††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Decisive role of electron-phonon coupling for phonon and electron
instabilities in transition metal dichalcogenides

Zishen Wang  Department of Physics, National University of Singapore, 117542
Singapore, Singapore  Centre for Advanced 2D Materials, National University of
Singapore, 117546 Singapore, Singapore  Chuan Chen  Institute for Advanced
Study, Tsinghua University, 100084 Beijing, China  Max-Planck Institute for
the Physics of Complex Systems, 01187 Dresden, Germany  Jinchao Mo  Department
of Physics, National University of Singapore, 117542 Singapore, Singapore  Jun
Zhou<EMAIL_ADDRESS>Institute of Materials Research &
Engineering, A*STAR (Agency for Science, Technology and Research), 138634
Singapore, Singapore.  Kian Ping Loh<EMAIL_ADDRESS>Centre for Advanced
2D Materials, National University of Singapore, 117546 Singapore, Singapore
Department of Chemistry, National University of Singapore, 117543 Singapore,
Singapore  Yuan Ping Feng<EMAIL_ADDRESS>Department of Physics, National
University of Singapore, 117542 Singapore, Singapore  Centre for Advanced 2D
Materials, National University of Singapore, 117546 Singapore, Singapore

###### Abstract

The origin of the charge density wave (CDW) in transition metal dichalcognides
has been in hot debate and no conclusive agreement has been reached. Here, we
propose an ab-initio framework for an accurate description of both Fermi
surface nesting and electron-phonon coupling (EPC) and systematically
investigate their roles in the formation of CDW. Using monolayer 1H-NbSe2 and
1T-VTe2 as representative examples, we show that it is the momentum-dependent
EPC softens the phonon frequencies, which become imaginary (phonon
instabilities) at CDW vectors (indicating CDW formation). Besides, the
distribution of the CDW gap opening (electron instabilities) can be correctly
predicted only if EPC is included in the mean-field model. These results
emphasize the decisive role of EPC in the CDW formation. Our analytical
process is general and can be applied to other CDW systems.

The formation of charge density wave (CDW) is a spontaneous symmetry breaking
process with periodic charge density modulations and lattice distortions below
a critical temperature ($T_{CDW}$) [1, 2]. However, the origin of CDW is a
long-standing problem, which attracts broad research interest in condensed
matter physics [3, 4, 5, 6]. The first mechanism, Fermi surface nesting (FSN),
was originated from Peierls’ model of an ideal one-dimensional (1D) metal
atomic chain. FSN relates to an elastic electronic scattering at the Fermi
surface [7]. The zero-energy electronic excitations screen the phonon
vibration at the CDW vector Q, inducing an abrupt phonon softening (known as
Kohn anomaly) [8]. However, the extension of FSN to real materials is not
ideal. With a few exceptions, FSN is argued to have limited power in inducing
CDW distortions in higher-dimensional systems [4, 5, 6, 9].

Instead, momentum-dependent electron-phonon coupling (q-EPC), which involves
an inelastic electronic scattering mediated by a phonon, is argued to be more
prevailing in higher-dimensional systems [4, 5, 6, 9]. In an electron-phonon
interaction dominant system, the electron field can be integrated out as a
perturbation to the free phonon field, softening phonon frequencies from their
bare values [see SI-III for details]. Thus, both FSN and q-EPC may soften
phonons to imaginary values (phonon instabilities) and induce CDW distortions.
However, quantitative studies of q-EPC are rare, and the reported method to
directly obtain q-EPC is a tight-binding model by merely using the electronic
band structure [10, 11, 12]. Besides, such a semi-empirical method is hard to
give rigorous results compared to the first-principles calculations.
Therefore, it is imperative to explore a general method to accurately describe
q-EPC in the CDW materials.

Another important feature of a CDW is the band gap opening (electron
instabilities) accompanied by the CDW distortions, which lowers the energy of
the system [2, 4, 13]. The location of the CDW gap in the Brillouin zone (BZ)
can be identified accurately by a band unfolding scheme [14, 15, 16], which
however requires the prior knowledge of the corresponding CDW structure and
cannot provide insight into the underlying mechanism. Thus, the driving force
of the CDW gap is still elusive. A convenient method that can give physical
insight into the gap opening is needed.

In this letter, we use monolayer 1H-NbSe2 and 1T-VTe2 (in short NbSe2 and
VTe2) as representative examples for the most common high-symmetry phases of
the transition metal dichalcogenides (TMDs). Another reason for such a choice
is the mechanisms of their CDWs are considered to be different. The electron-
phonon coupling (EPC) has been shown to be dominant in NbSe2 [15, 17, 18],
while FSN in VTe2 seems to be substantial, which leads to a peak in the static
Lindhard susceptibility [19, 20]. Thus, we conduct a comprehensive study of
their CDW properties from both phononic and electronic perspectives. In the
phononic part, we obtain the accurate q-EPC from the fully first-principles
calculations, which allows us to make a straightforward comparison between the
contributions from FSN and q-EPC in the CDW formation. Interestingly, the
q-EPC is shown to play the dominant role in designating the CDW vectors for
both NbSe2 and VTe2. While in the electronic part, we find the rigorous EPC
matrix elements are the key to predict the distribution of the CDW gaps by our
mean-field model. In addition, we argue that our analysis process, besides its
conciseness and accurateness, should be a general framework, which can be
easily exploited in other CDW systems.

Figure 1: (a) Top and side views of the crystal structure, (b) phonon
dispersion of monolayer 1H-NbSe2. (c, d) Same as panel (a, b) but for
monolayer 1T-VTe2. The insert figure in (b) displays the first BZ with
reciprocal lattice vectors.

Phonon instabilities and CDW mechanism.–TMDs have layered structures, in which
transition metal atoms are sandwiched by chalcogen atoms, forming trigonal
prisms (a 1H phase) or octahedrons (a 1T phase). The structures of 1H-NbSe2
and 1T-VTe2 are shown in Figs. 1(a) and 1(c), respectively. As the phonon
spectra for NbSe2 and VTe2 shown in Figs. 1(b) and 1(d), their longitudinal
acoustic (LA) phonon modes collapse at the CDW vectors, which triple the unit
cell for NbSe2 with $\mathbf{Q^{H}}=2/3\Gamma M$, and quadruple the unit cell
for VTe2 with $\mathbf{Q^{T}}=1/2\Gamma M$, in line with the CDW supercells
obtained from experiments [19, 21, 22, 23].

To account for the driving force of the phonon softening, the relationship
between the softened phonon frequency $\omega_{\mathbf{q}}$ and its bare
phonon frequency $\Omega_{\mathbf{q}}$ is described under the random phase
approximation (RPA) [See more details in SI-III]:

$\omega_{\mathbf{q}}^{2}=\Omega_{\mathbf{q}}^{2}-2\Omega_{\mathbf{q}}\chi_{\mathbf{q}}$
(1)

where $\chi_{\mathbf{q}}$ is the generalized static electronic susceptibility,
including both contributions from FSN and EPC, which is given by:

$\chi_{\mathbf{q}}=\sum_{\mathbf{k}}{\left|g_{\mathbf{k},\mathbf{k}+\mathbf{q}}\right|^{2}\frac{f(\varepsilon_{\mathbf{k}})-f(\varepsilon_{\mathbf{k}+\mathbf{q}})}{\varepsilon_{\mathbf{k}+\mathbf{q}}-\varepsilon_{\mathbf{k}}}}$
(2)

where $f(\varepsilon)$ is the Fermi-Dirac function of the eigenvalue
$\varepsilon$, $g_{\mathbf{k},\mathbf{k}+\mathbf{q}}$ is the electron-phonon
coupling matrix element that couples electronic states k and k+q with a phonon
of momentum q. According to eq. (1), the ordering vector is estimated from the
maximum of $\chi_{\mathbf{q}}$ [11, 12]. The $\chi_{\mathbf{q}}$ often reduces
to the static Lindhard susceptibility ${\chi^{\prime}}_{\mathbf{q}}$ under the
constant matrix element approximation
($|g_{\mathbf{k},\mathbf{k}+\mathbf{q}}|=1$):

${\chi^{\prime}}_{\mathbf{q}}=\sum_{\mathbf{k}}\frac{f(\varepsilon_{\mathbf{k}})-f(\varepsilon_{\mathbf{k}+\mathbf{q}})}{\varepsilon_{\mathbf{k}+\mathbf{q}}-\varepsilon_{\mathbf{k}}}$
(3)

which is a pure electron effect, and its peak reflects the electronic
instability by FSN [4]. Similarly, $\chi_{\mathbf{q}}$ can reduce to the q-EPC
${\bar{g}}_{\mathbf{q}}$ under the “constant fraction” approximation
($\frac{f(\varepsilon_{\mathbf{k}})-f(\varepsilon_{\mathbf{k}+\mathbf{q}})}{\varepsilon_{\mathbf{k}+\mathbf{q}}-\varepsilon_{\mathbf{k}}}=1$):

${\bar{g}}_{\mathbf{q}}=\sum_{\mathbf{k}}\left|g_{\mathbf{k},\mathbf{k}+\mathbf{q}}\right|^{2}$
(4)

which reflects a pure EPC effect. Focusing on the low energy interaction
around the Fermi level, we only consider the coupling between the lowest
phonon branch and the single electronic band which crosses the Fermi level
[see the pink bands in Figs. S1(b) and S1(e)]. Due to the complexity in
describing the EPC matrix elements, only static Lindhard susceptibility has
been widely used to understand the CDW formation, while the q-EPC which
however may play a more important role is ignored [4, 6].

