# Role of surface termination in the metal-insulator transition of V2O3(0001)
ultrathin films

Asish K. Kundu<EMAIL_ADDRESS>Surface Physics and Material Science
Division, Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata
700064, India  Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory, Upton, New York 11973, USA  Sukanta Barman
Surface Physics and Material Science Division, Saha Institute of Nuclear
Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India  Department of Physics,
Raja Peary Mohan College, Uttarpara, Hooghly 712258, India  Krishnakumar S. R.
Menon<EMAIL_ADDRESS>Surface Physics and Material Science Division,
Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064,
India

###### Abstract

Surface termination is known to play an important role in determining the
physical properties of materials. It is crucial to know how surface
termination affects the metal-insulator transition (MIT) of V2O3 films for
both fundamental understanding and its applications. By changing growth
parameters, we achieved a variety of surface terminations in V2O3 films that
are characterized by low energy electron diffraction (LEED) and photoemission
spectroscopy techniques. Depending upon the terminations, our results show MIT
can be partially or fully suppressed near the surface region due to the
different filling of the electrons at the surface and sub-surface layers and
change of screening length compared to the bulk. Across MIT, a strong
redistribution of spectral weight and its transfer from high-to-low binding
energy regime is observed in a wide-energy-scale. Our results show total
spectral weight in the low-energy regime is not conserved across MIT,
indicating a breakdown of ‘sum rules of spectral weight’, a signature of a
strongly correlated system. Such change in spectral weight is possibly linked
to the change in hybridization, lattice volume (i.e., effective carrier
density), and spin degree of freedom in the system that happens across MIT. We
find that MIT in this system is strongly correlation-driven where the
electron-electron interactions play a pivotal role. Moreover, our results
provide a better insight in understanding the electronic structure of strongly
correlated systems and highlight the importance of accounting surface effects
during interpretation of the physical property data mainly using surface
sensitive probes, such as surface resistivity.

##  I Introduction

Study of the metal-insulator transition in strongly correlated transition
metal oxides such as Ti2O3, VO2, V2O3, and NbO2 are still an active field of
research for the understanding and tuning of MIT for their potential
applications in intelligent windows and field-effect transistors Zhou  _et
al._ (2013); Yamamoto  _et al._ (2019); Brockman  _et al._ (2011). These
materials are known to undergo a temperature-dependent MIT in concurrence with
a structural transition Mott and Friedman (1974); Joshi  _et al._ (2019). In
addition to the structural transition, the magnetic transition also occur
simultaneously in V2O3 and is found to play an important role in MIT Trastoy
_et al._ (2018a). Numerous studies show, a small change of the crystal
structure, by Cr or Ti substitution Kuwamoto  _et al._ (1980); McWhan  _et
al._ (1969); Shin  _et al._ (1995); Lupi  _et al._ (2010) or applying pressure
Limelette  _et al._ (2003) can have a major effect on the MIT of bulk V2O3.
Other factors such as surface reconstruction and lattice defects can also
affect the MIT of these materials Wickramaratne  _et al._ (2019); Panaccione
_et al._ (2007). The breakdown of Mott physics at the surface of VO2 thin
films is reported by Wahila et al. Wahila  _et al._ (2020) due to the surface
reconstructions/terminations. For V2O3 thin-films, various types of surface
termination have been reported by several groups Dupuis  _et al._ (2003);
Pfuner  _et al._ (2005); Feiten  _et al._ (2015a) which can make the
understanding of the physics of MIT even more complicated.

At room temperature, V2O3 is a paramagnetic metal (PM) with a corundum
structure while at low temperature, below about $\sim$150-160 K, it undergoes
a transition to an antiferromagnetic insulating (AFI) phase accompanied by a
structural change from the trigonal (corundum) to a monoclinic one with a 1.4%
volume increase. Despite the extensive studies of MIT on V2O3 single crystal
and polycrystalline powder, contradictory results persist in thin-films Luo
_et al._ (2004); Dillemans  _et al._ (2014); Grygiel  _et al._ (2007).
Recently, Luo et al. Luo  _et al._ (2004) have reported a thickness-dependent
metal-insulator transition of V2O3 thin-film grown on Al2O3(0001) substrate.
They argue that the MIT with decreasing film thickness is due to the increase
of the $c/a$ lattice parameter ratio because of the substrate-induced strain
in the films. Their results show the thicker films (20 nm) remain in the
metallic phase down to 4 K. In contrast, Dillemans et al. Dillemans  _et al._
(2014) have shown, all their films with thickness ranging from 4$-$73 nm grown
on the same substrate (Al2O3) undergo temperature-dependent MIT. Further,
x-ray diffraction (XRD) studies in these films (thickness range of 10-100 nm)
do not show any systematic change of lattice parameters Grygiel  _et al._
(2007). Schuler et al. Schuler  _et al._ (1997) have addressed an impact of
the synthesis conditions on MIT and growth modes of the films. All these
together suggest the absence of MIT observed in ref. Luo  _et al._ (2004);
Grygiel  _et al._ (2007) can not be explained only by considering c/a ratio
change.

It is known that V2O3(0001) films generally contains various types of surface
termination which changes upon growth conditions Feiten  _et al._ (2015b, a);
Dupuis  _et al._ (2003); Pfuner  _et al._ (2005). Films grown at ultra-high
vacuum (UHV) in optimal growth condition favours the formation of vanadyl
(V=O) terminated surface Feiten  _et al._ (2015b, a). Further, annealing the
V=O terminated films at higher oxygen partial pressure gives rise to a
$(\sqrt{3}\times\sqrt{3})R30^{o}$ surface due to the removal of 1/3 or 2/3 of
the V=O groups from the surface layer Dupuis  _et al._ (2003); Pfuner  _et
al._ (2005). From the theoretical phase diagram Feiten  _et al._ (2015b) of
oxygen chemical potential vs. growth temperature, complete removal of V=O
groups is also possible at even higher oxygen partial pressure and temperature
which should reflect as a reconstructed O3 terminated (rec-O3) surface layer
Feiten  _et al._ (2015b); Kresse  _et al._ (2004). The structural model for
all these surface termination as obtained from density function theory (DFT)
Feiten  _et al._ (2015b) are shown in figure 1(e). Depending upon the
terminations, the electronic structure at the surface/sub-surface is expected
to deviate from its bulk and its impact should reflect on the electronic
properties and MIT. Pfuner et al. Pfuner  _et al._ (2005) have reported
suppression of MIT for the $(\sqrt{3}\times\sqrt{3})R30^{o}$ terminated
V2O3(0001) surface. It appears that during the interpretation of the
experimental data such as surface resistivity, an account of the surface
termination effect, in addition to the substrate-induced strain, is essential
to get the complete picture of MIT. However, it is not typically discussed how
breaking crystal symmetry via surface termination affects the MIT and detailed
systematic spectroscopic studies are barely available in the literature. In
general, in strongly correlated systems, the change of hybridization, lattice
volume (electron density), spin degree of freedom can show unusual spectral
evolution mainly in the low-energy scale Meinders  _et al._ (1993); Kohno
(2019); Rozenberg  _et al._ (1996), however, there is still little
understanding how MIT in V2O3 alters the low-energy physics.

Here, we study the effect of surface termination on the electronic structure
of V2O3 films and its impact on MIT. To achieve this goal, well ordered
V2O3(0001) surface has been grown on Ag(111) substrate with different oxygen
partial pressures and substrate temperatures, following post-growth
procedures. Our results show, at optimal growth condition, the V2O3(0001)
surface is terminated by the V=O groups while annealing the film at higher
oxygen partial pressure and temperature results in the
$(\sqrt{3}\times\sqrt{3})R30^{o}$ surface structure due to the removal of V=O
groups. Films directly grown at higher oxygen partial pressures and
temperatures results in a nearly rec-O3-like surface. Partial to full
suppression of MIT is observed by going from V=O to
$(\sqrt{3}\times\sqrt{3})R30^{o}$ and rec-O3-like surfaces. We also show that
MIT in this system is strongly correlation driven and it is associated with a
strong redistribution of orbital occupancy in the much wider energy scale than
previously thought. Further, we discuss the possible origin of unusual
spectral weight enhancement in the low-energy-scale that occurs when the
system is cooled below MIT.

##  II Results and discussion

###  II.1 Surface structure of V2O3 films

Figure 1: LEED pattern of V2O3 films grown on Ag(111) at various substrate
temperatures and oxygen partial pressures, along with structural models. (a)
Films grew at 600 K in presence of $P(O_{2})$= 2$\times$10-7 mbar followed by
UHV annealing at 773 K. The position of first-order diffraction spots from
Ag(111) (circle) and V2O3 (square) are marked. Images with enhanced contrast
and brightness are shown inside oval (red). (b) and (c) Post annealing of the
film at 673 K, $P(O_{2})$= 5$\times$10-7 mbar and 673 K, $P(O_{2})$=
6$\times$10-6 mbar, respectively. (d) Films directly grown at 673 K,
$P(O_{2})$= 6$\times$10-6 mbar. Growth temperatures and post-annealing
temperatures are denoted as GT and AT, respectively. (e) Side-view of the
structural model (gray (V) and blue (O)) for the vanadyl (V=O),
$(\sqrt{3}\times\sqrt{3})R30^{o}$, and rec-O3 terminated V2O3(0001) surface as
obtained from DFT Feiten  _et al._ (2015b). V=O groups are enclosed by the
dotted ovals. Three types of bulk termination can be realized by terminating
the surface at V (half-metal), V’ (full-metal), and O3 (bulk-O3) layers.
Dotted rectangle encloses trilayer structure on top of half-metal terminated
V2O3 surface. Trylayer has VO2 stoichiometry but structurally different than
the rutile VO2(100) (f). (g)-(i) Top-view of the half-metal, V=O and
$(\sqrt{3}\times\sqrt{3})R30^{o}$ terminated surface, respectively. Vanadyl
oxygens are enclosed by the yellow circles. Rhombus in (h) represents the
surface unit-cell of V=O terminated surface. Rhombus with solid and dotted
lines in (i) represents the surface unit cell corresponding to the removal of
1/3 and 2/3 of V=O groups, respectively.

