VandenBerg 2018). They have also been used to estimate the solar bolometric
correction in Gaia’s photometric system. Below we make a new attempt to
determine precise and accurate solar colours.

We use the sample of solar-analogue candidates selected from GSP-Spec from
Sect. 7 in order to estimate the colours of the Sun. As we demonstrated in
Sect. 7.4, these stars have reliable extinction estimates from GSP-Phot.
Consequently, the BP/RP spectra with very low extinction (according to GSP-
Phot) can be used to indirectly estimate the intrinsic continuum shape of the
Sun. Among GSP-Spec’s solar-analogue candidates, there are 682 with GSP-Phot’s
$A_{0}<0.001$ mag. Given these, we obtain an absolute magnitude of

$M_{G,\odot}=(4.614\pm 0.179)\,\textrm{mag}$ (5)

For this, we adopt the inverse parallax as a distance estimator because our
candidate selection requires very high parallax quality
($\frac{\varpi}{\sigma_{\varpi}}>20$). For comparison, a value of
$M_{G,\odot}=4.66$ is adopted for FLAME (Creevey et al. 2022, Sect. 4.3
therein). Given the 682 candidates with $A_{0}<0.001$ mag, we also obtain mean
colours and standard deviations of

$\displaystyle(G_{\mathrm{BP}}-G_{\mathrm{RP}})_{\odot}=(0.818\pm
0.029)\,\textrm{mag}$ (6) $\displaystyle(G_{\mathrm{BP}}-G)_{\odot}=(0.324\pm
0.016)\,\textrm{mag}$ (7) $\displaystyle(G-G_{\mathrm{RP}})_{\odot}=(0.494\pm
0.020)\,\textrm{mag}$ (8) $\displaystyle(G-J)_{\odot}=(0.969\pm
0.578)\,\textrm{mag}$ (9) $\displaystyle(G-H)_{\odot}=(1.292\pm
0.401)\,\textrm{mag}$ (10) $\displaystyle(G-K_{s})_{\odot}=(1.371\pm
0.351)\,\textrm{mag}$ (11) $\displaystyle(G-W_{1})_{\odot}=(1.449\pm
0.066)\,\textrm{mag}$ (12) $\displaystyle(G-W_{2})_{\odot}=(1.405\pm
0.065)\,\textrm{mag}$ (13)

where we restrict the AllWISE comparison to cases with $W_{1}>8$ mag in order
to avoid saturation. These colours are in excellent agreement with the values
$(G_{\mathrm{BP}}-G_{\mathrm{RP}})_{\odot}=0.82$ mag,
$(G_{\mathrm{BP}}-G)_{\odot}=0.33$ mag and $(G-G_{\mathrm{RP}})_{\odot}=0.49$
mag obtained in Casagrande & VandenBerg (2018) from Gaia DR2 passbands and
synthetic as well as observed spectra for the Sun. Their absolute magnitude of
$M_{G,\odot}=4.67$ mag is also consistent with our estimate. In order to do
this comparison with Gaia DR3 passbands and also include near-infrared
photometry, we take the Kurucz model sun_mod_001.fits from the CALSPEC
library212121https://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/calspec (Bohlin et al. 1995, 2014,
2020) and simulate its photometry using the pyphot
package222222http://mfouesneau.github.io/docs/pyphot. We obtain synthetic
colours of $(G_{\mathrm{BP}}-G_{\mathrm{RP}})_{\odot}=0.813$ mag,
$(G_{\mathrm{BP}}-G)_{\odot}=0.324$ mag and
$(G-G_{\mathrm{RP}})_{\odot}=0.490$ mag, which are again in excellent
agreement with our estimated colours. For color combinations with 2MASS, we
obtain $(G-J)_{\odot}=0.992$ mag, $(G-H)_{\odot}=1.320$ mag, and
$(G-Ks)_{\odot}=1.360$ mag, which are again in excellent agreement with our
candidates. Concerning AllWISE (Cutri et al. 2021), we obtain
$(G-W_{1})_{\odot}=1.380$ mag and $(G-W_{2})_{\odot}=1.301$ mag. These values
are slightly bluer than the values we estimate from the GSP-Spec candidates,
but are still within $1\sigma$ and $1.6\sigma$, respectively.

###  10.4 Ages of UCDs not seen by Gaia

Table 9:  Identification and parameters of UCDs without Gaia solutions in
binary systems with full-solution companions. The mass and ages indicate the
median, lower and upper confidence intervals, while the parallax shows the
value and uncertainty.

