# On the complex solution of the Schrödinger equation with exponential
potentials

Javier Garcia<EMAIL_ADDRESS>1Instituto de Física La Plata,
Consejo Nacional de Investigaciones Científicas y Técnicas, and Departamento
de Física, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina

###### Abstract

We study the analytical solutions of the Schrödinger equation with a repulsive
exponential potential $\lambda e^{-r}$, and that with an exponential wall
$\lambda e^{r}$, both with $\lambda>0$. We show that the complex eigenenergies
obtained for the latter tend either to those of the former, or to real
rational numbers as $\lambda\rightarrow\infty$. In the light of these results,
we explain the wrong resonance energies obtained in a previous application of
the Riccati-Padé method to the Schrödinger equation with a repulsive
exponential potential, and further study the convergence properties of this
approach.

##  I Introduction

Resonances are solutions of the Schrödinger equation that have complex
energies and present a purely outgoing behavior in the asymptotic region.
These states, introduced by Siegert to describe decaying nuclear states using
a stationary-state pictureSiegert (1939), arise, together with bound and
virtual states (i.e., states that grow exponentially in the asymptotic
region), as poles of the scattering matrix. They spark a lot of interest among
physicists and chemists, evidenced by the large amount of methods devised for
their computation. Most of them are based on the complex rotation method,
introduced by Aguilar, Balslev, and Combes Aguilar and Combes (1971); Balslev
and Combes (1971), which consists in performing a dilation on the coordinate
$U(\theta)rU^{-1}(\theta)=e^{i\theta}r$, by choosing the parameter $\theta$ in
such a way that the outgoing-wave state is transformed into one that
asymptotically decays to zero, as bound states do, and then using one of the
many numerical methods devised to compute the latter. The first applications
of the complex rotation method involved a variational approach, which remains
popular (e.g. Refs. Fernández and Garcia, 2013; Kuroś and Okopińska, 2015; Myo
and Katō, 2020) but also different techniques have been employed, such as
numerical integration methods Rittby, Elander, and Brändas (1981, 1982a);
Atabek, Lefebvre, and Jacon (1982); Atabek and Lefebvre (1983), dimensional
scaling Kais and Herschbach (1993), the Lanczos’ Tau method Midy, Atabek, and
Oliver (1993), spherical-box approaches Maier, Cederbaum, and Domcke (1980);
Zhou, Meng, and Zhao (2009), direct computation of the Jost function Sofianos
and Rakityansky (1997), gradient optimization Bai  _et al._ (2021), and
eigenvector continuation Yapa, Fossez, and König (2023), among others. There
is also a wide array of methods that do not involve a complex rotation;
examples of these include methods based on Siegert pseudo-states Tolstikhin,
Ostrovsky, and Nakamura (1997, 1998); Batishchev and Tolstikhin (2007); Čurík,
Tarana, and Horáček (2023), the complex absorbing potential Riss and Meyer
(1993); Müller, Santra, and Cederbaum (2003), Padé approximations of the
S-matrix Rakityansky, Sofianos, and Elander (2007), the real stabilization
method Zhang  _et al._ (2008), the coupled channels method Liang, Meng, and
Zhou (2015), and the Riccati-Padé method (RPM) Fernandez (1995); Fernández
(1995). The list of methods provided here is by no means comprehensive.

The RPM consists in expanding a modified logarithmic derivative of the
wavefunction in a Taylor series, and constructing Hankel determinants with the
expansion coefficients, the roots of which give rise to sequences that
converge rapidly towards both the bound states and resonances Fernández, Ma,
and Tipping (1989a, b). It was originally proposed for the computation of
bound states, but later it was found that it is also able to yield resonances
without resorting explicitly to a complex rotation Fernandez (1995); Fernández
(1995). Several applications to the computation of resonances ensued Fernández
(1996, 1997, 2008); Amore and Fernández (2008); Fernández (2012); Fernández
and Garcia (2016, 2018), which showed that it is able to compute them very
accurately and with comparatively little cost. For example, the resonances for
the Stark effect in the hydrogen atom computed by us in Ref. Fernández and
Garcia, 2018 are to our knowledge the most accurate available in literature.

As with bound states, there are very few systems whose resonances can be
computed analytically. One of such systems are the $s-$states of a particle
exposed to a decaying exponential potential, i.e., $V(r)=\lambda e^{-r}$, with
real $\lambda$. If $\lambda<0$, the system admits a finite number of bound
states Ma (1946); Haar (1946). On the other hand if $\lambda>0$, the potential
is not expected to hold bound or resonant states, but some of the
eigenfunctions, which correspond to complex eigenvalues, behave as such
Atabek, Lefebvre, and Jacon (1982). In addition, the repulsive exponential
potential has a set of virtual states. The authors of Ref. Atabek, Lefebvre,
and Jacon, 1982 studied the complex eigenstates of the complex exponential
potential both analytically and by means of the complex rotation method, and
soon after the eigenvalues were used as benchmarks to test a few of the other
numerical methods mentioned in the present work Kais and Herschbach (1993);
Midy, Atabek, and Oliver (1993); Fernandez (1995); Fernández (1996). Among
these is the RPM, which produced some baffling results: it yields complex
energies that are very close to those of the analytical solutions, but are not
quite the same, unlike the approaches of Refs. Atabek, Lefebvre, and Jacon,
1982; Kais and Herschbach, 1993; Midy, Atabek, and Oliver, 1993, which yielded
the correct ones. The wrong results provided by the RPM were later confirmed
in a more systematic study of the same problem Amore and Fernández (2008).
Ref. Fernández, 1996 also saw wrong results for the Schrödinger equation with
the one-dimensional potential $V(x)=(x^{2}-2J)\exp(-\lambda x^{2})$,
introduced by Moiseiev to model pre-dissociation resonances of diatomic
molecules Moiseyev, Certain, and Weinhold (1978), but it was later explained
in Ref. Fernández and Garcia, 2016 that the incorrect eigenvalues can also be
obtained by complex rotation, provided that the rotation angle is set to be
greater than the critical value $\theta_{{\rm crit}}=\pi/4$. In fact, such
eigenvalues had already been obtained and discussed by other authors Rittby,
Elander, and Brändas (1981, 1982a); Korsch, Laurent, and Möhlenkamp (1982);
Rittby, Elander, and Brändas (1982b). It is reasonable to expect that the
discrepancy between the exact complex eigenenergies for the repulsive
exponential potential and those obtained by the RPM can be explained in a
similar fashion to those for the potential introduced in Ref. Moiseyev,
Certain, and Weinhold, 1978. In the present work, we show that this assumption
is correct, and that the seemingly wrong resonances obtained by means of the
RPM are in fact complex solutions of the Schrödinger equation for a particle
exposed to an infinite exponential well, $V(r)=\lambda e^{r}$. Even though the
latter has been solved analyticallyAtabek and Lefebvre (1983), to our
knowledge, both spectra have not yet been compared.