Figure 1(a) shows the LEED image of 30 MLE V2O3 films on Ag(111), grown at a
substrate temperature of 600 K in presence of $P(O_{2})$= 2$\times$10-7 mbar
followed by UHV annealing at 773 K. The position of the first order
diffraction spots from Ag(111) (before deposition of films) and V2O3 are
marked. LEED pattern shows $(\sqrt{3}\times\sqrt{3})R30^{o}$ structure w.r.t
Ag(111) substrate. It is expected as the lattice parameter of the V2O3(0001)
surface (4.95 Å) is very close to the $\sqrt{3}$ times of Ag(111) lattice
parameter ($\sqrt{3}\times$2.89=5.0 Å) with only 1% lattice mismatch,
favouring the growth of epitaxial V2O3(0001) films. This structure can be
thought as (1$\times$1) structure w.r.t the V2O3(0001) basis vectors.
Throughout the paper, we will call this structure as (1$\times$1) structure
for the simplicity of our discussions. The (1$\times$1) structure have been
also observed by previous studies of V2O3 films on Au(111) and W(110)
substrate at similar growth conditions which was regarded as a V=O terminated
V2O3(0001) surface Dupuis  _et al._ (2003); Pfuner  _et al._ (2005). The V=O
terminated surface may be thought as an ideal half-metal terminated surface
(figure 1(g)) of V2O3(0001) with one additional oxygen atom bonded on top of
each outermost V atom (figure 1(h)). Annealing the V=O terminated surface in
presence of oxygen shows additional spots (superstructures) that form
$(\sqrt{3}\times\sqrt{3})R30^{o}$ structure w.r.t. (1$\times$1)-V2O3 as shown
in figure 1(b). These superstructure spots gain intensities upon further
annealing the sample at higher oxygen pressure as can be seen in figure 1(c).
The $(\sqrt{3}\times\sqrt{3})R30^{o}$ structure could originate by removing
the 1/3 or 2/3 amount of V=O groups from the V=O terminated surface Schoiswohl
_et al._ (2004); Feiten  _et al._ (2015a). The surface atomic configuration is
shown in figure 1(i), where the rhombus with solid and dotted lines represent
the surface unit cell corresponding to the removal of 1/3 and 2/3 of V=O
groups, respectively. Relatively lower intense and broad superstructure spots
in figure 1(b) compare to figure 1(c) may indicate that the removal of V=O
groups is less uniform and lower in numbers in figure 1(b) than in figure
1(c). Similar behavior has been reported by Schoiswohl et al. Schoiswohl  _et
al._ (2004) using combined scanning tunneling microscopy (STM) and LEED study.

Figure 1(d) shows the LEED pattern of films that has been directly grown at
higher temperature and oxygen partial pressure (673 K, $P(O_{2})$=
6$\times$10-6 mbar). The LEED pattern shows mainly (1$\times$1) structure but
additional broad spots with extremely faint intensities can be visible at
higher brightness/contrast (inside oval) that form
$(\sqrt{3}\times\sqrt{3})R30^{o}$ structure. Feiten et al. Feiten  _et al._
(2015b), have pointed out that both V=O and rec-O3 terminated surface exhibit
identical reflex patterns and indicated that the LEED-IV measurements might be
helpful to distinguish them. Further, DFT calculation Feiten  _et al._ (2015b)
also predicts that the rec-O3 surface termination favors at higher temperature
and oxygen partial pressure (see the structural model in figure 1(e)). Thus,
the presence of extremely low intense $(\sqrt{3}\times\sqrt{3})R30^{o}$
structure could mean that only locally 1/3 or 2/3 amount of V=O groups are
missing or left. In other words, this surface is either very close to the V=O
terminated or rec-O3 terminated surface.

Figure 2: Comparison of experimental LEED-IV curves with theory Feiten  _et
al._ (2015b) for V=O (GT: 600 K, $P(O_{2})$= 2$\times$10-7 mbar followed by
UHV annealing at 773 K) and rec-O3 (GT: 673 K, $P(O_{2})$= 6$\times$10-6 mbar)
terminated surface. Experiment (top panel) and theory (bottom panel). The peak
marked by ‘a’ shows a prominent two-peak structure for V=O terminated surface,
consistent with theory. Relative to the rec-O3 termination, the ‘b’ and ‘c’
features shifted towards higher energy for the V=O terminated surface. Arrows
at 124 eV indicate the appearance of a peak for the V=O terminated surface
which is absent for rec-O3.

To get more insight into the surface structure, we have performed experimental
LEED-IV measurements and compared them with the reported theoretical results
Feiten  _et al._ (2015b). The top panel in figure 2 shows the experimental
LEED-IV curves extracted from the first-order diffraction spots of V2O3
corresponding to the figure 1(a) (black) and figure 1(d) (red), respectively.
The bottom panel in figure 2 shows the simulated LEED-IV curves for the V=O
and rec-O3 terminated surfaces Feiten  _et al._ (2015b). Between the two
experimental curves, there are some significant differences can be observed
such as black spectra shows prominent two peak structure for the peak ‘a’,
while peaks ‘b’ and ‘c’ shifted towards the higher energy compared to the red
spectra. By comparing with the theoretical spectra, it appears that the
position of peaks in black experimental spectra better match with the V=O
terminated while the red spectra with the rec-O3 terminated surface. Although
it is not perfectly rec-O3 terminated surface as in that case it would show
perfectly ($1\times 1$) LEED pattern Feiten  _et al._ (2015b), but we will
call it rec-O3-like due to its similarities with the LEED-IV curves with the
rec-O3 terminated surface. We note that the position of peaks in the
experimental I-V curves are in agreement with the theoretical curves whereas
peak intensities differ. The experimental curves are not normalized by the
incident electron beam-current that could produce the observed intensity
differences. Unfortunately, we could not correct it as there were some
technical issues with the beam-current reader during I-V data acquisition. The
normalization can only change the intensity of the peaks, not the peak
features or their position. Thus, it will not affect the main claims of our
study.

###  II.2 Electronic structure and MIT of V2O3 films

Figure 3: XPS core-levels of O $1s$, V 2$p_{1/2}$ and V 2$p_{3/2}$ for V=O
(GT: 600 K, 2$\times$10-7 mbar), $(\sqrt{3}\times\sqrt{3})R30^{o}$ (AT: 673 K,
6$\times$10-6 mbar) and rec-O3 (GT: 673 K, 6$\times$10-6 mbar) terminated
surfaces. All the spectra are Shirley background subtracted.

Understanding the chemical composition of these films is essential as vanadium
may exist at different oxidation states (V+2 to V+5 ) depending upon the
growth conditions. The most common way to directly access the chemical state
of materials is to perform the XPS measurements. Figure 3 shows the O $1s$, V
2p1/2 and V 2p3/2 core-levels of V=O, $(\sqrt{3}\times\sqrt{3})R30^{o}$, and
rec-O3 terminated films. Despite the different surface structures observed in
LEED, all the XPS spectra show very similar spectral features, indicating that
they have a similar chemical composition in the bulk. We note that depending
upon the used photon energy, XPS can be surface or bulk sensitive as its
probing depth varies depending upon materials, the kinetic energy, and
photoemission angle of the photoelectrons being measured Brundle and Crist
(2020). Here, the kinetic energy of the V 2p photoelectrons measured using Al
Kα (hv=1486.6 eV) is $\sim$ 965 eV that corresponds to $\lambda$ $\sim$ 2.06
nm and probing depth 3$\lambda$ $\sim$ 6.2 nm. Where $\lambda$ is the
attenuation length (mean free path) of photoelectrons. As the probing depth,
$\sim$ 6.2 nm is comparable to the film thickness $\sim$ 6.5$-$7 nm, thus the
surface effect is expected to be suppressed here.

Figure 4: ARPES intensity plot and EDCs at $\bar{\Gamma}$ for various V2O3
films (30 MLE) below (red) and above (black) MIT. (a) and (b) are the ARPES
intensity plot for the V=O terminated surface at 300 K and 108 K,
respectively. (c) EDCs (at $\bar{\Gamma}$) as obtained from (a) and (b) are
plotted together. Inside dotted circle a Fermi edge-like feature (red spectra)
can be seen. (d) Annealing the V=O terminated film at 600 K, $P(O_{2})$=
5$\times$10-7 mbar for 10 min, (e) Annealing of the film (d) at 673 K,
$P(O_{2})$= 5$\times$10-7 mbar for 10 min. (f) Annealing of the film (e) at
673 K, $P(O_{2})$= 6$\times$10-6 mbar for 10 min. (g) EDCs at 300 K for
samples (c)-(f) plotted together. (h) Films directly grown at 673 K,
$P(O_{2})$= 6$\times$10-6 mbar (rec-O3). All the data were taken using He Iα
(21.2 eV) photons.