Here, to remedy this blemish, we applied density functional perturbation
theory (DFPT) to obtain the accurate element $g$, which is given by [24]:

$g_{\mathbf{k},\mathbf{k}+\mathbf{q}}={(\frac{\hbar}{2M\omega_{\mathbf{q}}})}^{1/2}\bra{\varphi_{\mathbf{k}+\mathbf{q}}}\partial_{\mathbf{q}}V\ket{\varphi_{\mathbf{k}}}$
(5)

where $\partial_{\mathbf{q}}V$ is the derivative of the electron-ion potential
with phonon frequency $\omega_{\mathbf{q}}$, $\varphi_{\mathbf{k}}$ is the
electronic wavefunction with wavevector k. For a direct comparison, we also
calculated the matrix element $g$ between electronic states k and k+q by the
tight-binding (TB) model [10, 11, 12], which is:

$g_{\mathbf{k},\mathbf{k}+\mathbf{q}}\propto\left(\mathbf{\nu}_{\mathbf{k}}-\mathbf{\nu}_{\mathbf{k}+\mathbf{q}}\right)\cdot\frac{\mathbf{q}}{|\mathbf{q}|}$
(6)

where $\mathbf{\nu}_{\mathbf{k}}$ is the electron velocity at k point in the
coupled band, $\frac{\mathbf{q}}{|\mathbf{q}|}$ is the longitudinal projection
as only LA phonons soften to zero. And this method succeeded in describing EPC
properties in bulk 2H-NbSe2 [11, 12] and monolayer 1T-VSe2 [25]. To
distinguish the two different sources of EPC matrix elements in the following
discussions, the related quantities obtained by the TB method are denoted as
${\bar{g}}_{\mathbf{q}}^{TB}$ and $\chi_{\mathbf{q}}^{TB}$, while the
quantities obtained by DFPT are used bare notation ${\bar{g}}_{\mathbf{q}}$
and $\chi_{\mathbf{q}}$.

Figure 2: (a) Static Lindhard susceptibility ${\chi^{\prime}}_{\mathbf{q}}$,
(b) q-EPC ${\bar{g}}_{\mathbf{q}}$, (c) generalized static electronic
susceptibility $\chi_{\mathbf{q}}$ and (d) the lowest phonon frequency
$\omega_{\mathbf{q}}$ in the first BZ of NbSe2. (e-h) Same as panel (a-d) but
for VTe2.

The calculated ${\chi^{\prime}}_{\mathbf{q}}$, ${\bar{g}}_{\mathbf{q}}$,
$\chi_{\mathbf{q}}$, and $\omega_{\mathbf{q}}$ for NbSe2 (upper panels) and
VTe2 (lower panels) are shown in Fig. 2. For NbSe2, the static Lindhard
susceptibility ${\chi^{\prime}}_{\mathbf{q}}$ has a broad plateau from
2/5$\Gamma M$ to 4/5$\Gamma M$ [see Fig. 2(a) and the blue line in Fig. 3(a)],
which indicates the weakness of FSN [18]. Nonetheless,
${\bar{g}}_{\mathbf{q}}$ shows a strong electron-phonon interaction near
2/3$\Gamma M$ for NbSe2 [see Fig. 2(b) and the red line in Fig. 3(a)]. In
addition, the topology of $\chi_{\mathbf{q}}$ is very similar to
${\bar{g}}_{\mathbf{q}}$ [Figs. 2(b) and 2(c)] in the NbSe2 first BZ,
indicating the dominated role played by the q-EPC in $\chi_{\mathbf{q}}$. Fig.
2(d) displays the softened phonon modes of NbSe2, which are concentrated in
the arch-like area [see the purple dashed line in Fig. 2(d)], in agreement
with the highland in $\chi_{\mathbf{q}}$ [see the purple dashed line in Fig.
2(c)]. More importantly, the peaks in ${\bar{g}}_{\mathbf{q}}$ and
$\chi_{\mathbf{q}}$ [see the red areas in Figs. 2(b) and 2(c)] are at
2/3$\Gamma M$, which is consistent with the dip in $\omega_{\mathbf{q}}$ [see
the blue areas in Fig.2(d)].

The EPC properties of bulk 2H-NbSe2 have been well-described by using the TB
method [11, 12], however, we find this method fails in addressing monolayer
1H-NbSe2. As for the generalized static electronic susceptibility
$\chi_{\mathbf{q}}^{TB}$, the overall topology in the first BZ [Fig. S3(c)]
cannot fit the phonon softening [Fig. 2(d)], which is in stark contrast to the
good match achieved by DFPT [Fig. 2(c)]. Besides, the peaks of
${\bar{g}}_{\mathbf{q}}^{TB}$ and $\chi_{\mathbf{q}}^{TB}$ at the $\Gamma M$
path [see the green line in Fig. 3(a) and the orange line in Fig. 3(b)] are
near 1/2$\Gamma M$, which predict the formation of $4\times 4$ CDW instead of
$3\times 3$ CDW. By comparing to the prominent peaks of
${\bar{g}}_{\mathbf{q}}$ and $\chi_{\mathbf{q}}$ at 2/3$\Gamma M$ obtained by
the DFPT [see the red line in Fig. 3(a) and the black line in Fig. 3(b)], we
conclude that the ab-initio based DFPT method is superior to the TB method in
obtaining the q-EPC and predicting the CDW vector in monolayer NbSe2.

Figure 3: Direct comparison of the role of EPC and FSN along $\Gamma M$ path
in NbSe2 (up panel) and VTe2 (down panel). (a) Comparing the difference among
${\bar{g}}_{\mathbf{q}}$ (red), ${\bar{g}}_{\mathbf{q}}^{TB}$ (green) and
${\chi^{\prime}}_{\mathbf{q}}$ (blue) in NbSe2, (b) Comparing the difference
between $\chi_{\mathbf{q}}$ (black) and $\chi_{\mathbf{q}}^{TB}$ (orange) in
NbSe2. (c, d) Same as panels (a, b), but for VTe2. Note that the unit of the
y-coordinate is arbitrary units (a.u.) for straightforward comparison.  Figure
4: Simulation of the (a) Fermi surface and (b) the spectral function along
$\Gamma K$ direction with constant $|g|$ of the NbSe2 CDW system. (c, d) Same
as (a, b), but with anisotropic $g$ in the simulation. Simulation of the (e)
Fermi surface and (f) the spectral function along $MK$ direction with constant
$|g|$ of the VTe2 CDW system. (g, h) Same as (e, f), but with anisotropic $g$
in the simulation. The brightness of the dots denotes the spectral weights.
The pink solid lines in (a, c, e, g) in the lower half of the BZ are the
corresponding non-CDW Fermi surfaces for comparison. The green arrows in (a)
indicate the partial CDW gaps opened by FSN, the blue arrows in (c, g)
indicate the full CDW gaps opened by FSN + EPC. The paths of the spectral
functions are chosen to better show the full CDW gaps.

For monolayer VTe2, ${\chi^{\prime}}_{\mathbf{q}}$ has a peak near 2/5$\Gamma
M$ [see Fig. 2(e) and the blue line in Fig. 3(c)], in line with previous works
[19, 20]. However, this peak does not correspond to the $4\times 4$ CDW
structure observed from the experiments [19, 23]. Besides, the profile of
$\chi_{\mathbf{q}}$ in the first BZ is again very close to
${\bar{g}}_{\mathbf{q}}$ [Figs. 2(f) and 2(g)], which both show maxima close
to 1/2$\Gamma M$, providing a powerful clue of phonon softening at
$\mathbf{Q^{T}}$. As shown in Fig. 2(h), the distribution of the softened
phonon frequency $\omega_{\mathbf{q}}$ of VTe2 shows hexapetalous flower-like
pattern [see the blue area in Fig. 2(h)], which match well with the “hot” area
in $\chi_{\mathbf{q}}$ [Fig. 2(g)]. One should note that the peak of
${\bar{g}}_{\mathbf{q}}$ is very sharp, which overwhelms the fluctuation of
${\chi^{\prime}}_{\mathbf{q}}$ in VTe2 [see the red line and the blue line in
Fig. 3(c)], leading to the correct 1/2$\Gamma M$ peak position of
$\chi_{\mathbf{q}}$ [see the black line in Fig. 3(d)]. However, the TB method
still cannot explain the phonon softening in VTe2. Although
${\bar{g}}_{\mathbf{q}}^{TB}$ shows a peak at 1/2$\Gamma M$, such peak is even
broader than the nesting peak at 2/5$\Gamma M$, leading to the incorrect peak
position of $\chi_{\mathbf{q}}^{TB}$ at 2/5$\Gamma M$ [see the green line in
Fig. 3(c) and the orange line in Fig. 3(d)]. Compared the distribution of
${\bar{g}}_{\mathbf{q}}^{TB}$ [Fig. S3(e)], the “sharpness” of
${\chi^{\prime}}_{\mathbf{q}}$ make the topology of $\chi_{\mathbf{q}}^{TB}$
closer to that of the ${\chi^{\prime}}_{\mathbf{q}}$ in the first BZ of VTe2
[Figs. 2(e) and S3(f)], which cannot explain the phonon softening shown in
Fig. 2(h). Therefore, with the help of the accurate EPC matrix element $g$
obtained by DFPT, our study clearly demonstrates that it is q-EPC rather than
FSN determines the phonon softening at correct positions and accounts for the
CDW formation in both NbSe2 and VTe2.

Electron instabilities and CDW gaps.–Having identified the domination of EPC
in phonon softening, we now study the momentum-dependent CDW gap (electron
instabilities) based on mean-field theory. The Hamiltonian of the CDW phase is
minimally described by including one band crossing the Fermi level and
electron-phonon interaction with phonon momentum Q.

$H_{mf}=\sum_{\mathbf{k}}{\varepsilon_{\mathbf{k}}c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}+\sum_{\mathbf{k},\mathbf{Q}}{2g_{\mathbf{k},\mathbf{k}+\mathbf{Q}}\Delta_{\mathbf{Q}}c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}+\mathbf{Q}}}}+h.c.$
(7)

Here, $c_{\mathbf{k}}^{\dagger}$ ($c_{\mathbf{k}}$) and
$\varepsilon_{\mathbf{k}}$ are creation (annihilation) operator and energy for
an electron with momentum k. $\Delta_{\mathbf{Q}}$ is the order parameter
which was approximated to a constant because of symmetry and small pocket
size. This Hamiltonian can then be used to calculate the spectral function of
the CDW phases [see SI-IV for more details about the theoretical background].