This work is organized as follows. In Sec. II, we review the analytical
solution of the Schrödinger equation for both problems, and we perform a
comparison between their eigenspectra. Then, in Sec. III, we employ an
efficient implementation of the RPM to improve upon the computations of Refs.
Fernandez, 1995; Fernández, 1996; Amore and Fernández, 2008, and we study the
rate of convergence of the roots of the Hankel determinants towards both kinds
of complex eigenvalues. We finally sum up our results and draw further
conclusions in Sec. IV.

##  II Analytical treatment

### The Schrödinger equation with a repulsive exponential potential

We first treat the Schrödinger equation with a repulsive exponential
potential,

$-\phi^{\prime\prime}(r)+\lambda e^{-r}\phi(r)=\varepsilon\phi(r),$ (1)

with $\lambda>0$ and $\phi(0)=0$. As discussed in the Introduction, this
problem was already studied in Ref. Atabek, Lefebvre, and Jacon, 1982 in great
detail, but for consistency, we briefly describe its analytical solution in
the following paragraphs.

By defining $\mu=\sqrt{-4\varepsilon}$, Eq. (1) is exactly solvable in terms
of the Bessel functions of the first kind $J_{\pm\mu}(z)$, where
$z=2\sqrt{-\lambda}e^{-r/2}\,$ Ma (1946); Atabek, Lefebvre, and Jacon (1982),
or, equivalently, in terms of the modified Bessel functions of the first kind
$I_{\pm\mu}(t)$, where $t=2\sqrt{\lambda}e^{-r/2}$. We prefer the latter since
for real $r$, $t$ is also real, whereas $z$ is imaginary. The general solution
of Eq. (1) can be written as $A_{\mu}I_{\mu}(t)+B_{\mu}I_{-\mu}(t)$, and the
condition that $\phi(r)$ behaves as an outgoing wave, i.e., $\phi(r)\sim
e^{ikr}$, with $k=i\mu/2$, ${\rm Re}(k)>0$ and ${\rm Im}(k)<0\,$ Atabek,
Lefebvre, and Jacon (1982), which implies ${\rm Re}(\mu)<0$ and ${\rm
Im}(\mu)<0$, requires that $B_{\mu}=0$, since for small $|t|$,
$I_{\pm\mu}(t)\sim(t/2)^{\pm\mu}/\Gamma(\mu+1)$. The condition $\phi(0)=0$
results in

$I_{\mu_{n}}(2\sqrt{\lambda})=0,$ (2)

which yields the eigenvalues $\varepsilon_{n}=-\mu_{n}^{2}/4$. Here
$n=0,1,\ldots$ serves as a label that orders them by increasing absolute
value. It is well known that the solutions of Eq. (2) are either real and
negative or come in complex conjugate pairs, with ${\rm Re}(\mu)<0$ Atabek,
Lefebvre, and Jacon (1982). For $\lambda\rightarrow 0$, they tend to negative
integers, and as $\lambda$ increases, pairs of solutions coalesce to form a
complex pair, one pair at a time. The solutions with real $\mu<0$ correspond
to virtual states, i.e., states that grow exponentially as
$r\rightarrow\infty$, and the complex solutions correspond to either
resonances, if ${\rm Im}(\mu)<0$, or growing states ${\rm Im}(\mu)>0$. Since
the roots of Eq. (2) come in complex conjugate pairs, every resonance comes
with an associated growing state. In the present work, the notation $\mu_{n}$
will refer only to resonances. If needed, the associated growing states will
be referred to by using complex conjugation, i.e. $\mu_{n}^{*}$.

The authors of Ref. Atabek, Lefebvre, and Jacon, 1982 were limited to solving
Eq. (2) for relatively small values of $\lambda$, since it was easier for them
to find its roots by means of the computational resources that were available
at that time. Contrarily, nowadays they are quite trivial to find with any
number of significant digits by using any modern computer algebra software, or
a multiprecision library such as mpmath mpmath development team (2023), which
we have used extensively in the present work.

### The Schrödinger equation with an infinite exponential well

We now focus our attention on the Schrödinger equation with an infinite
exponential well potential, i.e.,

$-\phi^{\prime\prime}(r)+\lambda e^{r}\phi(r)=\epsilon\phi(r).$ (3)

It is related to Eq. (1) by a change of variable $r\rightarrow-r$, and it is
also solvable in terms of Bessel functions, as was shown by Atabek and
Lefebvre in Ref. Atabek and Lefebvre, 1983. They posed Eq. (3) as a three-
parameter problem, with $V(r)=Ae^{\alpha(r-r_{0})}$, but by making the change
of variable $r\rightarrow r/\alpha$, setting $\lambda=A\exp\left(-\alpha
r_{0}\right)/\alpha^{2}$, and multiplying the energy by $\alpha^{2}$, Eq. (3)
is obtained without loss of generality. In the present work we would like to
provide an equivalent derivation of its analytical solution that we deem more
adequate for the ensuing discussion.