XPS spectra were fitted (not shown) to extract the exact peak position and
their full width at half maximum (FWHM). The details of fitting can be found
elsewhere in our previous study Kundu and Menon (2017). The obtained peak
position of O $1s$, V 2p1/2 and V 2p3/2 core-levels are 530.2 eV, 522.85 eV
and 515.6 eV, respectively. In the case of vanadium oxide, the most common way
to identify the exact phase is to measure the energy separation ($\Delta$)
between the O $1s$ and V 2p3/2 core-levels and their FWHM Mendialdua  _et al._
(1995). Here, the obtained value of $\Delta$ is 14.7 eV and FWHM of O $1s$ and
V 2p3/2 peaks are 1.3 eV and 4.05 eV, respectively. All these values lie in
the stoichiometric region of V2O3 Kundu and Menon (2017); Sawatzky and Post
(1979). The overall spectral line-shape is also in good agreement with the
V2O3 films Kundu and Menon (2017); Sawatzky and Post (1979); Kamakura  _et
al._ (2004). These results are in contrast with the DFT predictions Feiten
_et al._ (2015b), as the bulk phase of rec-O3 terminated films is expected to
be already in the region of V2O5 stoichiometry. This inconsistency may come
from the overestimation of the Coulomb correlation effects in DFT calculation
and the choice of exchange-correlation functional. In our experimental growth
conditions, it may be possible that the formation of a higher oxidation state
in bulk is just prevented by kinetic limitations.

Further, it is important to know how surface termination affects the
electronic structure and the MIT. Figure 4(a) and (b) show the ARPES intensity
plots within the V $3d$ region of V=O terminated V2O3 films, above and below
MIT temperatures, respectively. The energy distribution curves (EDCs) obtained
from figure 4(a) and (b) are shown in figure 4(c). In pure V2O3, V3+ has two
$3d$ electrons, occupying the triply degenerate $t_{2g}$ (a1g,
e${}_{g}^{\pi}$) orbitals, split into a lower e${}_{g}^{\pi}$ doublet and an
upper $a_{1g}$ singlet in the trigonal crystal field (CF) of PM phase. In the
low-temperature monoclinic phase, the additional low-symmetry CF component
lifts e${}_{g}^{\pi}$ degeneracy. Moreover, EDCs of V $3d$ region is dominated
by e${}_{g}^{\pi}$ and a1g orbital. In the paramagnetic phase (figure 4(a)),
the bands at $\sim$ -1.1 eV and $\sim$ -0.4 eV are arises from the lower
Hubbard band (LHB) and quasiparticle peak (QP), respectively, according to the
previous studies Mo  _et al._ (2003); Rodolakis  _et al._ (2009). The QP is
broad and less intense due to the lower photon energy (21.2 eV) used in our
experiment, where we are extremely sensitive to the surface. Previous studies
also show less intense QP at the surface than bulk due to the enhanced
correlation at the surface region relative to the bulk Mo  _et al._ (2003).
This is also in line with the DFT+ dynamical mean-field theory (DMFT) results,
where it has been shown that a dead layer forms below the surface of V2O3,
where quasiparticles are exponentially suppressed Borghi  _et al._ (2009).

In figure 4(c), a significant change in spectral weight near Fermi level (EF)
and its shifts towards the higher binding energy is observed when the sample
is cooled below the MIT temperature. The observed behavior is mainly caused by
the change in orbital occupancy in a1g and e${}_{g}^{\pi}$ orbital of V 3d
across MIT, as observed in x-ray absorption spectroscopy (XAS) measurements
Park  _et al._ (2000). By lowering the temperature from 300 K (PM) to 108 K
(AFI), it seems like the spectral weight is transferred from QP to the LHB
side and the QP and LHB are shifted by $\sim$ 200 meV and $\sim$ 80 meV,
respectively, towards higher binding energy. Total width (FWHM) of the V $3d$
region (and LHB) also changes from 1.85 eV (1.4 eV) to 1.54 eV (1.24 eV) by
going from PM to AFI phase. These values are obtained from fitting the spectra
(see supplementary material, figure S1). According to the Hubbard model,
decrease of LHB width ($W$) and its shift towards the higher binding energy,
implies that the on-site Coulomb interactions ($U$), as well as correlation
($U/W$) of the system, gets enhanced in the AFI phase compared to the PM
phase. This suggests, MIT in V2O3 is strongly correlation driven, where the
electron-electron correlation plays an important role. Although the Hubbard
model ignores the degeneracy of bands, which is crucial for understanding the
magnetic structures Castellani  _et al._ (1978), it seems well captures the
interplay between the electron-electron interactions and kinetic energy. The
change of $U$ and $W$ across the phase transition are most likely due to a
change of lattice parameter (change the hopping parameters) and effective
screening length. Additionally, in the AFI phase, the scattering rate is also
found to decrease which might sharpen the bandwidth as well Deng  _et al._
(2014). From PM to AFI transition, an increase of $U$ and decrease of $W$
values are also reported by Rozenberg et al. Rozenberg  _et al._ (1995) using
optical measurements, similar to our observation.

However, it is not clear whether the QP band is totally absent in the AFM
phase or it just shifts towards the higher energy with a redistribution of
spectral weight. Across MIT, similar spectral features like our results were
also reported for the V=O terminated V2O3 films Pfuner  _et al._ (2005) and
bulk single crystals Mo  _et al._ (2006). This apparent shift of the PM phase
spectrum across the MIT has been also found in a recent full-orbital DMFT
calculation Anisimov  _et al._ (2005), although the actual shape and peak
position of the theoretical spectrum has different from the experimental
spectra. Other DFT+DMFT calculations Lechermann  _et al._ (2018); Leonov  _et
al._ (2015) incorporating full charge self-consistency were unable to capture
the complex ‘two-peak structure of the LHB in the insulating (AFI) phase. We
note that all these calculations were performed without accounting the exact
spin-structure of the AFI phase of V2O3 and some of them only account the PM
crystal structure, which may produce the observed discrepancy between theory
and experiment. Trastoy et al. have also shown the importance of magnetic
structure in MIT of V2O3 using resistivity and magnetoresistance measurements,
combined with the theoretical calculations Trastoy  _et al._ (2018a).

Besides, in figure 4(c), we observe a small Fermi-edge-like feature (enclosed
by a dotted circle) at EF in the 108 K spectra which hinders the opening of a
full energy gap at EF. This feature is not resolved in the 300 K spectra as it
masks by the QP peak. By comparing with the theoretical layer-resolved density
of state calculation Kresse  _et al._ (2004), it appears that this edge-like
feature is most probably originating from the V=O groups at the surface of the
film. The theoretical data clearly shows the presence of a small density of
states (DOS) at EF, which arises from a VO-like surface layer. This feature
gains its intensity when measurements were performed using He IIα than He Iα
(see supplementary information, figure S2) that strongly suggest its surface-
related origin, as He IIα is more surface sensitive than He Iα. Furthermore,
as VO does not show temperature-dependent MIT thus, the DOS originating from
VO should remain at the same energy position both below and above MIT
temperatures, which agrees with our results. This peak (V=O related) forms a
Fermi-edge-like feature as it is cut by the Fermi Dirac distribution function
at EF. We also vary the substrate temperature, a different substrate such as
W(110), and oxygen partial pressure in a wide range but this feature was
always present with a slight variation of intensity, depending upon the growth
condition. According to Schoiswohl et al. Schoiswohl  _et al._ (2004), V2O3
films grown at optimal condition followed by UHV annealing produce large-area
films with fully covered V=O groups at the surface. In agreement, upon post-
annealing the film in UHV, we also observe the intensity of the Fermi-edge-
like feature and other V=O-related peaks are enhanced (see supplementary
information figure S3). Possibilities of Ag segregation from substrate or
formation of microcracks in the film can be ruled out, as no intensity
enhancement of Ag-related peaks is observed (figure S3). Our previous study
shows strong enhancement of Ag-related peak intensity when Ag segregates on V
islands Kundu and Menon (2018). However, our study can not completely rule out
the possibility of the presence of some precursor metallic phase even at 108
K, as the coexistence of microscopic metallic and insulating islands was found
in a wide temperature range for Cr-doped V2O3 sample Lupi  _et al._ (2010).
Moreover, It is very unlikely that the V=O surface layer with its different
geometry and electronic structure can support the MIT. Thus, the observation
of spectral change across MIT in figure 4(c) and opening of a partial energy
gap below MIT temperature, clearly suggest that the MIT in the bulk of the
V2O3 film is communicated through the V=O surface layer.

