17)
where we followed the same notation of the previous paragraph
and the subscript δ′ indicates the odd arrangement of “delta
prime” interactions. From Equation (17) we have the limits
lim
L→0
Ŵ
ν<1/2
δ′ ( (o)
1 , (o)
2 , k, − L, L) = 111 ; (18)
lim
L→0
Ŵ
ν=1/2
δ′ ( (o)
1 , (o)
2 , k, − L, L) =
1 2λ2k2
0 1
; (19)
lim
L→0
Ŵ
1/2<ν<1
δ′ ( (o)
1 , (o)
2 , k, −L, L) =
1 hk2
0 1
,
h ≡ +∞ ; (20)
where 111 indicates the 2 × 2 identity matrix. For ν ≥ 1, similarly
to Equation (20), the limit will result in matrices whose elements
have improper limits [albeit different from Equation (20)].
From Equations (18) to (20), it follows that the renormalized
odd arrangement of δ′ interactions converges to the free case
for ν < 1
2 . For ν = 1
2 the limit (Equation 19) seems, at
first, to result in a single (even) δ′ interaction. However, the
k2-dependence in the limiting transfer matrix would imply an
interaction distribution with singular order +3 in Equation (1)
and would violate the tenets of the distributional approach, which
allows a maximum singular order of +1 [39]—otherwise one
would have singular wave functions from which no physically
meaningful solutions of the Schrödinger equation can be
obtained5. Therefore, interactions such as Equation (19) are
not allowed by the distributional approach to general singular
interactions (nor are they allowed by the SAE method), and for
ν = 1
2 the zero range limit of a renormalized odd arrangement of
δ′ interactions does not converge to a well defined (in the sense
of distributions) singular interaction. In the same way, the k2-
dependence in Equation (20), and indeed for all ν > 1
2 , means
that for these values of ν the odd arrangement of δ′ interactions
does not converge to a generalized point interaction.
3. TUNNELING TIMES FOR BARRIERS
WITH WELL-DEFINED SYMMETRY
In this section we will consider the tunneling times for a non-
relativistic particle scattered by two point barriers. We are mainly
interested in the time associated with the transmission across the
double barrier as given by both a SWP clock, which runs only
when the particle is in this region, and by the phase time.
In order to calculate the SWP clock time, it is sufficient for
the purposes of this work to notice that the effect of the particle-
clock coupling is equivalent to introduce a weak potential V(x)
equal to a constant V0 (V0 ≪ E) in the region (x1, x2) and zero
otherwise (for an in depth account of the SWP clock see [12, 13]).
Introducing this coupling into the Schrödinger Equation (1) it
becomes
ψ′′
(x) + k2
ψ = s1 [ψ] (x) + s2 [ψ] (x) + θ(x − x1) θ(x2 − x)
V0 ψ(x) , (21)
where θ(x) is the Heaviside theta function.
Assuming a particle incident upon the barriers from the left,
the reflection and transmission amplitudes [R(k, q) and T(k, q),
respectively] can be immediately obtained from the formalism
developed in Section 2. Then, by considering a unit incident flux,
we have A1 = 1, A2 = R(k, q), C1 = T(k, q) and C2 = 0, and
using Equation (6) the incident and reflection amplitudes can be
written in terms of the matrix elements of Equation (7) as
5For details the reader is referred to Calçada et al. (in preparation).

R k, q =
−M21
M22
and T(k, q) =
e2i(θ1+θ2)
M22
, (22)
where in the last equation we have used the fact that det M =
e2i(θ1 + θ2), as can be promptly verified from the conditions on
the parameters in Equation (4) and from Equations (7) to (9).
For the purpose of calculating tunneling times the
transmission and reflection amplitudes given by Equation
(22), with k ≡
√
E and q ≡ k2 − V0, are more conveniently
expressed as
T(k, q) = |T(k, q)|ei[ϕT(k,q)−k(x2−x1)] and
R(k, q) = |R(k, q)|ei[ϕR(k,q)+2kx1] , (23)
where ϕT(k, q) is the phase difference, at any time t, between
the transmitted wave at x2 and the incident wave function at x1,
whereas ϕR(k, q) is the phase difference, at any time t, between the
reflected and the incident waves at x1. Then, the SWP clock times
for transmission and reflection (sometimes called the conditional
dwell times [21]) are given by [12, 13]
tT(R)
c (k) = −¯h
∂
∂V0
ϕT(R)(k, q)
V0=0
(24)
and satisfy the identity τD(k) = |T(k, k)|2tT
c (k) + |R(k, k)|2tR
c (k),
where τD(k) is the dwell time [13].