UCD | Parallax | SpT | Mass | Companion | SpT | Age | Lit. age
---|---|---|---|---|---|---|---
Code, Name | [mas] |  | [$M_{\jupiter}$] | Source_ID |  | [Ga] | [Ga]
A, 2MASSI J0025036+475919 | 22.8 $\pm$ 0.92 | L4:+L4:2 | $84.4_{83.7}^{87.4}$ | 392562179817077120 | F8 | $5.1_{4.2}^{5.9}$ | $4.4_{1.6}^{5.3}$
B, 2MASS J02233667+5240066 | 12.2 $\pm$ 1.11 | L1.53 | $80.6_{80.3}^{80.7}$ | 452046549154458880 | F5 | $2.9_{2.5}^{3.4}$ | $2.0_{1.7}^{2.8}$
C, 2MASS J06462756+7935045 | 53.6 $\pm$ 2.21 | L94 | $59.4_{51.9}^{64.4}$ | 1141280704422528128 | F7V | $2.2_{1.6}^{2.9}$ | $4.3_{2.5}^{6.0}$
D, HD 49197B | 29.9 $\pm$ 2.11 | L45 | $79.3_{78.9}^{79.6}$ | 952346742338146176 | F5 | $4.0_{3.1}^{4.9}$ | $2.8_{0.3}^{4.7}$
E, 2MASS J12173646+1427119 | 16.1 $\pm$ 0.71 | L13 | $82.3_{82.3}^{84.5}$ | 3921176983720146560 | F8 | $4.9_{4.2}^{5.7}{\dagger}$ | $1.8_{0.5}^{3.8}$
F, HD 118865B | 21.7 $\pm$ 0.91 | T56 | $67.5_{65.2}^{69.1}$ | 3663438298389132416 | F7V | $4.6_{3.9}^{5.4}{\dagger}$ | $3.2_{2.5}^{3.5}$
G, 2MASS J14165987+5006258 | 22.1 $\pm$ 0.87 | L5.58 | $75.6_{75.3}^{75.8}$ | 1508557582834745088 | G5 | $6.3_{5.3}^{7.3}$ | $5.8_{4.2}^{9.0}$
H, ULAS J142320.79+011638.2 | 29.4 $\pm$ 1.01 | T8p9 | $34.9_{30.3}^{38.4}$ | 3654496279558010624 | G1.5V | $5.6_{4.0}^{7.2}$ | $5.9_{4.6}^{11.2}$
I, Gl 564 C | 74.7 $\pm$ 21.31 | L4:10 | $77.4_{77.1}^{77.6}$ | 1265976524286377856 | F9IV-V | $5.3_{4.4}^{6.2}{\dagger}$ | $0.9_{0.6}^{4.7}$
J, Gl779B 244691 | 47.1 $\pm$ 6.11 | L4.511 | $71.0_{39.9}^{73.3}$ | 1821708351374312064 | G0V | $2.8_{0.5}^{3.8}$ | $3.2_{1.7}^{5.2}$
K, eps Indi C | 275.3 $\pm$ 3.012 | T613 | $36.5_{11.0}^{50.2}$ | 6412595290592307840 | K5V | $2.0_{0.2}^{4.3}$ | $4.6_{0.9}^{5.7}$

Notes: 1: Spectrophotometric distance, 2: Cruz et al. (2007), 3: Deacon et al.
(2014), 4: Loutrel et al. (2011), 5: Metchev & Hillenbrand (2004), 6:
Burningham et al. (2013), 7: Faherty et al. (2012), 8: Chiu et al. (2006), 9:
Pinfield et al. (2012), 10: Goto et al. (2002), 11: Liu et al. (2002), 12:
Weinberger et al. (2016), 13: Burgasser et al. (2006). The $\dagger$ indicates
FLAME ages derived using the GSP-Phot-$T{\mathrm{{}_{eff}}}$, while other ages
are derived using the GSP-Spec-$T{\mathrm{{}_{eff}}}$.

Another application of the Gaia astrophysical parameters is to constrain the
characteristics of faint UCDs that are beyond the mission magnitude limit but
in binary systems with brighter objects that are observed by Gaia. Once we
have identified a multiple system we assume that the UCD has the same chemical
composition, age, distance and, after allowing for orbital motion, proper
motions. In addition, if the movement due to the orbital motion is detected by
Gaia this will provide a constraint on the mass of the various components.
Brown dwarfs evolve and cool over time and their observational properties are
degenerate with age, mass and metallicity; binary systems are therefore
benchmarks for understanding these processes. Gaia will provide a large
homogeneous multi-parametric sample with intersecting constraints that will
tie down the UCD regime. For this illustrative discussion we concentrate on
the age parameter232323The BaSTI models (Hidalgo et al. 2018) were used to
derive the age and they span from the ZAMS to the tip of the red giant branch
for stellar masses between 0.5 and 10$\,{{M}}_{\odot}$.. We note however, that
more precise ages can be obtained by combining Gaia with other observational
data such as asteroseismology.