The general solution of Eq. (3) can be written
$A_{\nu}I_{\nu}(s)+B_{\nu}K_{\nu}(s)$, where $s=2\sqrt{\lambda}e^{r/2}$, and
$K_{\nu}$ is the modified Bessel function of the second kind, defined for
noninteger $\nu$ as $K_{\nu}(s)=\pi\csc(\nu\pi)[I_{-\nu}(s)-I_{\nu}(s)]/2$,
and for integer $n$, $K_{n}(s)=\lim_{\nu\rightarrow n}K_{\nu}(s)$. It could
also be written in terms of $I_{\pm\nu}(s)$, but we choose $K_{\nu}$ instead
since for $|s|\rightarrow\infty$ they admit the following asymptotic
expansions,

$\displaystyle I_{\nu}(s)$ $\displaystyle\sim\left(\frac{1}{2\pi
s}\right)^{1/2}e^{s}\left[1-\frac{4\nu^{2}-1}{8z}+\frac{(4\nu^{2}-1)(4\nu^{2}-9)}{2!(8z)^{2}}+\ldots\right]$
(4) $\displaystyle K_{\nu}(s)$
$\displaystyle\sim\left(\frac{\pi}{2s}\right)^{1/2}e^{-s}\left[1+\frac{4\nu^{2}-1}{8z}+\frac{(4\nu^{2}-1)(4\nu^{2}-9)}{2!(8z)^{2}}+\ldots\right],.$
(5)

Eq. (4) is valid only when $-\pi/2<{\rm arg}\,s<\pi$, whereas Eq. (5) is valid
in the range $-\pi<{\rm arg}\,s<\pi$. For real $r$ (and, consequently, real
$s$) the decaying eigenfunctions of Eq. (3) require $A_{\nu}=0$, and the
condition $\phi(0)=0$ implies

$K_{\nu}(2\sqrt{\lambda})=0,$ (6)

which yields the bound states. For $\lambda>0$, there is an infinite amount of
roots of Eq. (6), and all of them are all imaginary Bateman (1953); Bagirova
and Khanmamedov (2020); Krynytskyi, , and and (2021), which implies that the
spectrum is comprised of an infinite number of positive energies. The
eigenfunctions are strongly decaying for $r\rightarrow\infty$, since
$\phi(r)\sim\sqrt{\pi/\sqrt{\lambda}}\exp[-2e^{r/2}\sqrt{\lambda}-r/4]$.

We now study the solutions to Eq. (3) under a complex rotation of the form
$r=\rho e^{i\theta}$, with $(\rho,\theta)\in\mathbb{R}$. We focus on the
interval $(-\pi/2,\pi/2)$ since when $\theta=\pm\pi/2$, the asymptotic
behavior of the solutions changes drastically, as $r$ becomes imaginary. In
fact, as $|\theta|>\pi/2$, the real part of $r$ changes sign, and the problem
could be reinterpreted as that of Eq. (1), but with $r=\rho
e^{i\theta^{\prime}}$, and $\theta^{\prime}=\pi-\theta$; we may therefore
define $\theta_{\rm crit}=\pi/2$ as the critical complex rotation angle for
which the infinite well is transformed into the finite repulsive exponential
potential. The complex-rotated $s$ can be written as

$s=2\sqrt{\lambda}e^{\frac{1}{2}\rho\cos\theta}e^{\frac{1}{2}i\rho\sin\theta},$
(7)

and, since ${\rm arg}\,s=\rho\sin(\theta)/2$, it becomes apparent that not
only does ${\rm arg}\,s$ change with $\theta$, but also with $\rho$. When
$|\rho\sin\theta|>\pi$, Eq. (4) is no longer valid, and when
$|\rho\sin\theta|>2\pi$, Eq. (5) becomes invalid, as well. To circumvent this
problem, we resort to the analytical continuation formulae for the modified
Bessel functions DLMF ,

$\displaystyle I_{v}(se^{m\pi i})$ $\displaystyle=e^{m\nu\pi i}I_{\nu}(s)$ (8)
$\displaystyle K_{v}(se^{m\pi i})$ $\displaystyle=e^{-m\nu\pi i}K_{\nu}(s)-\pi
i\sin(m\nu\pi)\csc(\nu\pi)I_{\nu}(s),$ (9)

where $m=0,\pm 1,\pm 2,\ldots$. Eqs. (8) and (9) can be employed to extend the
validity of Eqs. (4) and (5) to regions where ${\rm arg}\,s$ is greater than
$\pi/2$. For example, if $\rho$ and $\theta$ are such that $\pi/2<{\rm
arg}\,s<3\pi/2$, then by setting $m=-1$ in Eqs. (8) and (9), the argument of
$se^{m\pi i}$ stays within the region $(-\pi/2,\pi/2)$. In general, for any
given value of $\rho$, an integer $m$ can be chosen such that ${\rm
arg}(se^{m\pi i})$ lies between $-\pi/2$ and $\pi/2$, both for positive and
negative $\theta$. If $\lambda$ is large enough, the RHS of Eq. (9) is
expected to behave as an asymptotically decaying function if

$-(1+2m)\pi<\rho\sin\theta<-(1-2m)\pi,$ (10)

where $m$ and $\theta$ have opposite signs. Eq. (6) is now generalized,

$\displaystyle e^{-m{\nu_{n}^{(m)}}\pi
i}K_{\nu_{n}^{(m)}}(2\sqrt{\lambda})-\pi
i\sin[m{\nu_{n}^{(m)}}\pi]\csc[{\nu_{n}^{(m)}}\pi]I_{\nu_{n}^{(m)}}(2\sqrt{\lambda})=0,$
(11)

where again we use $n$ as a label for the eigenvalues, according to their
increasing absolute value, and now the value of $m$ is added as a
parenthesized superindex. The energy of each state is computed as
$\epsilon_{n}^{(m)}=-[\nu_{n}^{(m)}]^{2}/4$. When $m=0$ we recover Eq. (6),
but when $m\neq 0$, a different set of complex solutions is obtained. Since
for real $x$, $[I_{\nu}(x)]^{*}=I_{\nu^{*}}(x)$, and
$[K_{\nu}(x)]^{*}=K_{\nu^{*}}(x)$, by taking the complex conjugate of Eq.
(11), it can be readily seen that $\nu_{n}^{(-m)}=[\nu_{n}^{(m)}]^{*}$.