Post annealing the V=O terminated films with increasing oxygen partial
pressure and temperature are shown in figure 4(d)-(f). It can be seen that MIT
is progressively weakened and completely suppressed by going from figure 4(d)
to figure 4(f). The complete suppression of MIT is also observed for the
rec-O3 like surface as shown in figure 4(h). In figure 4(e), change of
spectral weight across MIT is still observed but without a gap at EF, whereas
in figure 4(f), the spectral change is minimal and MIT is completely
suppressed. In figure 4(g), all the EDCs from the figure 4(c) to (f) films
taken at 300 K are plotted together. It is evident that with increasing oxygen
partial pressure and temperatures, the total area of the V 3$d$ region (0$-$3
eV) decreases about $\sim$ 16% (from figure 4(c) to (f)) which suggests a
decrease of 3$d$ electrons from 2 for V2O3 to the lower numbers, near the
surface layer. The decrease in electron counts suggests that the surface is
getting more oxygen-rich. A similar change of spectral features and decrease
of V 3$d$ area are reported for bulk V2-yO3 sample Kim  _et al._ (1998). These
suggest that the bulk MIT of V2O3 films are more effectively screen for the
$(\sqrt{3}\times\sqrt{3})R30^{o}$ and rec-O3 like oxygen-rich surfaces than
(1$\times$1) V=O terminated surface. Suppression of MIT near the surface
region can be understood by considering the structural change which occurs
during the formation of rec-O3 and $(\sqrt{3}\times\sqrt{3})R30^{o}$ surfaces,
where the removal of each V=O group exposes three oxygen atoms of the
underlying O3 plane (see the structural model in figure 1(g) and (i)). To
maintain thermodynamical equilibrium, the surface region, including subsurface
layers relaxes and undergoes rearrangements (figure 1(e)) Schoiswohl  _et al._
(2004), which strongly deviates from corundum bulk structure, whereas for V=O
surface, only the top layer gets affected while subsurface layers preserve
their bulk structure. Suppression of MIT is also observed in resistivity
measurements for the oxygen-rich sample Trastoy  _et al._ (2018b). Surface
termination act as a charge doping (here, hole doping) to the surface layers,
thus drastically alter the MIT characteristics by pushing the $d$-filling
further or closer to the Mott criterion Lee  _et al._ (2019).

We note that for rec-O3 and $(\sqrt{3}\times\sqrt{3})R30^{o}$ (underlying
surface below the removed V=O groups) structures, the surface has a stacking
sequence -V2O3-V-O3V3O3 which may be thought of as an O3V3O3 trilayer
structure with a formal VO2 stoichiometry on top of the single V terminated
bulk structure (see figure 1(e)). As VO2 is known to show MIT below 320 K Choi
_et al._ (2021), both the 300 K and 108 K EDCs corresponding to these
structures would show insulating behavior or at least the 108 K EDCs, as the
underlying structure is still V2O3 which only shows MIT at low temperature.
While our data show metallic behavior in both temperatures (figure 4(f) and
(h)). This behavior suggests that the chemical composition alone cannot
determine the MIT properties but a combination of chemical composition and
crystal structure does. The structural difference between the trilayer
stacking with VO2 composition and the bulk VO2 are shown in figure 1(e) and
(f), respectively.

Figure 5:  EDCs in a wide energy range around MIT. EDCs using He IIα (40.8 eV)
photons for (a) V=O and (b) $(\sqrt{3}\times\sqrt{3})R30^{o}$ (AT: 673 K,
$P(O_{2})$= 6$\times$10-6 mbar) structure. Inset of (a) shows EDCs of V=O
terminated surface using He Iα photons.

We also note that in figure 4(c), the integrated area within the energy
ranges, 0 to -3.0 eV is enhanced by 16.5% in the AFI (108 K) phase compared to
PM (300 K), indicating breakdown of ‘sum rules of spectral weight’ in that
energy regime which in turn suggests the presence of strong correlation in the
system Meinders  _et al._ (1993); Kohno (2019). Theoretically, the breakdown
of ‘sum rules of spectral weight’ in the low-energy scale has been found in
strongly correlated systems when a change of hybridization and/or electron
density are introduced in the system Meinders  _et al._ (1993); Kohno (2019).
A question may arise, what is the source of this extra spectral weight? To get
further insight, we have plotted EDCs in a wide energy window as shown in
figure 5(a). Not only in the lower energy regime, but a strong redistribution
of spectral weight is also observed within the whole energy range, 0 to -9.5
eV. Spectra taken at He Iα photons (inset of figure 5(a)) also shows similar
behavior. The main difference between He Iα and He IIα spectra is that
spectral weight within the energy ranges, -5 to -8 eV decreases in the AFI
phase for He Iα whereas increases for He IIα. The different behavior could be
associated with the kz dependence of spectral weight transfer Kohno (2019).
The estimated increase of spectral weight in the AFI phase within the whole
measured energy ranges are $\sim$ 3.5% and $\sim$ 6% for He Iα and He IIα,
respectively. Previous photoemission studies performed on the Cr-doped V2O3
sample, between the paramagnetic insulator (PI) and the crossover regime also
found spectral weight redistribution within the whole valence band region Mo
_et al._ (2004). In contrast, their results show that the V 3d and O 2p
spectral weight conserved separately within these two regions (PI and
crossover). Similar to our results, optics experiment also shows total
spectral weight across MIT is not conserved within energy range, 0 to 6 eV
above the Fermi energy Qazilbash  _et al._ (2008); Stewart  _et al._ (2012);
Rozenberg  _et al._ (1995). Interestingly, it can be seen that for the oxygen-
rich $(\sqrt{3}\times\sqrt{3})R30^{o}$ surface, the spectral weight change is
almost negligible as shown in figure 5(b). These results confirming that the
spectral weight enhancement and its redistribution are strongly related to the
MIT transition.

The change in lattice structure and volume that occurs across MIT may have
important consequences in the observed spectral weight renormalization
(redistribution), as it alters the hybridization strength between different
orbital as well as carrier density (carriers/volume) that strongly affect the
orbital occupancy. In general, electronic states (bands) are usually
considered invariant regardless of the electron density in a band picture.
However, in interacting systems, the spectral-weight distribution changes
depending on the electron density, and electronic states can even emerge or
disappear Kohno (2019); Meinders  _et al._ (1993). Furthermore, Kohno et al.,
Kohno (2019), have explicitly shown that the electronic excitations much away
from the Fermi level can also become dominant and exhibit significant
characteristics due to the change in electron density. Keeping in mind that,
as we are not externally doping carriers in the system then the total carriers
should be conserved but carrier density may change across MIT due to the
change in lattice volume. Thus, one feasible interpretation for an apparent
increase of spectral weight within 0 to -9.5 eV regime would be that the
transfer of spectral weight occurs from much deeper to the lower binding
energy side, due to the redistribution of orbital occupancy in the larger
energy regime. It can be clearly seen from the He Iα data (inset of figure 5)
that the spectral weight decreased in the higher binding energy side whereas
increased going towards the Fermi energy. Further, theoretical study
considering the antiferromagnetic ordering in V2O3 also find evidence of
spectral weight transfer from higher to lower binding energy side compared to
the PM phase Rozenberg  _et al._ (1996). All these together suggest that the
strong change in spectral weight across MIT is due to the complex interplay
between the charge, orbital and spin degree of freedom in this system.

We also note that MIT of V2O3(0001) films have been studied by several groups
by resistivity measurement Luo  _et al._ (2004); Dillemans  _et al._ (2014);
Grygiel  _et al._ (2007); Brockman  _et al._ (2011). Unlike bulk V2O3 single
crystals, in most of the thin-films, resistivity vs. temperature curve do not
show sharp transition Dillemans  _et al._ (2014); Grygiel  _et al._ (2007);
Brockman  _et al._ (2011) and in some cases, the system remains in the
metallic phase Luo  _et al._ (2004); Grygiel  _et al._ (2007). Substrate-
induced strain effect in the films has been considered as an origin of the
absence of the temperature-driven MIT Luo  _et al._ (2004); Grygiel  _et al._
(2007). In contrast, Dillemans et al. Dillemans  _et al._ (2014) have shown
that V2O3(0001) films grown on the same substrate (Al2O3) undergo temperature-
dependent MIT, questions the validity of strain-induced loss of MIT in this
system. Our results highlight, in addition to the epitaxial strain, there
should be surface termination effects, and combining these two can better
explain the observed resistivity change in a wide temperature scale or
suppression/absence of MIT in most of the thin-film systems.

###  II.3 Conclusion

In summary, we have grown ultrathin films of V2O3(0001) with various surface
terminations and characterized them using LEED and photoemission spectroscopy
techniques. We have shown that depending on the types of termination, MIT at
the surface region can be partially or fully suppressed. This suggests high
sensitivity of the Mott metal-insulator transition to the local environment,
consistent with recent theoretical predictions Lechermann  _et al._ (2018). We
have also shown, upon MIT, the spectral weight is not only shifted from QP to
the LHB, but a huge redistribution of spectral weight is observed within the
whole valence band. Total spectral weight was found to be not conserved within
the energy ranges, 0 to -9.5 eV which indicates that the spectral weight is
redistributed in much larger energy, suggesting the system is heavily
correlated. Such enhancement of spectral weight in the low-energy regime is
mainly due to the spectral weight transfer from higher to lower binding energy
regime due to the change in complex interplay between the charge, orbital and
spin degree of freedom in this system that occurs during MIT. We have also
shown that in addition to the other effects, taking into account surface
termination can better explain the anomaly observed in the resistivity data of
V2O3. We expect that similar behavior would be at play for other complex
oxides and the interpretation of physical properties data (mainly using
surface sensitive probe) requires more attention. Furthermore, the observed
complex structure of LHB in our experiment (in the AFI phase) is not
reproduced by the currently available theories, including DFT+DMFT. Thus, we
believe that our results might help in the further development of the
theoretical understanding of MIT in strongly correlated systems.