Similarly, the transmission and reflection phase times are
given by [10, 15]
tT(R)
p (k) = ¯h
∂
∂E
ϕT(R)(k, k) . (25)
Since the point interaction parameters θ1 and θ2 cause a simple
shift in the phase difference, and due to its independence of
the energy E and the perturbation potential V0, it follows
that both the clock and the phase times are independent of
these parameters—for even and odd arrangements, Equations
(13) and (15), there is, in fact, no change in phase due to θ1
and θ2 (an in-depth analysis of the consequences of the phase
factor, arising at the boundary conditions, for one-dimensional
quantum tunneling is presented in Furuhashi et al. [59]). In what
follows we consider only arrangements of point interactions with
well-defined symmetry under parity transformations, according
to Equations (13)–(16), and obtain explicit results for the clock
and phase times. The tunneling times are further analyzed for
special cases of one-parameter arrangements.
3.1. Even Arrangements
From Equation (22) we have that for even arrangements of two
point interactions the difference between the phases ϕR(k, q) and
ϕT(k, q) is given by
ϕR(k, q) − ϕT(k, q) = arg (−M21) = ±
π
2
, (26)
since in this case M21 is purely imaginary
M21 = i
sin(2Lq) −q2 a2 + b2k2 + c2 + d2k2
+ 2q cos(2Lq) ac + bdk2
2kq
. (27)
The result expressed in Equation (26) generalizes the well-known
result for symmetric (even) regular potentials [10] to the case of
an even arrangement of two point interactions. It follows that
in this case t(T)
p (k) = t(R)
p (k), and from Equations (26) together
with the relationship between dwell and SWP times [13] we have
t(T)
c (k) = t(R)
c (k) = τD(k).
The general expressions for the clock and phase times for an
arbitrary even double point interaction are
t(T)
c (k) = t(R)
c (k) =
Nc k
D k
, (28)
t(T)
p (k) = t(R)
p (k) =
1
k
Np k
D k
, (29)
where we defined the following functions
Nc k = 4k ac + bdk2
sin2
(2kL) − c2
− k2
a2
+ b2
k2
− d2
× sin(4kL) + 4kL k2
a2
+ d2
+ b2
k4
+ c2
,
D k = 2c2
k2
a2
+ d2
+ k4
a4
+ 2a2
b2
k2
+ 2d2
− 4ad
+ b2
k2
+ d2 2
+ 6 − cos(4kL) 2c2
k2
d2
− 3a2
+ k4
a4
+ 2b2
k2
a2
− 3d2
− 12a2
d2
+ 12ad + b4
k4
+ d4
− 2 + c4
+ 4k sin(4kL) −ck2
a3
− 2ad2
+ d
+ bk4
−2a2
d + a + d3
+ ac3
− b3
dk6
+ c4
,
Np k = − cos(4kL) bk4
a3
+ ab2
k2
− 6ad2
+ 5d + ck2
× −6a2
d + 5a + d3
+ c3
d − 2k sin(4kL) (2ad − 1)
× b2
k4
− c2
+ 2k2
a2
− d2
(ad − 1) + k4
× b a3
+ ab2
k2
+ 2ad2
+ d + 4L a2
+ b2
k2
+ d2
+ c3
d + 4c2
k2
L + ck2
2a2
d + a + d3
.
Even though Equations (28) and (29) represent the most
general expressions for the corresponding tunneling times with
a potential barrier given by an even arrangement of two point
interactions, for further analysis of the behavior of these times
it is necessary to consider specific interactions. Thus, let us now
consider three special and important cases of one-parameter even
arrangements—in all the cases below the parameter γ ∈ R is the
“strength” of the interaction.
(i) The first case to be considered is that of an even
arrangement of two delta (δ) interactions, which corresponds to
set θ = 0, a = d = 1, b = 0 and c = γ in Equations (28)–
(29). In this case, the clock and phase times for reflection and
transmission are given by
t(T)
c (k) = t(R)
c (k)
=
2k γ + 4k2L + 2γ2L − γ2 sin(4kL)
−2γk cos(4kL)
γ4 + 8k4 + 4γ2k2 − γ2 γ2 − 4k2 cos(4kL)
+4γ3k sin(4kL)
, (30)

t(T)
p (k) = t(R)
p (k)
=
γ3 + 8k4L + 4γk2 + 4γ2k2L − γ3 cos(4kL)
+2γ2k sin(4kL)
k γ4 + 8k4 + 4γ2k2 − γ2 γ2 − 4k2 cos(4kL)
+ 4γ3k sin(4kL)
.