As shown in Fig. 4(a), the simulated Fermi surface of the NbSe2 CDW structure
with constant EPC matrix elements $|g|$ (i.e.,
$|g|=\sum_{\mathbf{k},\mathbf{k}+\mathbf{Q}}|g_{\mathbf{k},\mathbf{k}+\mathbf{Q}}|/{N_{\mathbf{k},\mathbf{k}+\mathbf{Q}}}$,
where $N_{\mathbf{k},\mathbf{k}+\mathbf{Q}}$ is the number of $g$ in the
calculation.) reflects the FSN effect under the mean-field picture. The norm
of $g$ is used to avoid the arbitrary phase factor problem in the band basis
of EPC matrix element. It is clearly shown that each K pocket has 3 couples of
partial gaps [see the green arrows in Fig. 4(a)], where the spectral intensity
becomes blurred as only partial electronic states are left at the Fermi
surface. The partial gaps are at both sides of the $\Gamma K$ path,
corresponding to the most heavily nested points of NbSe2 [see the red points
in white circle in Fig. S6(c)]. The incorporation of anisotropic matrix
elements $g$ develop a more extensive gap opening on the Fermi surface [Fig.
4(c)], which considers the synergistic effects of FSN and EPC. Remarkably, the
full band-gapped sectors can be found on the K pockets, where the electronic
states on the Fermi surface are completely obliterated [see the blue arrow in
Fig. 4(c)]. Furthermore, the spectral function was plotted along the $\Gamma
K$ path, and we find the pure nesting effect cannot open a band gap along this
path [Fig. 4(b)], in contrast with the spectral function derived with the
anisotropic $g$, which obviously displays a full band gap close to the K point
[Fig. 4(d)]. Considering there is no experimental report on the Fermi surface
of the monolayer NbSe2 CDW state, the predicted CDW gap distribution is
compared with the unfolded Fermi surface of the simulated NbSe2 CDW ground
state, which displays a remarkable agreement with each other [14, 17].

It is known that VTe2 has triangular hole pockets, which has parallel sides to
provide good nesting condition [19, 20]. Such nesting will induce a peak in
the static Lindhard susceptibility and possibly open a gap at the heavily
nested point [19]. In the phononic part discussion, FSN as the mechanism of
the phonon softening in VTe2 has been excluded, we now turn to discuss its
relation to the CDW gap. Fig. 4(e) suggests that there is no obvious spectral
weight depletion on the Fermi surface, and no CDW gap can be opened at the
$MK$ path [Figs. 4(e) and 4(f)]. However, after considering the effect of the
anisotropic EPC matrix elements $g$, there is a strong suppression of the
spectral intensity near the M point [see the blue arrow in Fig. 4(g)]. No
electronic state can be found on the Fermi surface at the $MK$ path, which
indicates a full gap opening [Fig. 4(h)], consistent with the recent
experimental results [19]. Moving toward the $\Gamma$ point, the decreasing of
the gap size is accompanied by the full to partial gap transition, and finally
the gap is closed at the triangular K pocket apex [see Fig. 4(g)]. Such an
anisotropic gap distribution on the Fermi surface agrees well with the angle-
resolved photoemission spectroscopy (ARPES) measurements [19].

Discussion.–Although the CDW formation has been widely studied for decades,
the underlying mechanism is still under debate. Using NbSe2 and VTe2 as
examples, we explore the origin of their CDW orders by providing a
comprehensive ab-initio theoretical study. This study is vital because it is
difficult to reconcile FSN and EPC so far. The main features of this work
include an accurate description of the q-EPC in the whole BZ for the first
time, correctly calculating the generalized static electronic susceptibility,
the understanding of the CDW formation mechanism, and comparing the CDW gap
distribution by the mean-field model with or without incorporating the EPC
effects. All these results are self-consistent and emphasize the importance of
EPC.

We also address some puzzles surrounding the CDW properties of monolayer NbSe2
and VTe2. For monolayer NbSe2, a small CDW gap of 4 meV is obtained by the
molecular-beam epitaxy (MBE) grown samples with low $T_{CDW}\sim$ 25K [21].
While the mechanically exfoliated samples show higher $T_{CDW}\sim$ 145K
without a gap measurement [26, 27], and it is believed that a large $T_{CDW}$
corresponds to a large gap size [1, 2, 11, 16]. Previous experimental and
theoretical studies indicate that the suppression of the $3\times 3$ CDW order
in the MBE-grown NbSe2 samples is due to the charge transfer from the graphene
substrate [26, 28, 29]. Therefore, the intrinsic NbSe2 is expected to have a
larger gap size than 4meV, as revealed by the previous unfolded band structure
[14, 15] and our mean-field simulation. For monolayer VTe2, previous work
reported that there is no observable charge transfer between VTe2 samples and
the graphene substrate [20], hence, the graphene based VTe2 samples may be
close to the freestanding one, which should have similar properties including
the CDW gap opening. Our mean-field calculations successfully reproduce the
experimental ARPES results as expected [19]. Besides, the long parallel sides
of the triangular hole pockets provide a good condition for FSN, which was
thought to be the origin of the anisotropic CDW gap [19]. However, our work
excludes this hypothesis and emphasizes the significance of EPC in the CDW gap
opening by providing a convincible description.

In conclusion, using first-principles calculations and the mean-field theory,
we report a quantitative study of CDW properties in monolayer 1H-NbSe2 and
1T-VTe2. Our results confirm the validity of the EPC mechanism in a non-1D CDW
system, which supplies previous experimental and theoretical works. The
combined analysis of FSN and EPC in both phononic and electronic pictures
constructs a profound understanding of the CDW formation mechanism. We argue
that the same physics, in principle, should be applied to other higher-
dimensional CDW systems. Besides, the proposed analytical method can be
decorated by more accurate calculations (i.e., GGA+U, GW, hybrid function,
etc.) [24, 30, 31, 32], which allows further CDW studies in more complex
systems. Our work paves a general way to unravel the physical insights of the
CDW formation mechanism with phonon and electron instabilities, which can be
extended for the understanding of charge ordering in other transition metal
compounds [33], kagome metals [34] and high-temperature superconductors [35].

Acknowledgement.– Z.W. thanks Dr. Felix Flicker for fruitful discussions. Z.W.
also thanks Dr. Miguel Dias Costa and Yi-Ming Zhao for IT support. C.C.
acknowledges the support from Shuimu Tsinghua Scholar Program. This research
project is partially supported by the Ministry of Education, Singapore, under
its MOE AcRF Tier 3 Award MOE2018-T3-1-002, and MOE AcRF Tier 2 Award
MOE2019-T2-2-030. Numerical computations were supported by the National
Supercomputing Center (NSCC) Singapore and Center of Advanced 2D Materials
(CA2DM) HPC infrastructure.

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#  Non-stationary SQM/IST Correspondence and ${\cal CPT}/{\cal PT}$-invariant
paired Hamiltonians on the line

V.P<EMAIL_ADDRESS>and A.J.
<EMAIL_ADDRESS><EMAIL_ADDRESS>

$\,{}^{\spadesuit}$ Akhiezer Institute for Theoretical Physics of NSC KIPT
1 Akademicheskaya St., Kharkiv 61108, Ukraine
$\,{}^{\vardiamondsuit}$ Department of Physics & Technology, Karazin Kharkiv
National University,
4 Svobody Sq., Kharkiv 61022, Ukraine
$\,{}^{\varheartsuit}$ Usikov Institute of Radiophysics and Electronics
12 Ak. Proskury, Kharkiv 61085, Ukraine

###### Abstract

We fill some of existed gaps in the correspondence between Supersymmetric
Quantum Mechanics and the Inverse Scattering Transform by extending the
consideration to the case of paired stationary and non-stationary
Hamiltonians. We formulate the corresponding to the case Goursat problem and
explicitly construct the kernel of the non-local Inverse Scattering Transform,
which solves it. As a result, we find the way of constructing non-hermitian
Hamiltonians from the initially hermitian ones, that leads, in the case of
real-valued spectra of both potentials, to pairing of ${\cal
CPT/PT}$-invariant Hamiltonians. The relevance of our proposal to Quantum
Optics and optical waveguides technology, as well as to non-linear dynamics
and Black Hole Physics is briefly discussed.

PACS: 03.65.-w, 12.60.Jv, 02.30.lk, 02.30.zz

Keywords: Supersymmetric Quantum Mechanics, Inverse Scattering Transform,
${\cal PT}$ symmetric Quantum Mechanics

In Sweet Memory of Anatoly Pashnev, our collegue and friend

##  1 Introduction

Exact solvable models have always played a distinctive role, especially in the
realm of Quantum World. The use of various kinds of approximations in the
course of modeling quantum phenomena is quite justified for natural materials,
but it becomes a problem for designing their artificial analogues with
improved or preset characteristics. Here, the exact solvability becomes a
necessary theoretical tool, that allows one to predict the response of a
quantum system to an external source, and, thereby, to control its behavior in
different regimes of interaction with driving forces. An even more intriguing
situation is realized in the case of using, on the theoretical side,
techniques for generating new exactly solvable models that have principally
new properties and characteristics compared to the initial ones. Such new
properties can arise as due to changes in the type of interaction between the
internal ingredients of a quantum system, as well as due to the specifics of
its interaction with external sources and fields. A clear understanding, on
the theory side, of the processes occurred in the system is an additional,
important, though by no means the last, key to their successful implementation
in practice.

Currently, it has known a sufficient number of techniques for generating new
exact solutions from already known ones (see, for example, monographs and
paper collections [1, 2, 3, 4, 5, 6, 7]). One of these techniques, developed
since the late 40s in quantum scattering theory, is the inverse scattering
method [8, 9, 10, 11, 12], which leads to a remarkable and non-standard
result: restoring the potential from the known spectrum is an ambiguous task.
What was treated as a bug before became a very useful feature after, when it
were established new relations between exactly solvable potentials of Quantum
Mechanics, new interconnections between non-linear equations and their
solutions, symmetry structures behind and many more. After seminal papers on
solitons and non-linear dynamics [13, 14, 15], the inverse scattering method
was renamed into the Inverse Scattering Transform, so that we will follow this
nomenclature, irrespectively of the presence or absence of non-linearity in
the system.

Another profound method of generating new solutions with exactly solvable
dynamics is Supersymmetric Quantum Mechanics, whose development, after the
foundation in early 80s [16, 17, 18, 19, 20], revealed similarities to the
inverse scattering method. In particular, (almost) the same spectrum may
correspond to potentials that are completely different in shape. This
observation made it possible to put studies of quantum mechanical problems on
a more solid ground than before, and to establish exact solutions even in the
case when conventional approximations (WKB and others) failed. (See, e.g.,
Refs. [21, 22] in this respect.) The latter becomes extremely important in
engineering based on Quantum Optics communication networks [23, 24], as well
as in designing quantum field-effect diodes and transistors [25, 26]. In both
cases, the employment of the Supersymmetric Quantum Mechanics [27, 28, 29, 30,
31, 32] demonstrates principle advantages of the approach.