##  III Experimental

The Ag(111) substrate was cleaned by repeated cycles of Ar+ ion sputtering
(600 eV, 1 $\mu$A) for 15 min followed by annealing at 823 K for 20 min until
a sharp $p(1\times 1)$ LEED pattern was observed. The substrate preparation,
growth, and LEED measurements were performed in the preparation chamber with a
base pressure of 1$\times$10-10 mbar. Vanadium was evaporated from a well-
degassed water-cooled e-beam evaporator at a constant rate of 0.3 Å/min in
oxygen partial pressure range of P(O2)= $1\times 10$${}^{-7}-6\times 10$-6
mbar for the deposition of the V2O3 film, while during deposition, the
substrate temperature was varied between at 473$-$673 K. The rate of
V-deposition was calibrated with a water-cooled quartz crystal thickness
monitor. One monolayer equivalent (MLE) of vanadium oxide film is defined as
the atomic density corresponding to 1 ML vanadium on Ag(001), i.e,
1.75$\times$1019 atoms/m2, under oxygen atmosphere. We have grown 30 MLE films
that appear $\sim$ 6.5$-$7 nm as estimated from the attenuation of Ag 3d core-
level intensity. To achieve a high-quality surface, the film was annealed at
673$-$773 K for 10 min in UHV. The sample temperature was measured by a K-type
thermocouple, in contact with the Ag crystal. The crystalline quality of the
film and the symmetry directions were determined by a four-grid LEED apparatus
while a highly-sensitive Peltier-cooled 12-bit CCD camera was used to collect
the LEED images. Photoemission spectroscopic measurements, XPS, and ARPES were
performed in an analysis chamber having a base pressure better than
8$\times$10-11 mbar. ARPES experiments were performed using a combination of
VG SCIENTA-R4000WAL electron energy analyzer with a 2D-CCD detector and a high
flux GAMMADATA VUV He lamp attached to a VUV monochromator, which has been
described in detail elsewhere Mahatha and Menon (2010). The ARPES measurements
were performed using monochromatized ultraviolet He Iα(21.2 eV) and He IIα
(40.8 eV) resonance lines, while the XPS measurements were performed with Al
Kα monochromatic X-ray source (1486.6 eV) from VG SCIENTA. At room temperature
(RT), the total experimental energy resolution was about 100 meV for the UPS
(using He Iα and He IIα photons) while for the monochromatic XPS the energy
resolution was set to about 600 meV.

##  IV acknowledgement

The Micro-Nano initiative program of the Department of Atomic Energy (DAE),
Government of India, is acknowledged for generous funding and support for this
work. During writing the paper, A.K.K receives funding from US Department of
Energy, Office of Basic Energy Sciences, contract no. DE-SC0012704. The
authors thank T. Valla, A. Samanta, and S. K. Mahatha for the helpful
discussions.

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#  Rationally-extended Dunkl oscillator on the line

C Quesne
Physique NucléaireThéorique et Physique Mathématique, Université Libre de
Bruxelles,
Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
E-mail<EMAIL_ADDRESS>
###### Abstract

It is shown that the extensions of exactly-solvable quantum mechanical
problems connected with the replacement of ordinary derivatives by Dunkl ones
and with that of classical orthogonal polynomials by exceptional orthogonal
ones can be easily combined. For such a purpose, the example of the Dunkl
oscillator on the line is considered and three different types of rationally-
extended Dunkl oscillators are constructed. The corresponding wavefunctions
are expressed in terms of exceptional orthogonal generalized Hermite
polynomials, defined in terms of the three different types of $X_{m}$-Laguerre
exceptional orthogonal polynomials. Furthermore, the extended Dunkl oscillator
Hamiltonians are shown to be expressible in terms of some extended Dunkl
derivatives and some anharmonic oscillator potentials.

Keywords: quantum mechanics, Dunkl derivative, exceptional orthogonal
polynomials

##  1 Introduction

Since several decades, deformations of quantum mechanics have been an
intensive subject of research. In particular, the use of the reflection
operator $R$, which is defined by $Rf(x)=f(-x)$ and dates back to Wigner [1]
and Yang [2], has found a lot of applications. More generally, Dunkl operators
[3], which are sets of differential-difference operators associated with
finite reflection groups, are very useful in several fields of mathematics and
physics.

In mathematics, they play, for instance, an important role in the analysis of
multivariate special functions and integral transforms associated with root
systems [4, 5, 6]. In physics, their usefulness for bosonizing supersymmetric
quantum mechanics [7, 8] and generalizations thereof [9], describing anyons
[10], and constructing Dunkl supersymmetric orthogonal functions [11, 12], has
been stressed. They are building blocks for describing an exchange operator
formalism in Calogero-Moser-Sutherland models and their generalizations [13,
14, 15, 16, 17]. They also provide a useful tool for proving the
superintegrability of some models [18, 19].

Several exactly-solvable quantum systems wherein the ordinary derivatives are
replaced by Dunkl ones have been extensively studied. One may quote, for
instance, the Dunkl oscillators in one, two, and three dimensions [20, 21, 22,
23, 24], the Dunkl-Coulomb problem in two and three dimensions [25, 26, 27],
as well as the one-dimensional infinite [28] and finite [29] square wells.
Coherent states [30], a generalization of shape invariance in supersymmetric
quantum mechanics [31], Dunkl derivatives with two and three parameters [32,
33], and some relativistic systems [34, 35] have also been investigated.

During the last few years, a lot of research activity has also been devoted to
the construction of new exactly-solvable rational extensions of well-known
quantum potentials. Such rational extensions arose after the introduction of
exceptional orthogonal polynomials (EOPs) [36], generalizing the classical
orthogonal polynomials (COPs) and forming orthogonal and complete polynomial
sets although they admit some gaps in the sequence of their degrees in
contrast with the latter. $X_{1}$ EOPs were indeed shown to be related to the
Darboux transformation in the context of shape invariant potentials in
supersymmetric quantum mechanics [37, 38]. Infinite families of shape
invariant potentials were then constructed in relation to $X_{m}$ EOPs [39]
and multi-indexed families of $X_{m_{1}m_{2}\ldots m_{k}}$ EOPs, connected
with multi-step Darboux transformations, were considered [40, 41].

The purpose of the present paper is to combine the extensions of quantum
mechanics connected with the use of Dunkl operators and that of EOPs. To show
that this can be done, we plan to consider here the simplest Dunkl model,
namely the one-dimensional Dunkl oscillator, and to employ its connection with
the radial oscillator in order to construct some rationally-extended models.
For such a purpose, we are going to use the three known infinite families of
$X_{m}$-Laguerre EOPs related to one-step Darboux transformations [38, 42, 43,
44].

This paper is organized as follows. In section 2, we review the treatment of
the Dunkl oscillator on the line and its relation to the radial oscillator
problem. In section 3, we summarize the results previously obtained for the
three types of radial oscillator rational extensions. We apply the latter in
section 4 to build rationally-extended Dunkl oscillators. In section 5, we
show that the Hamiltonians of the latter can be expressed in terms of extended
Dunkl operators. Finally, section 6 contains the conclusion.

##  2 The one-dimensional Dunkl oscillator and its relation to the radial
oscillator

The one-dimensional Dunkl oscillator model is defined by the Hamiltonian [20]

$H_{\mu}=\frac{1}{2}(-D_{\mu}^{2}+x^{2})=\frac{1}{2}\left(-\frac{d^{2}}{dx^{2}}-\frac{2\mu}{x}\frac{d}{dx}+\frac{\mu}{x^{2}}(1-R)+x^{2}\right),\qquad-\infty<x<+\infty,$
(2.1)

where $D_{\mu}$ is the Dunkl derivative

$D_{\mu}=\frac{d}{dx}+\frac{\mu}{x}(1-R),\qquad\mu>-\frac{1}{2},$ (2.2)

and $R$ is the reflection operator.

The corresponding Schrödinger equation reads

$H_{\mu}\psi^{(\mu)}_{2n+\epsilon}(x)=\left(2n+\epsilon+\mu+\frac{1}{2}\right)\psi^{(\mu)}_{2n+\epsilon}(x),\qquad
n=0,1,2,\ldots,\qquad\epsilon=0,1,$ (2.3)

where the wavefunctions $\psi^{(\mu)}_{2n+\epsilon}(x)$, which are even for
$\epsilon=0$ and odd for $\epsilon=1$, can be expressed as

$\psi^{(\mu)}_{2n+\epsilon}(x)=e^{-\frac{1}{2}x^{2}}H^{\mu}_{2n+\epsilon}(x)$
(2.4)

in terms of generalized Hermite polynomials [20]. The latter are defined by

$H^{\mu}_{2n+\epsilon}(x)=(-1)^{n}\left(\frac{n!}{\Gamma(n+\mu+\epsilon+1/2)}\right)^{1/2}x^{\epsilon}L^{(\mu-1/2+\epsilon)}_{n}(x^{2}),$
(2.5)

in terms of Laguerre polynomials [45]. The wavefunctions (2.4) satisfy the
orthonormality property

$\int_{-\infty}^{+\infty}\left[\psi^{(\mu)}_{2n^{\prime}+\epsilon^{\prime}}(x)\right]^{*}\psi^{(\mu)}_{2n+\epsilon}(x)|x|^{2\mu}dx=\delta_{n^{\prime},n}\delta_{\epsilon^{\prime},\epsilon}.$
(2.6)