(31)
The expression above for the clock (dwell) time agrees with
previous results [21, 23, 24].
As previously mentioned, in this work we are particularly
interested in the behavior of the tuneling times in the opaque
regime (i.e., for |T|2
≪ 1 and |R|2
≃ 1). For the even
double-δ interaction this regime corresponds to take |γ| ≫ 1.
It is straightforward to verify from Equations (30) to (31) that
t(T)
c (k) → 0 and t(T)
p (k) → 0 in the extreme opaque limit
|γ| → ∞, which is usually taken as an indication of the existence
of the GHE [21, 23, 24]. However, Figure 1A shows that the
off-resonance values of both the phase and clock (dwell) times
clearly increase with the barrier separation—for the parameters
in Figure 1A the scattering is well inside the opaque regime—
thus, contradicting the GHE, according to which one would
expect the off-resonance times to be independent of the barrier
spacing.
In order to better understand the behavior of the tunneling
times when the interaction approaches the opaque limit, it is
important to consider their asymptotic expressions when |γ| >>
1. Up to the second order in γ−1 we obtain
t(T)
c (k) ∼
1
γ
2
2kL csc2
(2kL) − cot(2kL) , (32)
t(T)
p (k) ∼
1
γk
+
1
γ
2
2kL csc2
(2kL) − 2 cot(2kL) . (33)
These expressions indicate that for any finite γ, such that |γ| >>
1, both the clock and the phase times depend on the barrier
spacing L at the same order 1
γ
2
. For a fixed value of the strength
γ, and apart from the oscillating terms, such a dependence tends
to grow linearly with L in the same fashion for both times. Thus,
it is strictly only for γ = ∞ (at which point the tunneling
times lose their physical meaning because the barriers become
impenetrable) that the above expressions become independent of
L (since they are zero).
(ii) The second case to be considered is an even arrangement
of two “delta prime” (δ′) interactions, which corresponds to the
choices θ = 0, a = d = 1, b = γ and c = 0 in Equations
(28)–(29). The resulting clock and phase times are
t(T)
c (k) = t(R)
c (k)
=
2 γ + 2γ2k2L + 4L + γ2k sin(4kL)
−2γ cos(4kL)
k γ4k4 − 4γ3k3 sin(4kL) + 4γ2k2
−γ2k2 γ2k2 − 4 cos(4kL) + 8
, (34)
t(T)
p (k) = t(R)
p (k)
=
4γ + γ3k2 − γ3k2 cos(4kL) + 4γ2k2L
−2γ2k sin(4kL) + 8L
k γ4k4 − 4γ3k3 sin(4kL) + 4γ2k2
−γ2k2 γ2k2 − 4 cos(4kL) + 8
, (35)
The opaque regime is again given by |γ| ≫ 1, and in the limit
|γ| → ∞ both Equations (34) and (35) tend to zero. However,
similarly to the previous case, a plot of the phase and clock
times, for parameters well inside the opaque regime, shows that
off-resonance both times depend quasi-linearly on the barrier
spacing—see Figure 1B.
The asymptotic expressions for the times in Equations
(34) and (35), in the opaque regime |γ| >> 1 are given,
respectively, by
t(T)
c (k) ∼
1
γ
2
1
k4
2kL csc2
(2kL) + cot(2kL) , (36)
t(T)
p (k) ∼
1
γk3
+
1
γ
2
1
k4
2kL csc2
(2kL) + 2 cot(2kL) ,(37)
up to second order in 1/γ. From these expressions it is clear that
for any large, but finite, interaction strength γ the off-resonance
tunneling times tend to increase linearly with the barrier spacing.
(iii) The third special case that we consider here is an
even arrangement of two so-called δ(1) interactions [53], which
corresponds to choose θ = 0, a = γ, b = 0, c = 0 and d =
γ−1 in Equations (28)–(29). Notice that a single δ(1) interaction
does not have a defined symmetry under parity transformations
(contrary to the single δ and δ′ interactions, which are even under
FIGURE 1 | Tunneling times for even arrangements of two point interactions (in atomic Rydberg units). (A) Double-δ, with γ = 350 and k = 2.0. (B) Double
δ′ with γ = 120 and k = 1.0.