The purpose of this paper is to fill some of existed gaps in the
correspondence between Supersymmetric Quantum Mechanics (SQM) [16, 17, 18, 19,
20] and the Inverse Scattering Transformation (IST) [8, 9, 10, 11, 12, 33, 34,
35, 36, 37]. Both methods were linked [38, 39, 40, 41, 42, 43, 44, 45, 46]
soon after the SQM foundation and its early development, that formed the
ground for that’s what we now call the SQM/IST Correspondence. The
Correspondence has been well established for the stationary SQM and IST,
though some points of the non-stationary extension of the SQM/IST or, more
exotically, of the non-stationary extension just on one of its sides, have
still remained unexplored. Here we try to take a fresh look at old problems,
and to open a new avenue in studying exactly-solvable models.

The paper is structured as follows. To make the presentation self-consistent,
in two subsequent sections we briefly review main points of the SQM/IST
Correspondence in the stationary [38, 39, 40, 41, 42, 43, 44, 45, 46] and non-
stationary cases [52, 53, 54, 55, 56, 57, 58, 60, 61, 62]. Formally, the
material contained in this part of the paper is not new, except for the
complement to the Goursat problem for the non-stationary IST, that has not
been done before. Nonetheless, Sections 2 and 3 are important for
understanding the main subject of our studies, and they may be considered on
its own as a concise review on main ingredients of the SQM/IST Correspondence,
disparate parts of which are collected in one place.

Next, in Section 4, containing the main result of the paper, we formulate
rationales behind our proposal to modify the picture on the IST side. This
modification, inter alia, makes it possible to pair stationary and non-
stationary Hamiltonians, and to bridge the gap between ${\cal CPT}$ and ${\cal
PT}$ invariant [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]
models. We prove a Lemma, according to which one may formulate the Goursat
problem for the kernel of non-local transformations of wave functions,
including the expression for the new paired potential as well. The proposed
here exact form of the kernel for the non-stationary IST suggests a
“covariantization” of the time derivative in the Schrodinger equation for the
new paired Hamiltonian by a compensator field. Such a “covariantization” has
been used in problems with time-dependent boundary conditions, when the wave
function acquires a time-dependent geometric (i.e. the Berry-type) phase [78].
The compensator field is introduced to balance additional contribution coming
from the action of temporal derivative on the time-dependent phase. We compare
our construction with that of the non-stationary SQM/IST Correspondence, and
establish the relation of the proposed new kernel of the IST and its temporal
derivative to the constituents forming the SQM supercharge.

Discussion of the obtained results, possible directions of their application
and further development is given in the last section.

##  2 The 1D SQM/IST Correspondence: Stationary case

We begin with a review of the correspondence between the standard
Supersymmetric Quantum Mechanics (SQM) [16, 17, 19, 20] and the approach of
the Inverse Scattering Transform (IST) [9, 10, 11, 12, 33, 34, 35, 36],
developed by Abraham and Moses in [37]. (See also Ref. [47] for a review, and
Refs. therein.) Here we closely follow [38], where the correspondence between
the SQM and the IST was realized.

Specifically, the factorization of a SQM Hamiltonian requires of knowing a
particular solution $\partial_{x}W(x)$ to the Riccati equation

$\left(\frac{1}{2}\partial_{x}W(x)\right)^{2}\mp\frac{1}{2}\partial^{2}_{x}W(x)=V_{\pm}(x),$
(1)

where $V_{\pm}(x)$ are the potentials for (almost) isospectral Hamiltonians
$H_{\pm}=-\partial_{x}^{2}+V_{\pm}(x)$. If we fix the initial Hamiltonian as
$H_{+}$, when the stationary Schrodinger equation
$H_{+}\Psi_{0}=E_{0}\Psi_{0}$ is solved for

$\Psi_{0}(x)=N_{0}\,e^{-\frac{1}{2}W(x)},\qquad E_{0}=0.$ (2)

Such a wave function is associated with the ground state of the “bosonic”
Hamiltonian $H_{+}$. This Hamiltonian admits the factorization by supercharges
$Q^{\dagger}$ and $Q$, which are first order differential operators

$Q^{\dagger}=-\partial_{x}+\frac{1}{2}\partial_{x}W(x),\qquad
Q=\partial_{x}+\frac{1}{2}\partial_{x}W(x),$ (3)

with (derivative of) the superpotential $W(x)$. Then, the “bosonic” and
“fermionic” parts of the super-Hamiltonian are constructed as

$H_{+}=Q^{\dagger}Q,\qquad H_{-}=QQ^{\dagger},$ (4)

and the spectra of $H_{\pm}$ are the same, except for the ground state.

Now, let’s factorize the Hamiltonian $H_{-}$ with other supercharges,
$\tilde{Q}$ and $\tilde{Q}^{\dagger}$,

$\tilde{Q}=\partial_{x}+f(x),\qquad\tilde{Q}^{\dagger}=-\partial_{x}+f(x),\qquad
H_{-}=\tilde{Q}\tilde{Q}^{\dagger}.$ (5)

Taking the definition of $V_{-}(x)$ from eq. (1), we get

$f^{2}(x)+\partial_{x}f(x)=\left(\frac{1}{2}\partial_{x}W(x)\right)^{2}+\frac{1}{2}\partial^{2}_{x}W(x).$
(6)

Mielnik [48], after van Kampen [49], marked the general solution to the
Riccati equation (6) for $f(x)$. It comes as

$f(x)=\frac{1}{2}\partial_{x}W(x)-\frac{e^{-W(x)}}{\lambda+\int_{x}^{\infty}dz\,e^{-W(z)}}.$
(7)

Here there has been appeared a constant $\lambda$, whose role will be central
in what follows.

For now, we have established the correspondence between $Q$ and $\tilde{Q}$
supercharges through the “fermionic” part of the super-Hamiltonian $H_{-}$,
that is, $H_{-}=QQ^{\dagger}=\tilde{Q}\tilde{Q}^{\dagger}$. Let’s construct
the superpartner of $H_{-}$ in terms of new supercharges $\tilde{Q}$,
$\tilde{Q}^{\dagger}$:

$\tilde{H}_{+}=\tilde{Q}^{\dagger}\tilde{Q}=-\partial^{2}_{x}+V_{+}+2\partial_{x}\phi(x).$
(8)

Here, we have introduced

$\phi(x)=\frac{e^{-W(x)}}{\lambda+\int_{x}^{\infty}dz\,e^{-W(z)}},$ (9)

which obeys

$\partial_{x}\phi(x)-\phi^{2}(x)+\phi(x)\partial_{x}W(x)=0.$ (10)

Clearly, $\tilde{H}_{+}\neq H_{+}$, and $\tilde{H}_{+}$ corresponds to the
Hamiltonian with new potential

$\tilde{V}_{+}=V_{+}+2\partial_{x}\phi(x).$ (11)

But what about the spectra of these two Hamiltonians?

It turns out that the Hamiltonians $H_{+}$ and $\tilde{H}_{+}$ are also
(almost) isospectral, via their parent Hamiltonian $H_{-}$. Indeed, we can
write

$\tilde{H}_{+}\tilde{Q}^{\dagger}=\left(\tilde{Q}^{\dagger}\tilde{Q}\right)\tilde{Q}^{\dagger}=\tilde{Q}^{\dagger}\left(\tilde{Q}\tilde{Q}^{\dagger}\right)=\tilde{Q}^{\dagger}H_{-},$

that means that eigenstates of $\tilde{H}_{+}$ are determined by
$|\tilde{\psi}^{(+)}_{i}\rangle=\tilde{Q}^{\dagger}|\psi^{(-)}_{i}\rangle$,
where $|\psi^{(-)}_{i}\rangle$ are the eigenvalues of $H_{-}$. So that, the
energies (eigenvalues of $\tilde{H}_{+}$) will be given by $E_{i}$,
$i=1,\dots,N$. However, as in the case of $H_{+}$, one needs to add the ground
state $\tilde{Q}|\tilde{\psi}^{(+)}_{0}\rangle=0$ [48]. In the coordinate
representation, the corresponding wave function $\tilde{\Psi}_{0}$ looks as

$\tilde{\Psi}_{0}=\tilde{N}_{0}\,e^{-\frac{1}{2}W(x)}e^{\int_{0}^{x}dz\,\phi(z)}.$
(12)

It is straightforward to verify that $\tilde{H}_{+}\tilde{\Psi}_{0}=0$, hence
the energy of the ground state of $\tilde{H}_{+}$ is also
$\tilde{E}_{0}=E_{0}=0$.

Other eigenstates of two Hamiltonians $H_{+}$ and $\tilde{H}_{+}$ are related
to each other via

$|\tilde{\psi}^{(+)}_{i}\rangle=\tilde{Q}^{\dagger}Q|\psi^{(+)}_{i}\rangle,$
(13)

So that the normalizations of wave functions $\\{\Psi_{0},\psi^{(+)}_{i}\\}$
and $\\{\tilde{\Psi}_{0},\tilde{\psi}^{(+)}_{i}\\}$ are different. This
situation, when the eigenvalues of two spectra are equal, but the
normalizations of a finite number of point spectra wave functions are
different is well-known in the IST, and it has been considered, e.g., in [35,
36, 37].

According to Abraham & Moses, Ref. [37], the wave functions of two isospectral
Hamiltonians are related to each other via non-local transformation

$\tilde{\psi}^{(+)}_{i}(x)=\psi^{(+)}_{i}(x)+\int_{x}^{\infty}dy\,K(x,y)\psi^{(+)}_{i}(y),$
(14)

with the kernel of the transformation $K(x,y)$. Both eqs. (13) and (14) are
different realizations of
$|\tilde{\psi}^{(+)}_{i}\rangle=U|\psi^{(+)}_{i}\rangle$, with local (eq.
(13)) and non-local (eq. (14)) transformation operators $U$, respectively.