These results for the Dunkl oscillator on the line can be directly obtained
from well-known ones for the radial oscillator (see, e.g., ref. [38]). The
corresponding Schrödinger equation reads

$\left(-\frac{d^{2}}{dx^{2}}+V_{l}(x)\right)\phi^{(l)}_{n}(x)=E^{(l)}_{n}\phi^{(l)}_{n}(x),\qquad
n=0,1,2,\ldots.$ (2.7)

Here

$V_{l}(x)=\frac{l(l+1)}{x^{2}}+\frac{1}{4}\omega^{2}x^{2},\qquad 0<x<+\infty,$
(2.8)

where $l$ denotes the angular momentum quantum number and $\omega$ the
frequency,

$E^{(l)}_{n}=\omega\left(2n+l+\frac{3}{2}\right),$ (2.9)

and

$\phi^{(l)}_{n}(x)=\left(\frac{\omega}{2}\right)^{\frac{1}{2}\left(l+\frac{3}{2}\right)}\left(\frac{2n!}{\Gamma(n+l+3/2)}\right)^{1/2}x^{l+1}L^{(l+1/2)}_{n}(z)e^{-\frac{1}{2}z},\qquad
z\equiv\frac{1}{2}\omega x^{2},$ (2.10)

is such that

$\int_{0}^{+\infty}\left(\phi^{(l)}_{n^{\prime}}(x)\right)^{*}\phi^{(l)}_{n}(x)dx=\delta_{n^{\prime},n}.$
(2.11)

On starting from the radial oscillator equation (2.7), performing the
replacements $\omega\to 2$, $l\to\mu-1$, and multiplying the wavefunction by
$x^{-\mu}$, we indeed arrive at the equation

$\frac{1}{2}\left(-\frac{d^{2}}{dx^{2}}-\frac{2\mu}{x}\frac{d}{dx}+x^{2}\right)x^{-\mu}\phi^{(\mu-1)}_{n}(x)=\left(2n+\mu+\frac{1}{2}\right)x^{-\mu}\phi^{(\mu-1)}_{n}(x),$
(2.12)

agreeing with (2.3) for $\epsilon=0$. Similarly, for $\omega\to 2$, $l\to\mu$,
we get

$\frac{1}{2}\left(-\frac{d^{2}}{dx^{2}}-\frac{2\mu}{x}\frac{d}{dx}+\frac{2\mu}{x^{2}}+x^{2}\right)x^{-\mu}\phi^{(\mu)}_{n}(x)=\left(2n+\mu+\frac{3}{2}\right)x^{-\mu}\phi^{(\mu)}_{n}(x),$
(2.13)

giving back (2.3) for $\epsilon=1$. In the transformation, equation (2.10)
leads to equations (2.4) and (2.5) after multiplying the former by an extra
$1/\sqrt{2}$ factor, which takes the change of normalization from the half-
line to the whole line into account (the phase factor in (2.5) being
optional).

##  3 Rational extensions of the radial oscillator

The rationally-extended radial oscillators resulting from one-step Darboux
transformations read [38, 42, 43]

$\displaystyle V_{l,{\rm ext}}(x)$ $\displaystyle=V_{l}(x)+V_{l,{\rm rat}}(x)$
$\displaystyle=V_{l^{\prime}}(x)-2\frac{d^{2}}{dx^{2}}\log\varphi_{l^{\prime}m}(x)-C,$
(3.1)

where according to the choice made for $l^{\prime}$ and
$\varphi_{l^{\prime}m}(x)$, three different types of extension are obtained:

$\displaystyle{\rm I:}\quad$ $\displaystyle
l^{\prime}=l-1,\quad\varphi_{l-1,m}(x)\propto x^{l}e^{\frac{1}{4}\omega
x^{2}}L^{(l-1/2)}_{m}(-\omega x^{2}/2),\quad C=-\omega;$ $\displaystyle{\rm
II:}\quad$ $\displaystyle l^{\prime}=l+1,\quad\varphi_{l+1,m}(x)\propto
x^{-l-1}e^{-\frac{1}{4}\omega x^{2}}L^{(-l-3/2)}_{m}(\omega x^{2}/2),\quad
C=\omega,$ $\displaystyle\quad m<l+3/2;$ $\displaystyle{\rm III:}\quad$
$\displaystyle l^{\prime}=l+1,\quad\varphi_{l+1,m}(x)\propto
x^{-l-1}e^{\frac{1}{4}\omega x^{2}}L^{(-l-3/2)}_{m}(-\omega x^{2}/2),\quad
C=-\omega,$ $\displaystyle\quad m<l+3/2,\quad\text{$m$ even}.$

The rational term in (3.1) can be written as

$V_{l,{\rm
rat}}(x)=-2\omega\left\\{\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}+2z\left[\frac{\ddot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}-\left(\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}\right)^{2}\right]\right\\},$
(3.3)

where $z=\frac{1}{2}\omega x^{2}$, $\alpha=l+\frac{1}{2}$, a dot denotes a
derivative with respect to $z$, and

$g^{(\alpha)}_{m}(z)=\begin{cases}L^{(\alpha-1)}_{m}(-z)&\text{for type
I},\\\\[5.69046pt] L^{(-\alpha-1)}_{m}(z),\quad m<\alpha+1&\text{for type
II},\\\\[5.69046pt] L^{(-\alpha-1)}_{m}(-z),\quad m<\alpha+1,\quad\text{$m$
even}&\text{for type III}.\end{cases}$ (3.4)

The Schrödinger equations obtained by replacing $V_{l}(x)$ by $V_{l,{\rm
ext}}(x)$ in (2.7) assume different forms.

For type I, one gets

$\left(-\frac{d^{2}}{dx^{2}}+V_{l,{\rm ext}}(x)\right)\phi^{({\rm
I},l)}_{n}(x)=E^{({\rm I},l)}_{n}\phi^{({\rm I},l)}_{n}(x),$ (3.5)

where

$E^{({\rm I},l)}_{n}=\omega\left(2n-2m+l+\frac{3}{2}\right)$ (3.6)

and

$\phi^{({\rm I},l)}_{n}(x)={\cal N}^{({\rm
I},l)}_{n}\frac{x^{l+1}e^{-\frac{1}{4}\omega
x^{2}}}{L^{(l-1/2)}_{m}(-\frac{1}{2}\omega x^{2})}L^{{\rm
I},l+1/2}_{m,n}\left(\tfrac{1}{2}\omega x^{2}\right).$ (3.7)

Here $n=m,m+1,m+2,\ldots$ and $L^{{\rm I},l+1/2}_{m,n}\left(\tfrac{1}{2}\omega
x^{2}\right)$ is an $n$th-degree $X_{m}$-Laguerre EOP of type I. Such EOPs are
known to form an orthogonal and complete polynomial set although 0, 1, …,
$m-1$ degrees are missing [44]. The wavefunctions $\phi^{({\rm I},l)}_{n}(x)$
satisfy the orthonormality properties

$\int_{0}^{\infty}dx\left(\phi^{({\rm
I},l)}_{n^{\prime}}(x)\right)^{*}\phi^{({\rm
I},l)}_{n}(x)=\delta_{n^{\prime},n},$ (3.8)

with the choice

${\cal N}^{({\rm
I},l)}_{n}=\left(\frac{\omega}{2}\right)^{\frac{1}{2}\left(l+\frac{3}{2}\right)}\left(\frac{2(n-m)!}{(n+l+\frac{1}{2})\Gamma(n+l+\frac{1}{2}-m)}\right)^{1/2}$
(3.9)

for the normalization factor.

For type II, one gets

$\left(-\frac{d^{2}}{dx^{2}}+V_{l,{\rm ext}}(x)\right)\phi^{({\rm
II},l)}_{n}(x)=E^{({\rm II},l)}_{n}\phi^{({\rm II},l)}_{n}(x),$ (3.10)

where

$E^{({\rm II},l)}_{n}=\omega\left(2n-2m+l+\frac{3}{2}\right)$ (3.11)

and

$\phi^{({\rm II},l)}_{n}(x)={\cal N}^{({\rm
II},l)}_{n}\frac{x^{l+1}e^{-\frac{1}{4}\omega
x^{2}}}{L^{(-l-3/2)}_{m}(\frac{1}{2}\omega x^{2})}L^{{\rm
II},l+1/2}_{m,n}\left(\tfrac{1}{2}\omega x^{2}\right),$ (3.12)

provided $l>m-\frac{3}{2}$. Here $n=m,m+1,m+2,\ldots$ and $L^{{\rm
II},l+1/2}_{m,n}\left(\tfrac{1}{2}\omega x^{2}\right)$ is an $n$th-degree
$X_{m}$-Laguerre EOP of type II. Such EOPs have rather similar properties as
those of type I with 0, 1, …, $m-1$ degrees missing [44]. The wavefunctions
$\phi^{({\rm II},l)}_{n}(x)$ satisfy orthonormality properties similar to
(3.8) provided

${\cal N}^{({\rm
II},l)}_{n}=\left(\frac{\omega}{2}\right)^{\frac{1}{2}\left(l+\frac{3}{2}\right)}\left(\frac{2(n-m)!}{(n+l+\frac{3}{2}-2m)\Gamma(n+l+\frac{5}{2}-m)}\right)^{1/2}.$
(3.13)