Eq. (11) generalizes Eqs. (25), (28) and (32) of Ref. Atabek and Lefebvre,
1983. The equivalence of both approaches is made more apparent by substituting
the definition of $K_{\nu}$ in terms of $I_{\pm\nu}$ into Eq. (11) and
dividing by $e^{m\nu\pi i}$, whereas we obtain

$\csc(\pi\nu)\left[e^{m\nu\pi i}I_{-\nu}(2\sqrt{\lambda})-e^{-m\nu\pi
i}I_{\nu}(2\sqrt{\lambda})\right]=0,$ (12)

[in the comparison, note that $I_{\nu}(z)=e^{\mp\nu\pi i/2}J_{\nu}(ze^{\pm\pi
i/2})$]. The factor $\csc(\pi\nu)$ cannot be overlooked since it becomes
singular when $\nu$ is an integer number. It is also apparent from Eq. (12)
that if $\nu_{n}$ is a solution, $-\nu_{n}$ is one as well, which is to be
expected since only $\nu^{2}$ appears in the Schrödinger equation. To simplify
the upcoming discussions, we will refer to the resonances that correspond to
the complex solutions of Eq. (2) as “barrier resonances”, (even though the
repulsive exponential potential is not a barrier, the complex solutions of
that problem behave as it were) and to those that correspond to Eqs. (11) and
(12) as “well resonances”. We may as well refer to Eq. (1) as the “barrier
problem”, and to Eq. (3) as the “well problem”. The notation we employ to
distinguish between the solutions of Eqs. (1) and (3), as well as the one we
use to refer to the quantities computed in Sec. III, is summarized in Table 1.

Symbol |  Meaning
---|---
$\mu_{n}$ |  Solutions of Eq. (2), ordered by increasing absolute value. If $\mu_{n}$ is complex, ${\rm Im}\,\mu_{n}<0$.
$\nu_{n}^{(m)}$ |  Solutions of Eq. (11), ordered by increasing absolute value.
$\nu_{n^{*}}^{(m)}$ |  Solution of Eq. (11) that is closest to $\mu_{n}$.
$\varepsilon_{n}$, $\epsilon_{n}^{(m)}$, $\epsilon_{n^{*}}^{(m)}$ |  $-\mu_{n}^{2}/4$, $-[\nu_{n}^{(m)}]^{2}/4$, $-[\nu_{n^{*}}^{(m)}]^{2}/4$.
$\tilde{\varepsilon}_{n}$, $\tilde{\epsilon}_{n}^{(m)}$, $\tilde{\epsilon}_{n^{*}}^{(m)}$ |  Roots of $H_{D}^{d}$ closest to $\epsilon_{n}$, $\epsilon_{n}^{(m)}$, and $\epsilon_{n^{*}}^{(m)}$ .
$\Delta_{n},\Delta_{n}^{(m)},\Delta_{n^{*}}^{(m)}$ |  $-\log_{10}|\varepsilon_{n}-\tilde{\varepsilon}_{n}|$, $-\log_{10}|\epsilon_{n}^{(m)}-\tilde{\epsilon}_{n}^{(m)}|$, $-\log_{10}|\epsilon_{n^{*}}^{(m)}-\tilde{\epsilon}_{n^{*}}^{(m)}|$.
Table 1:  Summary of notation.

### Comparison of both spectra

We now focus on comparing the well and barrier resonances. To that end, we
have computed the well resonances for $\lambda$ ranging from $0.1$ to $12$,
and $m=1,2,3,4$. We have also computed the barrier resonances for the same
values of $\lambda$; both kinds are shown in Fig. 2 (the barrier resonances
are the same for each plot, since they do not depend on $m$).

Figure 1:  Real and imaginary parts of the well resonances for $m=1,2,3$, and
$4$ [Eq. (11), solid lines], and of the barrier resonances [Eq. (2), dashed
lines].

It can be seen that for increasing $\lambda$, when $m=1$, all of the well
resonances converge towards the growing-state barrier ones. For $m=2$, half of
the well resonances converge towards barrier resonances, whereas the other
half appear to converge to semi-integer numbers. Following this trend, for
$m=3$, one every three well resonances converge towards barrier resonances,
wheras the other two converge towards fractions of 3 (excluding integers), and
for $m=4$, one in four well resonances converge towards barrier resonances. It
is apparent that the well resonances can be divided into two kinds; the first
kind involves those solutions that converge towards barrier resonances,
whereas the second kind involves those well resonances that converge towards
fractional numbers, and are only present for $m>1$. We will not provide
rigorous proof, but the existence of both kinds of solutions can somewhat be
explained by analyzing Eq. (12). First, we note that, provided that $\lambda$
is large enough, because of Eq. (5), $K_{\nu}(2\sqrt{\lambda})$ is a complex
number of very small magnitude. In addition, for ${\rm Im}(\nu_{n}^{(m)})<0$
and $m>0$, $e^{-m\nu\pi i}$ has also a very small magnitude. Therefore, it can
be expected that the zeros of Eq. (11) will be close to those of
$\sin(m\nu\pi)\csc(\nu\pi)I_{\nu}(2\sqrt{\lambda})$. The latter can occur
either if $I_{\nu}(2\sqrt{\lambda})=0$, which implies that
$\nu_{n}^{(m)}\approx\mu_{n}$ (this explains the first kind of resonances), or
if $\sin(m\nu\pi)\csc(\nu\pi)=0$, which occurs whenever $\nu=k/m$, where $k$
is an integer that is not a multiple of $m$ (explaining the second kind).