parity)—see e.g., [39] for a parity-symmetry analysis for single
interactions. The clock and phase times, for the even arrangement
of δ(1)s, are
t(T)
c (k) = t(R)
c (k) =
γ2
k2
4 γ4 + 1 kL + γ4 − 1 sin(4kL)
γ8 + 6γ4 − γ4 − 1
2
cos(4kL) + 1
,
(38)
t(T)
p (k) = t(R)
p (k) =
4γ2L
k
γ4 + 1
γ8 + 6γ4 − γ4 − 1
2
cos(4kL) + 1
.
(39)
It should be noticed that the phase time is invariant under the
substitution γ → γ−1, while the clock time does not present
such symmetry. This follows from the absence of a defined parity
for δ(1), as noted above, and the fact that the change γ → γ−1
in effect merely swaps the location of the two (different) point
interactions. The phase time, being the “temporal delay” caused
by the interactions [60], is insensitive to this location swapping—
as long as the interbarrier distance remains the same. On the
other hand, since the clock (in reality a stop-watch) runs only
while there is a probability of finding the particle in between the
barriers, and since δ(1) is not even, such probability depends on
the order of the two interactions, and the clock (dwell) time is not
expected to be invariant under the substitution γ → γ−1.
The opaque regime for the even arrangement of δ(1)
interactions, similarly to the single δ(1) case, is given by two
different ranges of the interaction strength, namely, |γ| ≪ 1 and
|γ| ≫ 1 [61]6. In Figure 2 we present a typical plot of both times
in the regime |γ| ≪ 1 (plots for the regime |γ| ≫ 1 have similar
characteristics). As it can be seen, in the opaque regime the phase
and clock times are essentially identical for L 1.2, with the
differences being of the order γ2 (or 1/γ2 in the regime |γ|≫1)—
these minute differences between the two times in this regime,
which come from the term (γ2/k2)(γ4 −1) sin(4kL)/[γ8 + 6γ4 −
(γ4−1)2 cos(4kL) + 1] in Equation (38), can be seen in Figure 2B.
6In Gadella et al. [61] the δ(1) interaction is given by choosing a = 0 and identifying
γ = 1+mb
1−mb in their Equation (22). Thus, from Equation (23) of the same reference,
it follows that the transmission coefficient goes to zero (the opaque limit) when
b → −1/m and b → +1/m, which in our notation correspond to γ → 0 and
|γ| → ∞, respectively.
As follows from Figure 2A, even in the opaque regime both times
do depend on L, as was the case with the other arrangements
considered.
The dependence of the tunneling times on L is, once again,
clear when one considers the asymptotic behavior of the times in
the opaque region. For |γ| >> 1, and up to second order in 1
γ ,
the asymptotic expansion for the tunneling times are given by
t(T)
c (k) ∼
1
γ
2
1
k2
2kL csc2
(2kL) + cot(2kL) , (40)
t(T)
p (k) ∼
1
γ
2
2L csc2(2kL)
k
, (41)
and, from the above mentioned invariance of the phase time,
the asymptotic behavior of Equation(39) when γ << 1 can be
obtained from Equation (41) by simply changing γ → γ−1,
whereas the corresponding asymptotic behavior of the clock time
can be obtained from Equation (40) by changing γ → γ−1 and by
replacing a minus sign in front of the “cot” term. It follows from
Equations (40) to (41) that for any finite (conversely, different
of zero) γ the tunneling times will depend on L (again, this
dependence tends to be linear for off-resonance times).
3.2. Odd Arrangements
Let us now consider odd arrangements of two point interactions.
From Equations (15), (22), and (23) the difference between the
reflection and transmission phases is
ϕR(k, q) − ϕT(k, q) = arg (−M21) =
tan−1 k2 b2q2 + 1 − c2 − q2
2k bq2 + c
≡ ϕ0(k, q) , (42)
where we have defined the auxiliary phase ϕ0(k, q), which allow
us to introduce the auxiliary clock and phase times [62, 63] given,
respectively, by
t0
c (k) ≡ −¯h
∂
∂V0
ϕ0(k, q)
V0=0
and t0
p(k) ≡ ¯h
∂
∂E
ϕ0(k, k) .
With the help of these auxiliary times, the transmission and
reflection times are now related by
t(R)
c (k) = t(T)
c (k) + t0
c (k) and t(R)
p (k) = t(T)
p (k) + t0
p(k) . (43)
FIGURE 2 | (A) Tunneling times for an even arrangement of two δ(1) interactions, with γ = 0.0075 and k = 0.6 (in atomic Rydberg units). (B) Detail, for the same
parameters.