And, as it became clear after the results of [38], the kernel of the non-local
transformations (14) should be taken to be

$K(x,y)=-\frac{\Psi_{0}(x)\Psi_{0}(y)}{\Lambda+\int_{x}^{\infty}dz|\Psi_{0}(z)|^{2}},$
(15)

with a constant $\Lambda$, that (see [37] for details) makes it possible to
construct a new, isospectral, Hamiltonian with the potential

$\tilde{V}_{+}=V_{+}-2\partial_{x}K(x,x).$ (16)

Comparing two equations, eq. (16) and eq. (11), to each other, one may
conclude that $K(x,x)=-\phi(x)$. Hence, eq. (15) turns into eq. (9) as soon as
(cf. eq. (2)) $\lambda=\Lambda/N^{2}_{0}$. In addition, we have to check the
following equation on the kernel

$\partial^{2}_{x}K(x,y)-\partial^{2}_{y}K(x,y)+2\left[\partial_{x}K(x,x)\right]K(x,y)-\left(V_{+}(x)-V_{+}(y)\right)K(x,y)=0,$
(17)

which holds for the kernel (15). Other aspects of the SQM/IST correspondence,
including the general non-equivalence of the approaches, can be found, e.g.,
in [39, 40, 41, 42, 51]. Note that for confined potentials, with discrete
(point in the nomenclature of [37]) spectra, the equivalence of the Abraham-
Moses to the SQM approach was confirmed in [43, 46]. So that we will focus
just on this case.

Thus, different approaches to the construction of new Hamiltonians – SQM and
the IST – are inherently related. And we can use this correspondence between
two approaches in both directions. Here we have started with the SQM
Hamiltonian construction, and turned to the IST technique after. We can use
the reverse procedure equally. That is, we can begin with the IST approach
first, to make the conclusion on the structure of the paired SQM Hamiltonian
after that.

##  3 The 1D SQM/IST Correspondence: Non-stationary case

Elaborating on the construction of new isospectral Hamiltonians with time-
dependent potentials, we have to sum up the previous achievements in the
stationary case. First, there is the correspondence between two different
approaches – the Inverse Scattering Theory and Supersymmetric Quantum
Mechanics – that are used to generate a new isospectral Hamiltonian
($\tilde{H}_{+}$ in the case) to the initial one ($H_{+}$ in the used here
notation). This correspondence is exact for confined potentials that includes
the harmonic potential as well.

Second, the SQM/IST correspondence is realized quite differently on the both
sides. As we have noticed before, the relation between eigenstates (wave
functions) of $\tilde{H}_{+}$ and $H_{+}$ is established either by two
consequent local transformations with different supercharges (cf. eq.(13)), or
by the single non-local transformation with the integral kernel (cf. eq.
(14)). Any of these transformations include specific combinations of the
ground state wave functions, eqs. (9) or (15), with some unspecified, and
generally complex-valued, constant $\lambda$. (Recall, $\Lambda$ in (15) is
related to $\lambda$ in (9) via the square of the normalization constant
$N_{0}$.) The natural requirement that restricts this constant is

$\lambda+\int_{x}^{\infty}dz\,e^{-W(x)}\neq 0.$ (18)

For the rest, this constant can be chosen freely in the domain of the
normalizability of the $\tilde{H}_{+}$ wave functions [44]. We will turn to
the discussion of this aspect of $\lambda$ later on, after reviewing the non-
stationary extension of the SQM/IST correspondence.

And third, the considered in [38] SQM/IST correspondence works in the
stationary case. That is, we have the correspondence between isospectral
Hamiltonians, entering the stationary Schrodinger equations. Below, we will
extend this picture on two notable cases, when the correspondence is
established as between non-stationary Hamiltonians, as well as between
stationary and non-stationary Hamiltonians.

Let’s begin with the SQM/IST correspondence for two non-stationary isospectral
Hamiltonians. The time-dependent extension on the IST side and its relation to
the (S)QM approach has been elaborated in [52, 53]. There, it was proposed the
following extension of the kernel $K(x,y)$:

$K(x,y;t)=-\frac{\psi^{(+)}(x,t)(\psi^{(+)}(y,t))^{*}}{\lambda+\int^{\infty}_{x}dz\,|\psi^{(+)}(z,t)|^{2}}$
(19)

with constant $\lambda$. Using the developed in [37] technique (see Appendix
of [37] for details on the stationary case), one can set up the Goursat
problem as

$\left\\{\begin{array}[]{c}\tilde{V}_{+}(x,t)=V_{+}(x,t)-2\partial_{x}K(x,x;t),\\\
\\\
\partial^{2}_{x}K(x,y;t)-\partial^{2}_{y}K(x,y;t)+i\partial_{t}K(x,y;t)-(\tilde{V}_{+}(x,t)-V_{+}(y,t))K(x,y;t)=0.\end{array}\right.$
(20)

It is straightforward to verify that the kernel (19) satisfies the second
equation of (20) for wave functions of the complete Schrodinger equation with
$H_{+}$,

$\left(H_{+}-i\partial_{t}\right)\psi^{(+)}(x,t)=0,\qquad
H_{+}=-\partial^{2}_{x}+V_{+}(x,t).$ (21)

The new potential $\tilde{V}_{+}(x,t)$ enters the twinned with $H_{+}$
Hamiltonian $\tilde{H}_{+}=-\partial^{2}_{x}+\tilde{V}(x,t)$. Its wave
functions $\tilde{\psi}^{(+)}(x,t)$ also obey the non-stationary Schrodinger
equation, now with $\tilde{H}_{+}$:

$\left(\tilde{H}_{+}-i\partial_{t}\right)\tilde{\psi}^{(+)}(x,t)=0,\qquad\tilde{H}_{+}=-\partial^{2}_{x}+\tilde{V}_{+}(x,t).$
(22)

The SQM side of the story was developed in Refs. [52, 53, 54, 55, 56]. (See
also Refs. [57, 58, 60, 76, 61, 62] for the development and applications of
methods by Bagrov et al.) The supercharge $Q$, associated with $H_{+}$, is
formed from the superpotential $W(x,t)$, similarly to eqs. (3). Concerning the
supercharge $\tilde{Q}$, it is realized in the following way (see Refs. [58,
59] for the detailed analysis of the construction of [52, 53, 54, 55, 56]):

$\tilde{Q}=l(t)\left(\partial_{x}+f(x,t)\right),\qquad
f(x,t)=\frac{1}{2}\partial_{x}W(x,t)-\phi(x,t).$ (23)

The functions $l(t)$ and $\phi(x,t)$ are generally complex-valued functions.
Then, the intertwining relation, which is a paraphrase of the isospectrality,

$\left(i\partial_{t}-H_{-}\right)\tilde{Q}=\tilde{Q}\left(i\partial_{t}-\tilde{H}_{+}\right)$
(24)

makes it possible to derive the relation [52, 53] between the new and the old
potentials,

$\tilde{V}_{+}(x,t)=V_{+}(x,t)+2\partial_{x}\phi(x,t)-i\partial_{t}\ln l(t).$
(25)

And the function $\phi(x,t)$ is restricted to satisfy

$\partial_{x}\left[f^{2}(x,t)-\partial_{x}f(x,t)-\tilde{V}_{+}(x,t)\right]-i\partial_{t}f(x,t)=0.$
(26)

Note also that the requirement of real-valued potentials in (25) leads to the
following constraints:

$\partial_{t}\ln|l(t)|^{2}=-4\partial_{x}\left[\Im\text{m}\,\phi(x,t)\right]\qquad\leadsto\qquad\partial^{2}_{x}\left[\Im\text{m}\,\phi(x,t)\right]=0,$
(27)

so that the non-triviality of $l(t)$ requires a complex-valued $\phi(x,t)$. At
the same time, dealing with a real-valued potential to get real-valued
eigenvalues of the Hamiltonian is an excess requirement [63, 64, 65]. We will
turn to the more detailed discussion of this point in the next section.

For now let us note that, from the construction of the kernel and
supercharges, it is clear that we are dealing with non-stationary paired
Hamiltonians. The SQM/IST correspondence at the level of the
supercharge/kernel relation is established by comparing the right hand sides
of $\tilde{V}_{+}(x,t)$ in (20) and (25). As a result, we obtain a
relationship between the spatial derivative of the transformation kernel and
the derivatives, both spatial and temporal, of the basic ingredients of the
supercharge. Thus, in contrast to the stationary case, the SQM/IST
correspondence is realized indirectly, through the derivatives of the main
quantities.

However, the direct realization of the SQM/IST correspondence is restored
within a slightly different, but important, extension of the stationary case,
which will formally connect stationary and non-stationary Hamiltonians. Now,
we are turning to the assembling of this construction.

##  4 The 1D SQM/IST Correspondence: Stationary to non-stationary Hamiltonians
pairing

Let’s carry on with a brief discussion on the kernel structure for the
stationary case. Equation (15) can be generalized to

$K(x,y)=-\frac{C_{2}\Psi_{0}(x)\Psi_{0}(y)}{C_{1}+C_{2}\int^{\infty}_{x}dz\,|\Psi_{0}(z)|^{2}},\qquad
C_{1,2}=\text{const}\in\mathbb{C}.$ (28)

However, what matters is the ratio of these two constants, which we have
denoted as $\lambda=C_{1}/C_{2}$. Changing the $\lambda$ does not affect the
spectrum, affecting the potential and wave functions, respectively. Therefore,
we can instantly change the $\lambda$ in its value, within its range,
determined by relation (18). Or, we can relate any instant $\lambda$ changing
to some unique value of the time parameter. Smooth changes form a function of
time $\lambda(t)$ after that. Thus, we would like to investigate consequences
of using the following kernel template on the IST side:

$K(x,y;t)=-\frac{C_{2}\Psi_{0}(x)\Psi_{0}(y)}{C_{1}(t)+C_{2}\int^{\infty}_{x}dz\,|\Psi_{0}(z)|^{2}},\quad
C_{2}=\text{const},$ (29)

where $\Psi_{0}(x)$ is the ground state wave function of $H_{+}$ (cf. eq.
(2)).

With the kernel of non-local (integral) transformations of wave functions
(29), we expect that the supercharges from eqs. (5) will become functions of
time, via

$\phi(x,t)=\frac{e^{-W(x)}}{\lambda(t)+\int^{\infty}_{x}dz\,e^{-W(z)}}.$ (30)

Then, the time dependence of the $\tilde{H}_{+}$ ground state wave function,

$\tilde{\Psi}_{0}(x,t)=\tilde{N}_{0}e^{-\frac{1}{2}W(x)}e^{\int^{x}_{0}dz\,\phi(z,t)},$
(31)

can be treated as a time-dependent geometric phase, i.e.,

$\tilde{\Psi}_{0}(x,t)=\tilde{N}_{0}e^{-\frac{1}{2}W(x)}e^{i\Phi(x,t)},\quad
i\Phi(x,t)=\int^{x}_{0}dz\,\phi(z,t).$ (32)

Recall, wave functions with time-dependent geometric phase naturally appear in
quantum systems with time-dependent boundary conditions. And a by-product of
the consideration is the “covariantization” [78] of the non-stationary
Schrodinger equation in temporal direction.