From (2.9), (3.6), and (3.11), we note that type I or II extended radial
oscillators have exactly the same spectrum as that of the starting radial
oscillator $V_{l}(x)$. This is not the case for type III extensions, for which
one gets

$\left(-\frac{d^{2}}{dx^{2}}+V_{l,{\rm ext}}(x)\right)\phi^{({\rm
III},l)}_{n}(x)=E^{({\rm III},l)}_{n}\phi^{({\rm III},l)}_{n}(x),$ (3.14)

where

$E^{({\rm III},l)}_{n}=\omega\left(2n-2m+l+\frac{3}{2}\right)$ (3.15)

and

$\phi^{({\rm III},l)}_{n}(x)={\cal N}^{({\rm
III},l)}_{n}\frac{x^{l+1}e^{-\frac{1}{4}\omega
x^{2}}}{L^{(-l-3/2)}_{m}(-\frac{1}{2}\omega x^{2})}L^{{\rm
III},l+1/2}_{m,n}\left(\tfrac{1}{2}\omega x^{2}\right),$ (3.16)

provided $l>m-\frac{3}{2}$ and $m$ is even. Here $n=0,m+1,m+2,\ldots$ and
$L^{{\rm III},l+1/2}_{m,n}\left(\tfrac{1}{2}\omega x^{2}\right)$ is an $n$th-
degree $X_{m}$-Laguerre EOP of type III. For such EOPs, 1, 2, … $m$ degrees
are indeed missing although they again form an orthogonal and complete
polynomial set [44]. Orthonormality properties similar to (3.8) are satisfied
with the choice

${\cal N}^{({\rm
III},l)}_{n}=\left(\frac{\omega}{2}\right)^{\frac{1}{2}\left(l+\frac{3}{2}\right)}\times\begin{cases}\left(\frac{2}{\Gamma(l+\frac{3}{2}-m)m!}\right)^{1/2}&\text{if
$n=0$},\\\\[5.69046pt]
\left(\frac{2(n-m-1)!}{n\Gamma(n+l+\frac{3}{2}-m)}\right)^{1/2}&\text{if
$n=m+1,m+2,\ldots$}.\end{cases}$ (3.17)

##  4 Rational extensions of the Dunkl oscillator

To get rational extensions of the Dunkl oscillator from those of the radial
oscillator, we are going to proceed as shown in section 2 for the standard
problem. In all cases, we replace $\omega$ by 2, so that $z=\frac{1}{2}\omega
x^{2}$ becomes $z=x^{2}$. In the $\epsilon=1$ case, which is obtained by
introducing an operator $\frac{1}{2}(1-R)$ in the Hamiltonian, $l$ is replaced
by $\mu$, so that the rational part of the potential is defined in terms of
$\alpha=\mu+\frac{1}{2}$, whereas in the $\epsilon=0$ case, obtained by
introducing an operator $\frac{1}{2}(1+R)$, $l$ is replaced by $\mu-1$, so
that $\alpha-1$ will appear instead of $\alpha$ in the rational part of the
potential. As a result, the three possible extensions of $H_{\mu}$ are given
by

$\displaystyle H_{\mu,{\rm ext}}$
$\displaystyle=\frac{1}{2}\Biggl{\\{}-D_{\mu}^{2}+x^{2}-2\Biggl{[}\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}+2z\Biggl{(}\frac{\ddot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}-\biggl{(}\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\biggr{)}^{2}\Biggr{)}\Biggr{]}(1+R)$
$\displaystyle\quad-2\Biggl{[}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}+2z\Biggl{(}\frac{\ddot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}-\biggl{(}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}\biggr{)}^{2}\Biggr{)}\Biggr{]}(1-R)\Biggr{\\}},$
(4.1)

where $g^{(\alpha)}_{m}(z)$ is defined as in (3.4) with
$\alpha=\mu+\frac{1}{2}$ and $z=x^{2}$.

The corresponding Schrödinger equations now read

$H_{\mu,{\rm
ext}}\hat{\psi}^{(\mu)}_{2n+\epsilon}(x)=\left(2n-2m+\epsilon+\mu+\frac{1}{2}\right)\hat{\psi}^{(\mu)}_{2n+\epsilon}(x),$
(4.2)

where $n=m,m+1,m+2,\ldots$ for type I or II, or $n=0,m+1,m+2,\ldots$ for type
III and in all cases $\epsilon=0,1$. The corresponding wavefunctions
$\hat{\psi}^{(\mu)}_{2n+\epsilon}(x)$ result from those of the extended radial
oscillator after multiplication by $x^{-\mu}$ and a change of normalization
from the half-line to the whole line. They may therefore be written as

$\hat{\psi}^{(\mu)}_{2n+\epsilon}(x)=e^{-\frac{1}{2}x^{2}}\hat{H}^{\mu}_{2n+\epsilon}(x),$
(4.3)

where $\hat{H}^{(\mu)}_{2n+\epsilon}(x)$ are exceptional orthogonal
generalized Hermite polynomials, which we define by

$\displaystyle\hat{H}^{(\mu)}_{2n+\epsilon}(x)$
$\displaystyle=\left(\frac{(n-m)!}{(n+\mu+\epsilon-\frac{1}{2})\Gamma(n-m+\mu+\epsilon-\frac{1}{2})}\right)^{1/2}\frac{x^{\epsilon}}{L^{(\mu+\epsilon-3/2)}_{m}(-x^{2})}$
$\displaystyle\quad\times L^{{\rm
I},\mu-\frac{1}{2}+\epsilon}_{m,n}(x^{2}),\qquad\text{if
$n=m,m+1,m+2,\ldots$},$ (4.4)

for type I,

$\displaystyle\hat{H}^{(\mu)}_{2n+\epsilon}(x)$
$\displaystyle=\left(\frac{(n-m)!}{(n-2m+\mu+\epsilon+\frac{1}{2})\Gamma(n-m+\mu+\epsilon+\frac{3}{2})}\right)^{1/2}\frac{x^{\epsilon}}{L^{(-\mu-\epsilon-1/2)}_{m}(x^{2})}$
$\displaystyle\quad\times L^{{\rm
II},\mu-\frac{1}{2}+\epsilon}_{m,n}(x^{2}),\qquad\text{if
$n=m,m+1,m+2,\ldots$},$ (4.5)

for type II, and

$\displaystyle\hat{H}^{(\mu)}_{2n+\epsilon}(x)$
$\displaystyle=\frac{x^{\epsilon}}{L^{(-\mu-\epsilon-1/2)}_{m}(-x^{2})}$
$\displaystyle\quad\times\begin{cases}\left(\frac{1}{\Gamma(\mu-m+\epsilon+\frac{1}{2})m!}\right)^{1/2}&\text{if
$n=0$},\\\
\left(\frac{(n-m-1)!}{n\Gamma(n-m+\mu+\epsilon+\frac{1}{2})}\right)^{1/2}L^{{\rm
III},\mu-\frac{1}{2}+\epsilon}_{m,n}(x^{2})&\\\ \qquad\text{if
$n=m+1,m+2,\ldots$}&\end{cases}$ (4.6)

for type III (with $m$ even). Note that the $\mu$ values have to be restricted
to $\mu>\frac{1}{2}$ for type I or $\mu>m-\frac{1}{2}$ for type II or III (in
order to get $\mu-\frac{1}{2}+\epsilon>0$ or $\mu-\frac{1}{2}+\epsilon>m-1$,
respectively).

##  5 Rationally-extended Dunkl oscillators in terms of extended Dunkl
derivatives

In the present section, we plan to show that the rationally-extended Dunkl
oscillator Hamiltonian (4.1) can be rewritten in a simpler way

$H_{\mu,{\rm ext}}=\frac{1}{2}\left(\hat{D}_{\mu}^{2}+x^{2}+G(x)\right)$ (5.1)

in terms of an extended Dunkl derivative

$\hat{D}_{\mu}=D_{\mu}+F(x)R=\frac{d}{dx}+\frac{\mu}{x}(1-R)+F(x)R,$ (5.2)

defined in terms of some odd function $F(x)$, while $G(x)$ in (5.1) is some
even function. The two functions $F(x)$ and $G(x)$ will of course be different
for the three types of extensions.