In Table 2, we show $\nu_{0^{*}}^{(m)}$, where the index $n^{*}$ refers to the
solution of Eq. (11) that is closest to $\mu_{n}$, for several values of
$\lambda$ and for increasing $m$.

$\lambda=1/2$ | $\lambda=2$
---|---
$m$ | ${\rm Re}(\nu)$ | ${\rm Im}(\nu)$ | $m$ | ${\rm Re}(\nu)$ | ${\rm Im}(\nu)$
1 | -1.708889402333520 | 0.313848239102419 | 1 | -2.19998150521571 | 1.47380332298928
2 | -1.746877069032750 | 0.289471834435959 | 2 | -2.19996056123564 | 1.47382974620043
3 | -1.744703781423560 | 0.280605627380309 | 3 | -2.19996055822964 | 1.47382974508387
4 | -1.743009480401320 | 0.281164740911281 | 4 | -2.19996055822965 | 1.47382974508356
5 | -1.743125645499290 | 0.281441559500668 | $\infty$ | -2.19996055822965 | 1.47382974508356
6 | -1.743171729577290 | 0.281420065685151 |  |  |
7 | -1.743167735582120 | 0.281412353572321 |  |  |
8 | -1.743166449785450 | 0.281413091451232 | $\lambda=10$
9 | -1.743166585008590 | 0.281413305242469 | $m$ | ${\rm Re}(\nu)$ | ${\rm Im}(\nu)$
10 | -1.743166620466710 | 0.281413280623280 | 1 | -2.91772003768616 | 4.57704829026394
11 | -1.743166616009740 | 0.281413274758419 | 2 | -2.91772003768629 | 4.57704829026402
12 | -1.743166615042500 | 0.281413275561234 | $\infty$ | -2.91772003768629 | 4.57704829026402
13 | -1.743166615186450 | 0.281413275720256 |  |  |
14 | -1.743166615212510 | 0.281413275694549 |  |  |
15 | -1.743166615207940 | 0.281413275690295 | $\lambda=100$
16 | -1.743166615207250 | 0.281413275691106 | $m$ | ${\rm Re}(\nu)$ | ${\rm Im}(\nu)$
17 | -1.743166615207390 | 0.281413275691218 | 1 | -4.32572713644441 | 17.46003535736490
18 | -1.743166615207410 | 0.281413275691193 | $\infty$ | -4.32572713644441 | 17.46003535736490
19 | -1.743166615207400 | 0.281413275691190 |  |  |
20 | -1.743166615207400 | 0.281413275691191 |  |  |
$\infty$ | -1.743166615207400 | 0.281413275691191 |  |  |
Table 2:  $\nu_{1^{*}}^{(m)}$, and $\mu_{0}$ (denoted with $m=\infty$) for
different values of $\lambda$. See Table 1 for the definition of each symbol.

It can be seen that the distance between $\nu_{n^{*}}^{(m)}$ and $\mu_{n}$
becomes smaller with increasing $\lambda$, but also with $m$. We further
demonstrate the latter by showing in Fig. 2 the computation of
$-\log_{10}|\mu_{n}-\nu_{n^{*}}^{(m)}|$, for $n=1,\ldots,4$, and several
values of $\lambda$ ranging from $1/2$ to $10$.

Figure 2:  Logarithmic difference between the roots of Eqs. (2) and (11).

It can be seen there that for fixed $\lambda$ and increasing $m$, the distance
between the well and barrier resonances decreases exponentially. In fact, one
can show that

$\nu^{(m)}-\mu\approx
i\,\frac{K_{\mu}}{\pi\dot{I}_{\mu}}\,e^{-i\mu\pi(2m-1)},$ (13)

where $\dot{I}_{\mu}=(\partial I_{\nu}/\partial\nu)|_{\mu=\nu}$, by expanding
the LHS of Eq. (11) a Taylor series about $\nu=\mu$ [where $\mu$ is a root of
Eq. (2)], keeping the linear term, and assuming that the term involving
$\dot{I}_{\mu}$ is much larger (in the asymptotic sense) than those involving
$K_{\mu}$ and $\dot{K}_{\mu}$. Although Eq. (13) is not proved rigorously, we
have performed several computations that confirm it numerically.

Going back to the barrier problem [Eq. (1)], one may wonder if a similar set
of solutions such as those of Eq. (11) may be discovered through complex
rotation, since if $r=\rho e^{i\theta}$ is substituted in the definition of
$t$, it follows that ${\rm Arg}\,t=-\rho\sin(\theta)/2$; i.e., ${\rm Arg}\,t$
changes with $\rho$, thereby modifying the asymptotic behavior of the solution
depending on the region where the boundary condition is imposed, just as in
the case of the well problem. The answer is that it would not, since, even
though an equation analogous to (10) might be posed for the barrier problem,
the analytical continuation of $I_{\nu}$ is proportional to the same function
[see (8)]; the set roots of an equation equivalent to (11) are then the same
for every $m$.

##  III Riccati-Padé method

The purpose of the present section is to apply the RPM to both Eqs. (1) and
(3). The application of the RPM to central-force problems was presented in
Ref. Fernández, Ma, and Tipping, 1989a, but for completeness we briefly review
it here.