It should be noticed that, contrary to the case of even potentials,
for odd potentials (and potentials without a defined parity) the
clock times in general do not coincide with the dwell time [63].
Below we consider two cases of odd arrangements of point
interactions, the first built from two δ and the second built from
two δ′ interactions. We note that it is not possible to build odd
arrangements with two δ(1) point interactions.
(i) We first consider the odd arrangement of two δ
interactions, which corresponds to set b = 0 in Equation
(16). Then, the strength of the interaction is c ≡ γ and the
transmission and auxiliary times are given, respectively, by
t(T)
c (k) =
2N1(k) − N2(k)
2D(k)
, (44)
t(T)
p (k) =
N1(k) − N2(k)
D(k)
; (45)
t0
c (k) =
−2k
γ
1
D0(k)
; (46)
t0
p(k) =
γ
k
1
D0(k)
, (47)
where we have defined the functions
N1(k) = 2kL γ2
+ 2k2
;
N2(k) = γ2
sin(4kL) ;
D(k) =
1
2
γ4
+ 8k4
+ 4γ2
k2
− γ2
γ2
+ 4k2
cos(4kL) ;
D0(k) = 4k2
+ γ2
.
The corresponding reflection times follow directly from these
expressions together with Equation (43).
The opaque regime is characterized by |γ| ≫ 1, and it is
straightforward to see that all times in Equations (44)–(47) tend
to zero in the extreme opaque limit |γ| → ∞. Figure 3A shows
a plot of the transmission times for an interaction strength well
inside the opaque regime (γ = 80). The difference between the
clock and phase times, tc − tp = N2(k)/ 2D(k) , behaves as
1/γ2 in this regime and it is not noticeable in the figure, except
for very small L. Figure 3A shows that both the off-resonance
phase and clock times increase with the interbarrier spacing in
this opaque regime; in addition, similarly to the cases previously
considered, the behavior of these times is essentially linear. Again,
this dependence on L for any finite interaction strength is made
clear by the asymptotic expansion of the times for |γ| >> 1. Up
to second order in 1
γ , these asymptotic expansions are:
t(T)
c (k) ∼
1
γ
2
2kL csc2
(2kL) − cot(2kL) ; (48)
t(T)
p (k) ∼
1
γ
2
2kL csc2
(2kL) − 2 cot(2kL) ; (49)
t0
c (k) ∼ 0 ; (50)
t0
p(k) ∼
1
γk
, (51)
and it is clear that, strictly, only for γ = ∞ there is no L
dependence.
(ii) The second possibility we will consider for an odd
arrangement of two point interactions is to set b = γ and c = 0 in
Equation (16), which corresponds to an odd arrangement of two
δ′ interactions. In this case, the transmission and auxiliary clock
and phase times are given by
t(T)
c (k) =
2M1(k) + M2(k)
2k (k)
, (52)
t(T)
p (k) =
M1(k) + M2(k)
k (k)
; (53)
t0
c (k) =
−2
k3γ
1
0(k)
; (54)
t0
p(k) =
γ
k
1
0(k)
, (55)
where
M1(k) = 2L γ2
k2
+ 2 ;
M2(k) = γ2
k sin(4kL) ;
(k) =
1
2
γ4
k4
+ 4γ2
k2
− γ2
k2
γ2
k2
+ 4 cos(4kL) + 8 ;
0(k) = 4 + k2
γ2
.
As before, the opaque regime is characterized by γ≫1 and in the
extreme opaque limit, |γ| → ∞, all the times above go to zero.
FIGURE 3 | Tunneling times for odd arrangements of two point interactions (in atomic Rydberg units). (A) Double-δ, with γ = 80 and k = 1.5. (B) Double-δ′,
with γ = 60 and k = 1.2. In both cases tp − tc ∼ O(1/γ2) and the times essentially overlap for the values of γ used.

In what regards the dependence of the transmission tunneling
times with L, in the opaque regime, and its consequences to the
GHE, the behavior of the times is similar to all the previous
cases considered, as can be seen from Figure 3B, which clearly
shows that the off-resonance times tend to increase linearly with
L (for this interaction tp − tc = M2(k)/ 2 (k) ∼ O( 1
γ2 )
in the opaque regime. Hence the apparent coincidence of the
times in Figure 3B). Once again, this behavior can be understood
from the asymptotic expansions of the times for |γ| >> 1,
which are
t(T)
c (k) ∼
1
γ
2
1
k4
2kL csc2
(2kL) + cot(2kL) ; (56)
t(T)
p (k) ∼
1
γ
2
1
k4
2kL csc2
(2kL) + 2 cot(2kL) ; (57)
t0
c (k) ∼ 0 ; (58)
t0
p(k) ∼
1
γk3
. (59)
and which depend on L for any finite γ (the expansions
for the auxiliary times are not necessary in the
analysis of the GHE and are presented only for
completeness).