Taking the latter into account, let us formulate the considered problem in
more general terms.

Lemma. Suppose that two sets of wave functions, $\psi_{\varepsilon}(x)$ and
$\Psi(x,t)$, are related to each other via non-local transformations

$\Psi(x,t)=e^{-i\varepsilon
t}\psi_{\varepsilon}(x)+\int_{x}^{\infty}dy\,K(x,y;t)\,e^{-i\varepsilon
t}\psi_{\varepsilon}(y).$ (33)

The kernel of these transformations

$K(x,y;t)=-\frac{C_{2}\psi(x,t)\psi^{*}(y,t)}{C_{1}(t)+C_{2}\int^{\infty}_{x}dz|\psi(z,t)|^{2}},$
(34)

is constructed out of the stationary wave functions
$\psi(x,t)=e^{-i\varepsilon t}\psi_{\varepsilon}(x)$, which obey the
Schrodinger equation

$\left(H_{0}-i\partial_{t}\right)\psi(x,t)=0,\qquad
H_{0}=-\partial^{2}_{x}+V(x).$ (35)

If the wave function $\Psi(x,t)$ obeys the generalization of the Schrodinger
equation,

$\left(H-iD_{t}\right)\Psi(x,t)=0,\qquad H=-\partial^{2}_{x}+U(x,t),$ (36)

in which $D_{t}=\partial_{t}+A_{t}$ is a “covariantization” of the time
derivative by a “compensator” field $A_{t}$, determined by $D_{t}K(x,y;t)=0$,
then the relation between the paired potentials and the Master equation for
the kernel (the Goursat problem) looks as follows:

$\left\\{\begin{array}[]{c}U(x,t)=V(x)-2\partial_{x}K(x,x;t)+iA_{t},\\\ \\\
\partial^{2}_{x}K(x,y;t)-\partial^{2}_{y}K(x,y;t)+iD_{t}K(x,y;t)-\left(\left[U(x,t)-iA_{t}\right]-V(y)\right)K(x,y;t)=0.\end{array}\right.$
(37)

Proof. Let’s act with $(H-iD_{t})$ operator on the wave function (33). This
operation can be split in two parts. One of them is

$\left(H-iD_{t}\right)\psi(x,t)=\left(-\partial^{2}_{x}+U(x,t)-i\left(\partial_{t}+A_{t}\right)\right)e^{-i\varepsilon
t}\psi_{\varepsilon}(x)$ $=\left(U(x,t)-iA_{t}-V(x)\right)e^{-i\varepsilon
t}\psi_{\varepsilon}(x),$ (38)

where we have used eq. (35). The other part of eq. (36) contains

$\left(H-iD_{t}\right)\int_{x}^{\infty}dy\,K(x,y;t)e^{-i\varepsilon
t}\psi_{\varepsilon}(y)=\left(-\partial^{2}_{x}+U(x,t)-iD_{t}\right)\int_{x}^{\infty}dy\,K(x,y;t)e^{-i\varepsilon
t}\psi_{\varepsilon}(y)$
$=\left(-\partial^{2}_{x}+U(x,t)\right)\int_{x}^{\infty}dy\,K(x,y;t)e^{-i\varepsilon
t}\psi_{\varepsilon}(y)-i\int^{\infty}_{x}dy\,\underbrace{\left(D_{t}K(x,y;t)\right)}_{=0}e^{-i\varepsilon
t}\psi_{\varepsilon}(y)$
$-i\int^{\infty}_{x}dy\,K(x,y;t)\left(\partial_{t}+A_{t}\right)e^{-i\varepsilon
t}\psi_{\varepsilon}(y)$
$=\left(-\partial^{2}_{x}+U(x,t)-\varepsilon\right)\int_{x}^{\infty}dy\,K(x,y;t)e^{-i\varepsilon
t}\psi_{\varepsilon}(y)-i\int^{\infty}_{x}K(x,y;t)A_{t}e^{-i\varepsilon
t}\psi_{\varepsilon}(y).$

By use of

$\left(-\partial^{2}_{x}+U(x,t)-\varepsilon\right)\left(\psi_{\varepsilon}(x)+\int^{\infty}_{x}dy\,K(x,y)\psi_{\varepsilon}(y)\right)=\left(U(x,t)-V(x)\right)\psi_{\varepsilon}(x)+\psi_{\varepsilon}(x)\frac{d}{dx}K(x,x)$
$+\psi_{\varepsilon}(x)\underbrace{\left(\frac{\partial}{\partial
x}K(x,y)\Big{|}_{y=x}+\frac{\partial}{\partial
y}K(x,y)\Big{|}_{x=y}\right)}_{=\frac{d}{dx}K(x,x)}$
$-\int^{\infty}_{x}dy\,\left(\partial^{2}_{x}K(x,y)-\partial^{2}_{y}K(x,y)-U(x,t)K(x,y)+V(y)K(x,y)\right)\psi_{\varepsilon}(y),$
(39)

we finally arrive at

$\left(H-iD_{t}\right)\Psi(x,t)=e^{-i\varepsilon
t}\Bigg{[}\left(U(x,t)-iA_{t}-V(x)+2\frac{d}{dx}K(x,x;t)\right)\psi_{\varepsilon}(x)$
$-\int^{\infty}_{x}dy\,\left(\partial^{2}_{x}K(x,y;t)-\partial^{2}_{y}K(x,y;t)+iA_{t}K(x,y;t)-\left(U(x,t)-V(y)\right)K(x,y;t)\right)\psi_{\varepsilon}(y,t)\Bigg{]}=0.$

Equating to zero the l.h.s. of the latter equation, we get the system of
equations (37). $\blacksquare$

In the notation of the paper, we have proved that the usage of the kernel (29)
results in the construction of the new, paired to $H_{+}$, effective
Hamiltonian $\tilde{H}_{+}$, extended by the interacting with the
“compensator” $A_{t}$ term. This field is fixed by the requirement of having
the “covariantly constant” kernel, i.e.,

$\partial_{t}K(x,y;t)+A_{t}K(x,y;t)=0.$ (40)

Then, for $K(x,y;t)$ (34),

$\partial_{t}K(x,y;t)=-\frac{\partial_{t}C_{1}(t)}{C_{1}(t)+C_{2}\int^{\infty}_{x}dz\,\psi_{\lambda}(z)^{2}}K(x,y;t)=-A_{t}K(x,y;t)$
(41)

so that the relation between the initial and the final, after the IST,
potential in our case (with the kernel (29)) becomes

$\tilde{V}_{+}(x,t)=V_{+}(x)-2\partial_{x}K(x,x;t)+\frac{i\partial_{t}C_{1}(t)}{C_{1}(t)+C_{2}\int^{\infty}_{x}dz\,|\Psi_{0}(z)|^{2}}\,.$
(42)

Comparing the latter expression with that of eq. (25), we can establish the
direct correspondence between the IST and the SQM quantities:

$K(x,x;t)=-\phi(x,t),\qquad\partial_{t}\ln
l(t)=-\frac{\partial_{t}C_{1}(t)}{C_{1}(t)+C_{2}\int^{\infty}_{x}dz\,|\Psi_{0}(z)|^{2}}.$
(43)

Therefore, the usage of the special IST kernel (29) makes it possible: 1) to
construct a new paired time-dependent potential from the initially stationary
one; 2) to realize the SQM/IST correspondence directly in the non-stationary
case, with the special kernel of the IST (34).

Another interesting feature behind our proposal can be figured out from the
analysis of the potential (42). The requirement to have a real-valued
potential $\tilde{V}(x,t)$ leads to the following relation between the kernel
and the “compensator” field:

$\Re\text{e}A_{t}(x,t)=2\partial_{x}\left(\Im\text{m}\,K(x,x;t)\right).$ (44)

Hence, the kernel $K(x,y;t)$ (34) has to be complex-valued, to provide the
non-triviality of $A_{t}(x,t)$. For the IST from a stationary system to a non-
stationary one, it means that the parameter $\lambda(t)=C_{1}(t)/C_{2}$ has to
be a complex-valued function of time. Then, according to (33) and (34), wave
functions of the paired Hamiltonian are principally non-stationary ones, in
the sense that they are non-factorizable in time and space coordinates, and
they are complex functions. If so, they correspond to scattering states, so
that the IST (33) with the kernel (34) may, for instance, transform bound
states to scattering states.

Even more interesting situation takes place when we omit the restriction on
the new scattering potential to be a real-valued function of time and space
coordinates. In this case we still have a possibility to operate with
Hamiltonians having real-valued energy spectra. It happens, for example, for
the so-called ${\cal P}{\cal T}$-invariant Hamiltonians [64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74], properties of which are nicely reviewed in Refs.
[75], [76] and [77].

Let’s apply the ${\cal P}{\cal T}$ transformations to the potential (42).
Under the ${\cal P}{\cal T}$ transformations, $V_{+}(x)\rightarrow V_{+}(-x)$.
For $K(x,y;t)$ we have (recall, $C_{2}$ is a constant)

$K(x,y;t)\rightarrow-\frac{C_{2}\psi_{\varepsilon}(-x)\psi_{\varepsilon}(-y)}{C_{1}(-t)+C_{2}\int_{-x}^{-\infty}d(-z)\,\psi_{\varepsilon}(-z)^{2}}=-\frac{C_{2}\psi_{\varepsilon}(x)\psi_{\varepsilon}(y)}{C_{1}(-t)-C_{2}\int_{x}^{\infty}dz\,\psi_{\varepsilon}(z)^{2}}$
$=-\frac{C_{2}\psi_{\varepsilon}(x)\psi_{\varepsilon}(y)}{C_{1}(-t)-C_{2}\int_{x}^{\infty}dz\,\psi_{\varepsilon}(z)^{2}}\,.$

Hence, ${\cal P}{\cal T}\left[K(x,y;t)\right]\rightarrow-K(x,y;t)$ for
$C_{1}(-t)=-C_{1}(t)$, that means ${\cal P}{\cal
T}\left[\partial_{x}K(x,y;t)\right]\rightarrow\partial_{x}K(x,y;t)$.