The RPM can be applied to any central-force problem, i.e.,

$-\psi^{\prime\prime}(r)+\left[V(r)+\frac{l(l+1)}{r^{2}}-E\right]\psi(r)=0,$
(14)

with $\psi(0)=0$, and $\psi(\infty)=0$, where the potential $V(r)$ can be
expanded in a Laurent series of the form $\sum_{j=-1}^{\infty}v_{j}r^{j}$. In
that case, $\psi(r)\sim Cr^{l+1}$ for $r\rightarrow 0$, and its logarithmic
derivative is singular at origin, $\psi^{\prime}(r)/\psi(r)\sim(l+1)/r$;
therefore it must be regularized. The resulting regularized function,
$f(r)=(l+1)/r-\psi^{\prime}(r)/\psi(r)$, can then be expanded in a Taylor
series about the origin, $f(r)=\sum_{j=0}^{\infty}f_{j}r^{j}$. It is a
solution to the Riccati equation,

$f^{\prime}(r)-f^{2}(r)+\frac{2(l+1)}{r}f(r)-\left[E-V(r)\right]=0,$ (15)

from which a recurrence relation can be found for the $f_{j}$ coefficients,

$f_{j+1}=\frac{1}{2l+j+3}\left[\sum_{i=0}^{j}f_{i}f_{j-i}-v_{j}+E\delta_{j0}\right],$
(16)

with $f_{0}=-v_{-1}/(2l+2)$. $f(r)$ can also be approximated by a Padé
approximant of order $[M/N]$, i.e., a quotient of polynomials of orders $M$
and $N$ that are typically chosen in such a way that
$[M/N](r)-f(r)=\mathcal{O}(r^{M+N+1})$. If the condition
$[M/N](r)-f(r)=\mathcal{O}(r^{M+N+2})$ is imposed instead, it leads to a
system of equations that has nontrivial solutions if

$|H(D,d)|=\begin{vmatrix}f_{d+1}&f_{d+2}&\ldots&f_{d+D}\\\
f_{d+2}&f_{d+3}&\ldots&f_{d+D+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
f_{d+D}&f_{d+D+1}&\ldots&f_{2D+d-1}\end{vmatrix}=0.$ (17)

Here we have defined $d=M-N$ and $D=N+1$. The Hankel determinants [the
determinants of $H(D,d)$] can be computed efficiently by resorting to the
following recurrence relation,

$H_{D}^{d}=\frac{H_{D-1}^{d}H_{D-1}^{d-2}-(H_{D-1}^{d+1})^{2}}{H_{D-2}^{d-2}},$
(18)

where $H_{D}^{d}=|H(D,d)|$, with $H_{0}^{d}=1$.

The RPM is closely related to the process of sending a movable pole at
infinity Abbasbandy and Bervillier (2011). This pole can be moved through any
ray in the complex plane, and for this reason the solutions of Eq. (17) yield
approximations to the eigenvalues of Eq. (14) whose corresponding
eigenfunctions that asymptotically decay through different regions of the
complex plane. Concordantly, the roots of Eq. (17) persist if a complex
rotation of the variable is performed Fernández and Garcia (2016). Another
important characteristic of the RPM is that $H_{D}^{d}$ typically exhibits a
great number of roots in the neighborhood of each eigenvalue; this makes it
easier to differentiate the roots that approximate an eigenvalue from those
that are spurious, but it also complicates the task to find the optimal
sequence of roots that converges to each eigenvalue by means of iterative
methods.

### Application to the present problem

In the present work we have employed two different approaches to compute the
roots of Eq. (17). The first of them is to compute an expression for
$H_{D}^{d}$ analytically, and solve for $E$ using a polynomial root-finding
algorithm; in this way, all of the roots of $H_{D}^{d}$ can be found for a
given $D$, but it is only applicable to small $D$ values, since the degree of
$H_{D}^{d}$ grows rapidly. The second one is to use the recurrence relation
(18) to compute $H_{D}^{d}$ and its derivative numerically, and to employ the
Newton-Raphson (NR) to find its roots. It has the advantage of being extremely
efficient, but only one root can be computed at a time, and a good initial
value of the energy is required for the NR method to converge. Since the
Hankel determinants present clusters of roots that approach the eigenvalues,
it is not guaranteed that the NR method will find the optimal one, but with
large $D$, even the suboptimal sequences of roots provide a good approximation
to the eigenvalues. We are typically able to compute determinants with $D\leq
30$ with the first approach before running out of memory, whereas with the
second one, we have reported results with $D>800$ Fernández and Garcia (2017),
and even larger determinants are possible, provided that the starting points
for the NR method are sufficiently accurate. We would like to mention here
that we have utilized the GiNaC library Bauer, Frink, and Kreckel (2002);
Vollinga (2006) for the implementation of the second approach.

By setting $V(r)=\lambda e^{\pm r}$, and $l=0$, Eq. (14) yields Eq. (1) or
(3). The choice of the sign in $V(r)$ is irrelevant since, as stated earlier,
the roots of $H_{D}^{d}$ remain unmodified. Since the results obtained by the
RPM do not typically change with $d$, we have set $d=0$ in all of our
computations. We have computed all of the roots of $H_{30}^{0}$, and selected
the ones that coincide with the exact eigenvalues of both Eqs. (1) and (3);
these are shown, for $\lambda=1/2$, in Table 3, and, for $\lambda=10$, in
Table 4. It can be appreciated that some of them approach the bound states,
and others approach the barrier and well resonances, but none of them are
close to any of the virtual states. For both values of $\lambda$, the bound
states and resonances corresponding to $m=1$ are found with more significant
digits than those with $m=2$, and in the latter case, there is also one root
that corresponds to an $m=3$ resonance. Among the results presented in Table 3
is included the seemingly wrong ground state for the exponential barrier,
$E\approx-0.705450568055028+0.268165964871580\,i$, found in Ref. Amore and
Fernández, 2008. This value coincides with $-[\nu_{1^{*}}^{(1)}]^{2}/4$, where
$\nu_{1^{*}}^{(1)}$ is taken from Table 2. The first root with $m=1$, and the
second one with $m=2$ of Table 3 may also be considered as approximations
towards $\varepsilon_{0}(\lambda=1/2)\approx-0.73985-0.2452i$, whereas the
third entries of Table 4 for $m=1,2$ may be considered as approximations
towards $\varepsilon_{0}(\lambda=10)\approx 3.10907-6.6772i$. We also note
here that the roots of Eq. (17) are either real or come in complex conjugate
pairs, since the $v_{j}$ coefficients of Eq. (16) are real, the $f_{j}$
coefficents are polynomials on $E$ with real coefficients, and so is
$H_{D}^{d}$; therefore, the same analysis performed here is valid for the
growing-state solutions.