4. CONCLUDING REMARKS
An analysis of the symmetry under parity transformation and
of the tunneling times was provided for a one-dimensional non-
relativistic particle scattered by a double barrier composed of two
general point potentials.
Our general analysis of the symmetry shows that it is possible
to construct potentials with well-defined parity, both odd and
even, for such double barriers of point interactions. This is in
stark contrast with the case of a single point interaction, for
which no odd interaction exists [37, 39, 64]—as illustrated in
Section 2 by the fact that the limit L → 0 of a renormalized
δ-dipole interaction, an odd arrangement of two δ [35], is
either trivial or results in a point interaction which is even or
has no defined parity. Furthermore, we extended Šeba’s [35]
procedure to the case of a renormalized odd arrangement of δ′
interactions and found that in the limit L → 0 this arrangement
is either trivial (for ν < 1/2) or it does not converge to a
general singular interaction (for ν ≥ 1/2). In other words,
for ν ≥ 1/2 such limit is not of the form (Equation 2)
[equivalently, (Equation 4)] and has no self-adjoint extension
as well.
We then proceeded to calculate the quantum tunneling times
for the double point barriers with well defined parity. We
calculated the phase and SWP clock times for three special
cases of even arrangements [pairs of δ, δ′ or δ(1) satisfying
the constraints (Equations 13 and 14)] as well as for two
odd arrangements of double point barriers [pairs of δ or
δ′ satisfying (Equation 15)–(Equation 16)]. Even though the
reflection tunneling times were also provided for completeness,
our main concern was the behavior of the transmission times in
the opaque regime and the existence of the so-called generalized
Hartman effect [16].
We found that, for all interactions considered, both the phase
and SWP clock times approach zero in the extreme opaque limit.
However, in all cases, for any finite values of the interaction
strength (for the even arrangement of δ(1)s, also γ = 0) the
tunneling times do depend on the interbarrier spacing, and such
dependence is essentially linear for off-resonance values of the
parameters. It is only strictly for the interaction strength at the
extreme opaque value (|γ| = ∞ or γ = 0 for the even double-
δ(1); |γ| = ∞ for all other interactions) that the times do not
depend on L, since in this case their off-resonance value is zero.
However, for these extreme values of the parameters the very
concept of tunneling time looses its meaning, since there is no
transmission (the barriers become impenetrable and thus the
solutions in both sides of each barrier are separable). Therefore,
for the potentials analyzed the GHE is just a mathematical artifact
of such an extreme limit and devoid of any physical meaning. As a
final remark we stress that even if the GHE is an artifact, its more
striking consequence, which is the apparent superluminality of
tunneling, is not precluded—in fact, superluminality does not
require GHE (or HE), and the slow growth of the tunneling times
with respect to the interbarrier distance in the opaque regime
could result, for sufficiently separated barriers, in an apparent
superluminal transmission. However, any apparent violation of
causality is avoided by interpreting the superluminal phase and
clock times respectively in terms of wave packet reshaping and
weak measurements [21, 23] (for a recent treatment of tunneling
times from the point of view of weak measurement theory see
[65]). It should also be noticed that apparent superluminality is
more properly addressed in a relativistic theory. The above results
on the GHE reinforce similar conclusions reached in Winful [1],
Lunardi and Manzoni [20], Aharonov et al. [21] and Lunardi
et al. [24]—an analysis of the GHE is, unavoidably, case-by-
case and, thus, this work adds to the body of results clarifying
such effect.
AUTHOR CONTRIBUTIONS
JL and LM contributed equally and in all stages of this work. ML
and EN contributed in the calculation of tunneling times.
ACKNOWLEDGMENTS
This work was partially supported by the Minnesota Space
Grant Consortium, CNPq/Brazil (JL, Grant PDE 233512/2014-
4), NSF STEP grant (# 0969568), Concordia College’s Office
for Undergraduate Research and a Concordia College sabbatical
support grant.

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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2016 Lee, Lunardi, Manzoni and Nyquist. This is an open-access
article distributed under the terms of the Creative Commons Attribution License
(CC BY). The use, distribution or reproduction in other forums is permitted,
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