Upon the action of ${\cal P}{\cal T}$ operators on the imaginary part of a
complex-valued potential (42) we obtain

${\cal P}{\cal
T}\left[\Im\text{m}\,\tilde{V}_{+}(x,t)\right]=-\frac{\partial_{t}C_{1}(-t)}{C_{1}(-t)-C_{2}\int^{\infty}_{x}dz\,\psi_{\varepsilon}(z)^{2}}\,.$
(45)

And if $C_{1}(-t)=-C_{1}(t)$, the $i\Im\text{m}\,\tilde{V}_{+}(x,t)$ stays to
be ${\cal P}{\cal T}$ invariant.

Therefore, the restriction on $C_{1}(t)$ to be an odd function of time
provides, together with the even parity of the potential $V_{+}(x)$, the
invariance of the paired potential under the ${\cal P}{\cal T}$
transformations. And the ${\cal P}{\cal T}$ invariance is not a drawback for
the factorization and application of Supersymmetry to the consideration. (Cf.,
e.g., Refs. [68, 69, 72, 73, 74, 76] in this respect.) Clearly, the
supercharge of the paired Hamiltonian,
$\tilde{Q}=l(t)\left(Q-\phi(x,t)\right)$, is constructed out of the initial
even-parity Hamiltonian supercharge $Q$, and the determined by eqs. (43)
functions with $C_{1}(-t)=-C_{1}(t)$. Hence, in the case of real-valued energy
Hamiltonians, we have the SQM/IST Correspondence between

  1. i)

stationary and non-stationary Hamiltonians;

  2. ii)

bound, scattering and tunnelling states;

  3. iii)

${\cal CPT}$-invariant and ${\cal PT}$-invariant Hamiltonians.

Early, relations between ${\cal CPT/PT}$-invariant Hamiltonians were
established merely for non-stationary systems. (See, e.g., [74, 76], and Refs.
therein.)

Furthermore, even if all the restrictions on the potential (like the
requirement of dealing with real-valued or ${\cal PT}$-invariant potentials)
will be omitted, we will take another possibility to construct paired complex-
valued Hamiltonians. As it is known from classical electrodynamics, choosing
specific boundary conditions leads to the appearance of quasi-normal modes
with complex-valued frequencies. The complex-valued frequencies correspond to
complex energies even for a real-valued potential. Therefore, the pairing
between real-valued (stationary) and complex-valued (non-stationary)
potentials may be realized as a result of imposing the specific boundary
conditions. The quasi-normal modes play, in particular, an important role in
waveguides, resonators, and in Black Hole Physics as well. So that, the
proposed here mechanics of generation of new exactly-solvable potentials may
also find its natural application in these domains of study.

##  5 Discussion and final remarks

In summary, we propose a new mechanism for generating paired potentials based
on a very specific time-dependent extension of the IST kernel or the SQM
supercharge. The standard generalization of the Abraham-Moses construction
[37] to the non-stationary case [52, 53, 54] assumes the replacement of
stationary wave-functions with non-stationary ones. At the same time, the
integration constant, coming from the general solution to the Riccati equation
[48, 49] and entering either the IST kernel or the SQM supercharge, remains
unchanged. Our proposal is based on the opposite consideration, when the
integration constant becomes a function of time, while the rest of the IST
kernel or the SQM supercharge remains the same as in the stationary case. So
that, if the standard non-stationary generalization of the Abraham-Moses
construction pairs time-dependent Hamiltonians, our approach admits a non-
standard and non-trivial pairing of time-independent and time-dependent
Hamiltonians.

Other non-trivial features of our proposal consist in the correspondence
between real-valued and complex-valued potentials, since the elaborated here
procedure of the paired Hamiltonian construction results, generally, in the
appearance of a complex-valued potential after the IST employment. If the
complex-valued potential is not constrained, the eigenvalues of a complex-
valued Hamiltonian are usually complex. And it can be viewed in different
ways. For instance, the complex part of energy values may indicate the
dissipation in real physical systems, that describes the natural energy loss,
e.g., in Quantum Optics, or in optical fibers. Another way to deal with
complex-valued energies is to consider effects of tunnelling through barriers
suggested by the potential shape, or by specific boundary conditions. Note
that the role of boundary conditions in the twinning of Hamiltonians with
real-valued and complex-valued potentials becomes fundamental, since setting
the specific boundary conditions is a most natural way to generate quasi-
normal modes with complex-valued frequencies. Therefore, following the
proposed here recipe, one may construct, on the ground of a hermitian
stationary exactly-solvable Hamiltonian, the effective exactly-solvable time-
dependent Hamiltonian for a system with the energy dissipation.

The requirement to have real-valued energies in the spectrum of a complex-
valued Hamiltonian leads us to the ${\cal PT}$-invariant quantum mechanics.
(See, e.g., [75, 76, 77] and Refs. therein.) We established the set of
constraints for the initial hermitian time-independent Hamiltonian potential
and the kernel of the non-local IST, that leads to the time-dependent complex-
valued paired potential. The standard SQM and, partially, the ${\cal
PT}$-invariant SQM gained in popularity in Solitons Theory [79, 80], Optical
Waveguides [27, 30], models of Analogue Black Holes [81], diffusion theory
[82, 83] and many more. (See, for instance, Ref. [84] for a review on modern
trends in SQM.) So we hope that the obtained here outcomes of our work will
give an impact on investigations in the mentioned as well as in other branches
of modern science and technology.

Finally, let’s pay special attention to the application of our results in
Black Hole Physics. We have mentioned the Quasi-Normal Modes, which, together
with the phenomenon called Superradiation, play a very important role in
modern studies of Black Holes. A time ago it was established the
isospectrality of the Regge-Wheeler and Zerilli equations for spin-2
perturbations over a specified gravitational background [85]. However
recently, in [86], it was observed that this phenomenon does not properly work
for a general gravitational background. It would be interesting to
investigate, on the ground of the obtained by us results, the
(non-)isospectrality between equations for spin-2 perturbations over
stationary and non-stationary gravitational background with formally complex-
valued effective potentials. We hope to report on this and other outcomes of
our work in future communications.

Acknowledgements. Different aspects related to the subject of the paper and
much more were once discussed with Anatoly Pashnev. We had the opportunity and
the privilege to know him and to work with him. And we never forget his
kindness and supportive spirit. His unexpected and premature passing was a
great lost for us. We dedicate this article to his sweet memory.

The work of A.J.N. is supported in part within the Cambridge-NRFU 2022
initiative ”Individual research (developments) grants for researchers in
Ukraine (supported by the University of Cambridge, UK)”, project
№2022.02/0052.

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A new class of exactly-solvable potentials by means of the hypergeometric
equation

Wei Yang

College of Science, Guilin University of Technology, Guilin, Guangxi 541004,
China

ABSTRACT

We obtained a new class of exactly-solvable potentials by means of the
hypergeometric equation for Schrödinger equation, which different from the
exactly-solvable potentials introduced by Bose and Natanzon. Using the new
class of solvable potentials, we can obtain the corresponding complex PT-
invariant potentials. This method can also apply to the other Fuchs equations.

Emails<EMAIL_ADDRESS>
##  1 Introduction

The exact solutions to the Schrödinger equation play crucial roles in quantum
physics. It is well-known that there are several potentials can be exactly
solved, as examples one can site the harmonic oscillator, the Coulomb, the
Morse [1], Pöschll-Teller [2], Eckart [3] potentials and so on. The reason why
these potentials are exact solvable is that the Schrödinger equation in these
potentials can be transformed by into either the hypergeometric or the
confluent hypergeometric equation. Here we focus on hypergeometric equation to
construct a new class of the solvable potentials within the framework of the
non-relativistic Schrödinger equation. Similar works have been carried out [4,
5, 6, 7]. Using the new class of solvable potentials that we have constructed,
we can easily obtain complex PT-invariant potentials, these potentials is a
hot issue of recent study[8, 9, 10, 11].

This work is organized as follows. In Section 2, We present the basic method
of our argument. In Section 3 as an illustration we take the hypergeometric
equation to construct the soluble potentials. In Section 4 we will give some
conclusions for this paper.

##  2 Basic methods

As we know, the stationary Schrödinger equation for a particle of energy $E$
in a potential $V(r)$ has the form

$\displaystyle\psi^{\prime\prime}(r)+(E-V(r))\psi(r)=0\,.$ (1)

Through choosing a transformation of the variable $z=z(r)$, the equation
become

$\displaystyle\psi_{zz}+{\frac{\rho_{z}}{\rho}}\psi_{z}+{\frac{E-V}{\rho^{2}}}\psi=0\,.$
(2)

Where $\rho={\frac{dz}{dr}}$, and apply the transformation $\psi=f(z)u(z)$. we
can obtain the following differential equation

$\displaystyle
u_{zz}+(2{\frac{f_{z}}{f}}+{\frac{\rho_{z}}{\rho}})u_{z}+({\frac{f_{zz}}{f}}+{\frac{f_{z}}{f}}{\frac{\rho_{z}}{\rho}}+{\frac{E-V}{\rho^{2}}})\psi=0\,.$
(3)

Compared with our target equation

$\displaystyle u_{zz}+g(z)u_{z}+h(z)u=0\,.$ (4)

we have

$\displaystyle g(z)=2{\frac{f_{z}}{f}}+{\frac{\rho_{z}}{\rho}}\,$ (5)
$\displaystyle
h(z)={\frac{f_{zz}}{f}}+{\frac{f_{z}}{f}}{\frac{\rho_{z}}{\rho}}+{\frac{E-V}{\rho^{2}}}\,.$
(6)

Integrating the equation (5) allows us to obtain

$\displaystyle f(z)=\sqrt{c_{1}}\rho^{-1/2}e^{\int g(z)dz/2}\,.$ (7)

Substitution of this equation into (6) yields

$\displaystyle
E-V=\rho^{2}(h-{\frac{g_{z}}{2}}-{\frac{g^{2}}{4}})+{\frac{1}{2}}\\{z,r\\}\,.$
(8)

Where the Schwarzian derivative given as

$\displaystyle\\{z,r\\}={\frac{z^{\prime\prime\prime}(r)}{z^{\prime}(r)}}-{\frac{3}{2}}({\frac{z^{\prime\prime}(r)}{z^{\prime}(r)}})^{2}=\rho\rho_{zz}-{\frac{1}{2}}\rho_{z}^{2}\,.$
(9)

Therefore, on the basis of the solvable differential equation (4), we are able
to construction of the solvable potentials for the original Schrödinger
equation. This problem will become how to find that new variable $z(r)$ and
function $f(z)$ satisfy equation (5) (6).