$m$ | ${\rm Re}(\epsilon)$ | ${\rm Im}(\epsilon)$
---|---|---
0 | $3.2292159472536048534$ | $0.0$
0 | $7.0837240758343799367$ | $0.0$
0 | $11.541280962123897449$ | $0.0$
0 | $16.53405783500778465$ | $0.0$
0 | $27.94592416939$ | $0.0$
0 | $41.0616$ | $0.0$
0 | $48.206$ | $0.0$
1 | $-0.70545056805502837410$ | $0.26816596487157970576$
1 | $-2.0145028385826182272$ | $-0.91594563985411650792$
1 | $-3.17882146596656262$ | $-2.84627255601162901$
1 | $-4.19709693935568$ | $-5.32673986864807$
1 | $-5.07091771510$ | $-8.26819182266$
1 | $-5.80277763$ | $-11.6168375$
1 | $-6.395162$ | $-15.33562$
1 | $-6.850$ | $-19.40$
2 | $-0.0624600582$ | $-0.00235480490$
2 | $-0.7419463$ | $0.2528359$
2 | $-0.5549276$ | $0.04161432$
2 | $-1.522$ | $-0.1790$
Table 3:  Complex eigenvalues obtained with $H_{30}^{0}=0$, for $\lambda=1/2$. Results coincide with the exact ones up to the last digit reported. They have been truncated at 20 significant digits. $m$ | ${\rm Re}(\epsilon)$ | ${\rm Im}(\epsilon)$
---|---|---
0 | $24.095880341888706123$ | $0.0$
0 | $39.078924487590608756$ | $0.0$
0 | $54.342383405484864190$ | $0.0$
0 | $70.173203793408790845$ | $0.0$
0 | $86.630901797549645582$ | $0.0$
0 | $103.722769025200$ | $0.0$
0 | $121.4400060068$ | $0.0$
0 | $139.768519$ | $0.0$
0 | $158.693$ | $0.0$
1 | $-10.243001240262833149$ | $6.9319297923706305098$
1 | $-3.7481264282889514022$ | $8.0918634524566962810$
1 | $3.1090702082731894910$ | $6.6772727549801459614$
1 | $-16.525883780702648862$ | $4.1357639513958152182$
1 | $-22.63511276702947437$ | $0.09542171991804396736$
1 | $-28.583717704014$ | $-4.97259427564364$
1 | $-34.3769116736$ | $-10.93053096967$
1 | $-40.017132$ | $-17.682645$
1 | $-45.506$ | $-25.1582$
2 | $-0.0624998$ | $-0.000000526971$
2 | $-3.74812643$ | $8.09186345$
2 | $3.109070208273$ | $6.677272754981$
3 | $-3.748126$ | $8.091863$
Table 4:  Complex eigenvalues obtained with $H_{30}^{0}=0$, for $\lambda=10$.
Results coincide with the exact ones up to the last digit reported. They have
been truncated at 20 significant digits.

It is expected that for increasing $D$, the accuracy of these results will
improve, but also that there will be roots of $H_{D}^{0}$ that approach the
well resonances with $m\geq 4$. Since the computation of the whole set of
roots of $H_{D}^{d}$ becomes expensive for $D>30$, in order to study how
rapidly the roots of $H_{D}^{d}$ converge towards both the barrier and well
resonances, we have resorted to our NR-based approach. With it, we have
computed $\tilde{\epsilon}_{n}^{(m)}$, which is the root of $H_{D}^{0}$
closest to $\epsilon_{n}^{(m)}$, and $\tilde{\varepsilon}_{n}$, the root of
$H_{D}^{0}$ closest to $\varepsilon_{n}$, for $D=5,10,\ldots,500$,
$m=0,\ldots,4$, $n=0,\ldots,7$, and $\lambda=1/2,10$. We have employed the
exact resonance energies computed from Eqs. (2) and (11) as a starting point
for the NR method. Instead of showing the tens, or even hundreds, of
coincident significant digits between the exact eigenvalues and their RPM
approximations, we illustrate how accurate the latter are by means of the
quantities
$\Delta_{n}^{(m)}(D)=-\log_{10}|\tilde{\epsilon}_{n}^{(m)}(D)-\epsilon_{n}^{(m)}|$,
and $\Delta_{n}(D)=-\log_{10}|\tilde{\varepsilon}_{n}(D)-\varepsilon_{n}|$,
both of which provide a good estimate of the number of correct decimal places.
The plots of $\Delta_{n}^{(m)}(D)$ and $\Delta_{n}(D)$ are shown in Fig. 3;
the latter labeled as $m=\infty$.

Figure 3:  $\Delta_{n}^{(m)}(D)$ for $\lambda=1/2$ and $\lambda=10$.
$m=\infty$ denotes $\Delta_{n}(D)$, instead.

It can be seen there that $\Delta_{n}^{(m)}(D)$ are straight lines whose
slopes depend on $m$, but not on $n$. The position of these straight lines
does depend on $n$, and we have found that in general they move to the right
when $n$ increases. Both findings are not surprising: on one hand, for
increasing $n$, the exact eigenfunction is expected to have a larger number of
nodes, which increments the number of $f_{j}$ coefficients that are needed to
represent its logarithmic derivative correctly. On the other, for increasing
$m$, the boundary conditions are moved further to the right (i.e., to larger
$|r|$), which also increases the number of coefficients needed. As is made
apparent by our computations, the effect of the latter is more noticeable. The
plots of $\Delta_{n}(D)$ show that the roots of $H_{D}^{d}$ also converge
towards the barrier resonances, albeit more slowly. Instead of being straight
lines, they are curves that are concave everywhere, except for a number of
plateaus which is different for each state. Convergence rate is lower for
higher $n$, meaning that the top curve in the $\lambda=10$ plot corresponds to
$\Delta_{0}$, the curve below it corresponds to $\Delta_{1}$, and so on. To
ease the discussion of these results, we have compiled some logarithmic
differences between well and barrier resonances, i.e.,
$-\log_{10}|\nu_{n^{*}}^{(m)}-\mu_{n}|$, in Table 5.