On inserting definition (5.2) in (5.1) and comparing the result with equation
(4.1), one finds that $F(x)$ must satisfy the first-order differential
equation

$F^{\prime}(x)=2\Biggl{\\{}\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}+2z\Biggl{[}\frac{\ddot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}-\Biggl{(}\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\Biggr{)}^{2}\Biggr{]}\Biggr{\\}}-2\Biggl{\\{}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}+2z\Biggl{[}\frac{\ddot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}-\Biggl{(}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}\Biggr{)}^{2}\Biggr{]}\Biggr{\\}},$
(5.3)

where a dash denotes a derivative with respect to $x$, while a dot stands for
a derivative with respect to $z=x^{2}$. In terms of such a function, $G(x)$
will then be given by

$\displaystyle G(x)$
$\displaystyle=-2\Biggl{\\{}\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}+2z\Biggl{[}\frac{\ddot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}-\Biggl{(}\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\Biggr{)}^{2}\Biggr{]}\Biggr{\\}}-2\Biggl{\\{}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}+2z\Biggl{[}\frac{\ddot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}-\Biggl{(}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}\Biggr{)}^{2}\Biggr{]}\Biggr{\\}}$
$\displaystyle\quad+\frac{2\mu}{x}F(x)-F^{2}(x).$ (5.4)

Solving equation (5.3) is straightforward and leads to

$F(x)=2x\left(\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}-\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}\right).$
(5.5)

On inserting (5.5) in (5.4), one gets

$G(x)=4(\alpha-1)\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}-4\alpha\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}-4z\left(\frac{\ddot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}+\frac{\ddot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}\right)+8z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}.$
(5.6)

To obtain more explicit results for $F(x)$ and $G(x)$, one has to use
definition (3.4) of $g^{(\alpha)}_{m}(z)$ in terms of Laguerre polynomials, as
well as some known properties of the latter, as summarized in the appendix.
The results read

$F(x)=\begin{cases}2x\left(\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha-1)}_{m}}-\frac{g^{(\alpha+1)}_{m-1}}{g^{(\alpha)}_{m}}\right)&\text{for
type I},\\\\[5.69046pt]
2x\left(-\frac{g^{(\alpha-2)}_{m-1}}{g^{(\alpha-1)}_{m}}+\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha)}_{m}}\right)&\text{for
type II},\\\\[5.69046pt]
2x\left(\frac{g^{(\alpha-2)}_{m-1}}{g^{(\alpha-1)}_{m}}-\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha)}_{m}}\right)&\text{for
type III},\end{cases}.$ (5.7)

and

$G(x)=\begin{cases}-4(\alpha-2+m)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}}+4(\alpha-1+m)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha)}_{m}}&\text{for
type I},\\\\[5.69046pt]
4(\alpha-m)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}}-4(\alpha+1-m)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha)}_{m}}&\text{for
type II},\\\\[5.69046pt]
-4(\alpha-m)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}}+4(\alpha+1-m)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha)}_{m}}&\text{for
type III}.\end{cases}$ (5.8)

For $m=1$, for instance, one gets as usual the same results for types I and
II, namely

$\displaystyle F(x)$
$\displaystyle=\frac{4x}{2x^{2}+2\mu-1}-\frac{4x}{2x^{2}+2\mu+1},$
$\displaystyle G(x)$
$\displaystyle=-\frac{4(2\mu-1)}{2x^{2}+2\mu-1}+\frac{4(2\mu+1)}{2x^{2}+2\mu+1},$
(5.9)

with $\mu>\frac{1}{2}$. For $m=2$, the results read

$\displaystyle F(x)$
$\displaystyle=\frac{8x(2x^{2}+2\mu+1)}{(2x^{2}+2\mu+1)^{2}-2(2\mu+1)}-\frac{8x(2x^{2}+2\mu+3)}{(2x^{2}+2\mu+3)^{2}-2(2\mu+3)},$
$\displaystyle G(x)$
$\displaystyle=-\frac{8(2\mu+1)(2x^{2}+2\mu-1)}{(2x^{2}+2\mu+1)^{2}-2(2\mu+1)}+\frac{8(2\mu+3)(2x^{2}+2\mu+1)}{(2x^{2}+2\mu+3)^{2}-2(2\mu+3)},$
(5.10)

with $\mu>\frac{1}{2}$ for type I,

$\displaystyle F(x)$
$\displaystyle=\frac{8x(2x^{2}+2\mu-3)}{(2x^{2}+2\mu-3)^{2}+2(2\mu-3)}-\frac{8x(2x^{2}+2\mu-1)}{(2x^{2}+2\mu-1)^{2}+2(2\mu-1)},$
$\displaystyle G(x)$
$\displaystyle=-\frac{8(2\mu-3)(2x^{2}+2\mu-1)}{(2x^{2}+2\mu-3)^{2}+2(2\mu-3)}+\frac{8(2\mu-1)(2x^{2}+2\mu+1)}{(2x^{2}+2\mu-1)^{2}+2(2\mu-1)},$
(5.11)

with $\mu>\frac{3}{2}$ for type II, and

$\displaystyle F(x)$
$\displaystyle=\frac{8x(2x^{2}-2\mu+3)}{(2x^{2}-2\mu+3)^{2}+2(2\mu-3)}-\frac{8x(2x^{2}-2\mu+1)}{(2x^{2}-2\mu+1)^{2}+2(2\mu-1)},$
$\displaystyle G(x)$
$\displaystyle=-\frac{8(2\mu-3)(2x^{2}-2\mu+1)}{(2x^{2}-2\mu+3)^{2}+2(2\mu-3)}+\frac{8(2\mu-1)(2x^{2}-2\mu-1)}{(2x^{2}-2\mu+1)^{2}+2(2\mu-1)},$
(5.12)

with $\mu>\frac{3}{2}$ for type III.

It is worth observing that in the transformation, the harmonic oscillator
potential $\frac{1}{2}x^{2}$ becomes an anharmonic one
$\frac{1}{2}[x^{2}+G(x)]$, which still vanishes at $x=0$ and goes to $+\infty$
at $x\to\pm\infty$, because $G(0)=G(\pm\infty)=0$.

##  6 Conclusion

In the present work, we have shown that it is possible to combine the
extensions of exactly-solvable quantum mechanical problems connected with the
replacement of the ordinary derivative by a Dunkl one and with that of COPs by
EOPs. For such a purpose, we have explicitly constructed three different types
of rationally-extended Dunkl oscillators on the line, whose wavefunctions can
be expressed in terms of exceptional orthogonal generalized Hermite
polynomials, defined in terms of the three different types of $X_{m}$-Laguerre
EOPs.

It is rather obvious that the method described here could be used to generate
rationally-extended Dunkl oscillators associated to multi-indexed
$X_{m_{1}m_{2}\ldots m_{k}}$-Laguerre EOPs, as well as rationally-extended
Dunkl oscillators in more than one dimension.

Considering similar constructions for some other quantum potentials and a
possible use of supersymmetric techniques would also be interesting problems
for future study.

## Acknowledgment

The author was supported by the Fonds de la Recherche Scientifique - FNRS
under Grant Number 4.45.10.08.

## Data availability statement

No new data were created or analysed in this study.

## Appendix. Proof of equations (5.7) and (5.8)

In the proof of equations (5.7) and (5.8), we shall use the following well-
known properties of Laguerre polynomials

$\displaystyle\left(z\frac{d^{2}}{dz^{2}}+(\alpha+1-z)\frac{d}{dz}+n\right)L^{(\alpha)}_{n}(z)=0,$
(A.1) $\displaystyle\frac{d}{dz}L^{(\alpha)}_{n}(z)=-L^{(\alpha+1)}_{n-1}(z),$
(A.2) $\displaystyle
z\frac{d}{dz}L^{(\alpha)}_{n}(z)=nL^{(\alpha)}_{n}(z)-(n+\alpha)L^{(\alpha)}_{n-1}(z),$
(A.3) $\displaystyle
L^{(\alpha-1)}_{n}(z)=L^{(\alpha)}_{n}(z)-L^{(\alpha)}_{n-1}(z).$ (A.4)

Equation (5.7) directly results from equations (5.5) and (A.2) for the
specific forms of $g^{(\alpha)}_{m}(z)$ given in (3.4).

To prove equation (5.8) by starting from equation (5.6), one first eliminates
the second-order derivatives with the help of equation (A.1), thereby leading
to

$G(x)=\begin{cases}4(2\alpha-2+z)\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}+4z\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}-8m+8z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}&\text{for
type I},\\\\[5.69046pt]
-4z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}-4(2\alpha+z)\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}+8m+8z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}&\text{for
type II},\\\\[5.69046pt]
4z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}-4(2\alpha-z)\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}-8m+8z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}&\text{for
type III}.\end{cases}$ (A.5)

On applying equations (A.2), (A.3), and (A.4), one gets

$\displaystyle
4(2\alpha-2+z)\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}=4(2\alpha-2)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha-1)}_{m}}+4m-4(m+\alpha-2)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}},$
$\displaystyle
4z\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}=4m-4(m+\alpha-1)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha)}_{m}},$
(A.6) $\displaystyle
8z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}=-8(\alpha-1)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha-1)}_{m}}+8(\alpha-1+m)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha)}_{m}},$

in the type I case,

$\displaystyle-4z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}=-4m+4(m-\alpha)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}},$
$\displaystyle-4(2\alpha+z)\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}=8\alpha\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha)}_{m}}-4m+4(m-\alpha-1)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha)}_{m}},$
(A.7) $\displaystyle
8z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}=-8(m-\alpha)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}}-8\alpha\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha)}_{m}},$

in the type II case, and

$\displaystyle
4z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}=4m-4(m-\alpha)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}},$
$\displaystyle-4(2\alpha-z)\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}=-8\alpha\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha)}_{m}}+4m-4(m-\alpha-1)\frac{g^{(\alpha)}_{m-1}}{g^{(\alpha)}_{m}},$
(A.8) $\displaystyle
8z\frac{\dot{g}^{(\alpha-1)}_{m}}{g^{(\alpha-1)}_{m}}\frac{\dot{g}^{(\alpha)}_{m}}{g^{(\alpha)}_{m}}=8(m-\alpha)\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha-1)}_{m}}+8\alpha\frac{g^{(\alpha-1)}_{m-1}}{g^{(\alpha)}_{m}},$

in the type III case. It only remains to sum all the terms to get the final
result (5.8).

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