##  3 Hypergeometric equation

In this section, we application the methods to hypergeometric equation

$\displaystyle
z(1-z)u^{\prime\prime}(z)+[\gamma-(\alpha+\beta+1)z]u^{\prime}(z)-\alpha\beta
u(z)=0\,.$ (10)

Obviously, in this case $g(z)={\frac{\gamma-(\alpha+\beta+1)z}{z(1-z)}}$ and
$h(z)=-{\frac{\alpha\beta}{z(1-z)}}$.

In order to determine the form of $f$, we follow the Ishkhanyan et al [6] who
discussed the reduction of the Schrödinger equation to a rather large class of
target equations by a transformation of $\rho(z)$, and $\rho(z)$ is a
polynomial, which all the roots should necessarily coincide with the singular
points of the target equation to which the Schrödinger equation is reduced.
Then, the coordinate transformation of $f$ need the form $f=z^{p}(1-z)^{q}$,
so the equation (5) have solution

$\displaystyle\rho=c_{1}z^{\gamma-2p}(1-z)^{\alpha+\beta+1-\gamma-2q}\,.$ (11)

Substitution of this equation (11) and $h(z)=-{\frac{\alpha\beta}{z(1-z)}}$
into (6), we can easily figure out that the Schrödinger equation have solvable
potentials only if $a=\gamma-2p$ and $b=\alpha+\beta+1-\gamma-2q$ satisfy some
of these discrete values $a=0,1,2$, $b=0,1,2$, and in these cases the
functional form of $z(r)$ and $V(r)$ are likely to be simple.

Case:1 consider $a=0,b=1$, so $\rho={\frac{dz}{dr}}=c_{1}(1-z)$, therefore

$\displaystyle z=1+c_{2}e^{-c_{1}r}\,.$ (12)

From equation (8), we can get

$\displaystyle
E-V=C-A{\frac{1}{(c_{2}+e^{c_{1}r})^{2}}}-B{\frac{1}{c_{2}+e^{c_{1}r}}}\,.$
(13)

Where the coefficients are given

$\displaystyle A={\frac{c_{1}^{2}c_{2}^{2}}{4}}\gamma(\gamma-2)$
$\displaystyle
B={\frac{c_{1}^{2}c_{2}}{2}}(\alpha\gamma+\beta\gamma+\gamma-\gamma^{2}-2\alpha\beta)$
$\displaystyle C=-{\frac{c_{1}^{2}}{4}}(\alpha+\beta-\gamma)^{2}\,.$ (14)

Case:2 consider $a=1/2,b=1/2$ , $\rho=c_{1}z^{1/2}(1-z)^{1/2}$, therefore

$\displaystyle z=1-\sin(1/2(c_{1}r+c_{2}))^{2}\,.$ (15)

So equation (8) tells us that

$\displaystyle
E-V=C-A\csc(c_{1}r+c_{2})^{2}-B\cot(c_{1}r+c_{2})\csc(c_{1}r+c_{2})\,.$ (16)

Where the coefficients are given

$\displaystyle
A={\frac{c_{1}^{2}}{4}}(2(\alpha+\beta)^{2}+1)-c_{1}^{2}(\alpha+\beta-\gamma+1)\gamma$
$\displaystyle
B={\frac{c_{1}^{2}}{2}}(\alpha+\beta-1)(\alpha+\beta+1-2\gamma)$
$\displaystyle C={\frac{c_{1}^{2}}{4}}(\alpha-\beta)^{2}\,.$ (17)

Case:3 consider $a=1/2,b=1$ , $\rho=c_{1}z^{1/2}(1-z)$ , therefore

$\displaystyle z=\tanh^{2}({\frac{c_{1}r-c_{2}}{2}})\,.$ (18)

So we have

$\displaystyle
E-V=C-A\operatorname{sech}^{2}[(c_{1}r-c_{2})/2]-B\operatorname{csch}^{2}[(c_{1}r-c_{2})/2]\,.$
(19)

Where the coefficients are given

$\displaystyle A=-{\frac{c_{1}^{2}}{16}}(4(\alpha-\beta)^{2}-1)$
$\displaystyle B={\frac{c_{1}^{2}}{16}}(4\gamma^{2}-8\gamma+3)$ $\displaystyle
C=-{\frac{c_{1}^{2}}{4}}(\alpha+\beta-\gamma)^{2}\,.$ (20)

Case:4 consider $a=1,b=0$, $\rho=c_{1}z$, therefore

$\displaystyle z=c_{2}e^{c_{1}r}\,.$ (21)

we have

$\displaystyle
E-V=C-A{\frac{1}{(c_{2}e^{c_{1}r}-1)^{2}}}-B{\frac{1}{c_{2}e^{c_{1}r}-1}}\,.$
(22)

Where the coefficients are given

$\displaystyle A=-{\frac{c_{1}^{2}}{4}}((\alpha+\beta-\gamma)^{2}+1)$
$\displaystyle
B={\frac{c_{1}^{2}}{2}}(\alpha^{2}+\beta^{2}+\gamma(1-\alpha-\beta)-1)$
$\displaystyle C=-{\frac{c_{1}^{2}}{4}}(\alpha-\beta)^{2}\,.$ (23)

Case:5 consider $a=1,b=1/2$ , $\rho=c_{1}z(1-z)^{1/2}$ , therefore

$\displaystyle z=\operatorname{sech}^{2}({\frac{c_{1}r+c_{2}}{2}})\,.$ (24)

we have

$\displaystyle
E-V=C-A\operatorname{csch}^{2}(c_{1}r+c_{2})-B\coth(c_{1}r+c_{2})\operatorname{csch}(c_{1}r+c_{2})\,.$
(25)

Where the coefficients are given

$\displaystyle
A={\frac{c_{1}^{2}}{4}}[4(\alpha^{2}+\beta^{2})-4(\alpha+\beta)+2\gamma-1]$
$\displaystyle B={\frac{c_{1}^{2}}{2}}(2\alpha-\gamma)(2\beta-\gamma)$
$\displaystyle C=-{\frac{c_{1}^{2}}{4}}(\gamma-1)^{2}\,.$ (26)

Case:6 consider $a=1,b=1$ , $\rho=c_{1}z(1-z)$, therefore

$\displaystyle z=1-{\frac{c_{2}}{e^{c_{1}r}+c_{2}}}\,.$ (27)

we have

$\displaystyle
E-V=C-A{\frac{e^{2c_{1}r}}{4(e^{c_{1}r}+e^{c_{2}})^{2}}}-B{\frac{e^{c_{1}r}}{4(e^{c_{1}r}+e^{c_{2}})}}\,.$
(28)

Where the coefficients are given

$\displaystyle A={\frac{c_{1}^{2}}{4}}[(\alpha-\beta)^{2}-1]$ $\displaystyle
B={\frac{c_{1}^{2}}{2}}(2\alpha\beta+\gamma-\alpha\gamma-\beta\gamma)$
$\displaystyle C=-{\frac{c_{1}^{2}}{4}}(\gamma-1)^{2}\,.$ (29)

So far we have discussed six cases, which represent six possible types of
potential energy. The other three possible cases $a=0,b=0$; $a=0,b=1/2$ and
$a=1/2,b=0$ cannot be considered as solvable potential due to the lack of an
energy terms.

Further, by applying the relationship between the hypergeometric function and
the Riemannian $P$-function and its transformation formula, we can obtain
other solutions of different forms.

It is worth noting that in case 2 if we set $c_{1}=i,c_{2}=\pi/2$, then

$\displaystyle z=(1-i\sinh r)/2\,.$ (30)

And

$\displaystyle E-V=C-A\operatorname{sech}^{2}r+iB\operatorname{sech}r\tanh r$
$\displaystyle
A=-{\frac{1}{4}}(2(\alpha+\beta)^{2}+1)+(\alpha+\beta-\gamma+1)\gamma$
$\displaystyle B=-{\frac{1}{2}}(\alpha+\beta-1)(\alpha+\beta+1-2\gamma)$
$\displaystyle C=-{\frac{1}{4}}(\alpha-\beta)^{2}\,.$ (31)

This is nothing but the complex PT-invariant potential energy discussed by
Ahmed et al [8, 9]. Similarly, we can construct others solvable PT-invariant
potentials. Set $c_{1}$ is pure imaginary number and $c_{2}$ is real number in
all sex cases, we will find these potentials are all complex PT-invariant
potentials.

If set $g(z)=0$ in the equation (5), then we can solve
$f(z)=\sqrt{c_{1}}\rho^{-1/2}$, and also assume that $f(z)$ have the form
$f=z^{p}(1-z)^{q}$, then we will arrive at the Bose solvable potentials [4].
These potentials have been studied generally [6, 5, 12, 13]. In a similar way,
we can also construct the corresponding complex PT-invariant potentials.

##  4 Conclusion

In this work, we construct a new class of solvable potentials by means of the
hypergeometric equation, where the Schrödinger equation is rewritten as a
hypergeometric equation through taking a similarity transformation. we
consider $g(r)\neq 0$, and suppose $f=z^{p}(1-z)^{q}$, such that $p,q$
dependent $\alpha,\beta,\gamma$, this method provides us a new perspective to
construct solvable potentials for Schrödinger equation. Our method unlike
previously let $g(r)=0$ or used the invariant identity of the target equation,
this method need $f(z)=\sqrt{c_{1}}\rho^{-1/2}$, cause $f(z)$ to be a fixed
function. In a similar way, computing the other Fuchs equations with our
method, such as the Heun equation, can also construct corresponding solvable
potentials[14].

## Acknowledgement

This work was supported by the Guangxi Scientific Programm Foundation under
grant No. 2020AC20014, the Scientific Research Foundation of Guilin University
of Technology under grant No. GUTQDJJ2019206.

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