Focusing first on the curve for $\Delta_{0}$, we note that, when $D<15$, it
coincides with $\Delta_{0^{*}}^{(1)}$, which suggests that the solution being
obtained is a well resonance, whose eigenvalue coincides with that of a
barrier resonance up to approximately 12 decimal places. At $D=15$, a small
region is reached where incrementing $D$ does not seem to improve accuracy; in
that region, $\Delta_{0}\approx 12.5$ coincides with $-{\rm
log}_{10}|\varepsilon_{0}-\epsilon_{0^{*}}^{(1)}|$, implying that
$\tilde{\epsilon}_{0^{*}}^{(1)}$ is a more accurate representation of
$\epsilon_{0^{*}}^{(1)}$ than $\epsilon_{0^{*}}^{(1)}$ is of
$\varepsilon_{0}$. When $D>25$, convergence seems to pick up, and the plot of
$\Delta_{0}$ becomes a concave curve with no additional plateaus; i.e., for
$D>25$ and $m>1$, $\Delta_{0}>\Delta_{0^{*}}^{(m)}$. The curve for
$\Delta_{1}$ is similar, but presenting two plateaus; before the first
plateau, it coincides with the curve for $\Delta_{1^{*}}^{(1)}$. Then, at the
first plateau, located approximately between $D=15$ and $D=40$,
$\Delta_{1}=8.75$, in coincidence with the entry for $m=1$, $n=0$ of Table 5.
After that, $\Delta_{1}$ adopts the form of $\Delta_{1^{*}}^{(2)}$, until it
reaches a second plateau where $\Delta_{1}\approx 17.53$ [c.f. Table 5],
located in a region where $65<D<80$. Finally, it assumes its concave shape,
where $\Delta_{1}>\Delta_{1^{*}}^{(m)}$, for $m>2$. Similar explanations can
be devised for the curves for $\Delta_{2}$, which seems to present four
plateaus (all included in Table 5), and for $\Delta_{3}$, which shows about 5
plateaus, all of them also included in the same table. The plateaus observed
in Fig. 3 are quite probably an artificial byproduct of our method to find the
roots of $H_{D}^{d}$. It is very likely that there are sequences of roots of
the Hankel determinants that coincide with the concave part of the curves we
described, but do not have the plateaus nor the parts that match exactly with
the curves for well resonances; we are just not able to find them since we do
not have good starting points for the NR method.

$m$ | $n$
---|---
0 | 1 | 2 | 3
1 | 12.39 | 8.75 | 5.60 | 2.70
2 | 24.88 | 17.53 | 11.22 | 5.46
3 | 37.37 | 26.30 | 16.85 | 8.22
4 | 49.86 | 35.07 | 22.47 | 10.97
5 | 62.34 | 43.85 | 28.10 | 13.73
Table 5:  $-\log_{10}|\varepsilon_{n}-\epsilon_{n^{*}}^{(m)}|$ for the four
states with $\lambda=10$.

The results presented above suggest that the roots of the Hankel determinants
converge towards both the well and barrier resonances. The authors of Ref.
Amore and Fernández, 2008 only found the well resonances, since they performed
computations with $D\leq 20$, which is not enough to discover the barrier
ones.

##  IV Conclusions

We have studied the analytical solutions to the Schrödinger equation with a
repulsive exponential potential, as well as those to the Schrödinger equation
with an infinite exponential well, and shown that the complex spectra
associated to both problems are related as some of the eigenvalues of the
latter tend to those of the former when the potential parameter $\lambda$
increases. Those well resonances that do not tend towards barrier ones do so
instead to real fractional numbers. Starting from any of the two problems, if
a complex rotation $r=\rho e^{i\theta}$ is performed, then $\theta_{\rm
crit}=\pi/2$ separates one set of resonances from the other. These findings
are similar to those of Refs. Rittby, Elander, and Brändas, 1981, 1982a;
Korsch, Laurent, and Möhlenkamp, 1982; Rittby, Elander, and Brändas, 1982b
regarding the potential proposed by MoiseievMoiseyev, Certain, and Weinhold
(1978), which had two sets of resonances that were uncovered by complex
rotation, separated by $\theta_{\rm crit}=\pi/4$. In both cases, even though
the asymptotic behavior of the solutions is markedly different, the
eigenvalues remain quite close to each other.

Regarding the RPM, we have found a satisfactory explanation for the seemingly
wrong results obtained in Refs. Fernández, 1995, 1996; Amore and Fernández,
2008: all the resonances that are deemed wrong there are in fact well
resonances. We have computed the roots of the Hankel determinants and analyzed
how they converge towards both the well and barrier resonances, our findings
suggesting that there are sequences of them that converge independently to
both of their kinds. Contrarily, we were not able to obtain any of the virtual
states by means of this method. All of the above suggests that it is only
possible to obtain approximations to eigenvalues that are discoverable by a
complex rotation, even though the RPM does not require to perform one
explicitly. The results obtained in the present work therefore provide
evidence that the convergent sequences of roots of the Hankel determinants
always correspond to eigenfunctions that decay asymptotically through some
path in the complex plane.

## Acknowledgments

This work was funded by Consejo Nacional de Investigaciones Científicas y
Técnicas (CONICET).

## Conflict of interest statement

The author has no conflicts to disclose.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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