To identify a potential list of objects with a high probability to be in a
binary system we used the positional and kinematical criteria given by Eq. 2
in Smart et al. (2019) and the list of known UCDs from that study. When the
faint UCD did not have a measured parallax we used its spectro-photometric
distance. We found 8 UCDs without Gaia DR3 5 parameter solutions that are in
binary systems in the FGKM sample, while also in the regime of reliable ages
(see Fouesneau et al. 2022). We added a further three interesting targets with
reliable ages here because they were rejected from the FGKM sample for failing
on only one of the criteria: A ipd_frac_multi_peak = 22; and B and F
classprob_dsc_combmod_binary $>0.99$.

These 11 UCDs are listed in Table 9 with name, adopted parallax, spectral type
and mass along with the companion Gaia source_id, age and the median published
ages with 16% and 84% percentiles.

The number of literature age estimates vary from 6 to 46 for each target and
are from varied sources: model comparisons (Holmberg et al. 2009; Casagrande
et al. 2011), chromospheric activity (Pace 2013; Metchev & Hillenbrand 2004),
or Galactic kinematics (Gontcharov 2012). The published age percentiles often
indicate uncertainties of a factor of 2 or a large portion of the age of the
Galaxy indicating the current difficulty in determining ages for stars. In
Figure 36 we show the Gaia vs the median published values from Table 9. When
available, we used the values based on the GSP-Spec $T{\mathrm{{}_{eff}}}$:
age_flame_spec these are denoted by the filled circles. The open circles are
age_flame which are based on GSP-Phot $T{\mathrm{{}_{eff}}}$. For most of the
stars we find general agreement with the literature, with the worst agreements
for systems E, I and K. For E, the $T{\mathrm{{}_{eff}}}$ from both GSP-Phot
and GSP-Spec agree to within 25 K and we would therefore trust its age if the
star is within the regime of models that were used. For I and K we find
significant disagreements between the GSP-Phot and GSP-Spec
$T{\mathrm{{}_{eff}}}$, and this could indicate a possible issue with
age_flame. We discuss each of the systems individually in the next section.

The interpolated masses of the UCDs are estimated from a comparison to the
illustrated tracks in Figure 37 taken from Baraffe et al. (2015) for stars and
Phillips et al. (2020) for brown dwarfs assuming the age of the companion star
from this work, and these are reported in Table 9.

Figure 36:  Literature ages versus FLAME ages for companions of UCDs not seen
by Gaia. Filled circles correspond to age_flame_spec (i.e. using the
$T{\mathrm{{}_{eff}}}$ from GSP-Spec) and open circles are age_flame. The
error bars represent the 16% and 84% percentiles. Figure 37:  Evolutionary
tracks and UCD locations in the H-band absolute magnitude versus age diagram,
adopting the companion age. The tracks are colour-coded by mass. The dashed
lines indicate the stellar to substellar transition zone (from 0.072 to 0.075
M⊙).

####  10.4.1 Notes on individual systems

2MASSI J0025036+475919 (A) is an L4+L4 binary in a multiple system with the
spectroscopic binary HD 2057 (Reid et al. 2006, and references therein) and
another component 11″ from the primary (Gaia EDR3 392562179817297536). Lithium
absorption has been detected in the combined spectra of the secondary
indicating it has an age less than 1.0 Ga (Cruz et al. 2007; Faherty et al.
2010; Filippazzo et al. 2015) which is much lower than the primary age
indicated here. This is one of the widest binary systems ($\sim$10 000 AU)
with an ultracool component but in the range of other systems of similar total
mass. The difference in age estimates of the primary and secondary is not
easily reconcilable. One possible solution is that it is not a binary system;
the agreement of high proper motions is a strong constraint, however, the
spectroscopic distance is very uncertain as the binary nature of the secondary
requires an assumption of the component flux contributions. Another
possibility is that the primary age estimate is high due to its binary nature.

2MASS J02233667+5240066 (B) was first noted to be in a common proper motion
system with HIP 11161 in Deacon et al. (2014). The primary has been shown to
have acceleration terms (Kervella et al. 2022; Brandt 2021) but the separation
with the UCD is large (41″) and the primary has now been resolved by Gaia into
two components and it is listed as a spectroscopic binary in the non-single
stars orbital solution results. It also has a very high
classprob_dsc_combmod_binary (¿0.99). The observed acceleration is therefore
due to the primary binarity and not the UCD. Using age_flame_spec we find a
mass of $80\,M_{\jupiter}$ which defines the end of the stellar main sequence.

2MASS J06462756+7935045 (C) was indicated as being in a binary system with HD
46588 based on a high common proper motion (Loutrel et al. 2011). It is an L9
brown dwarf, one of the few known at the L/T transition in wide binary
systems. These allow constraints on their astrophysical properties. The
age_flame_spec is lower by 1$\sigma$ than the primary literature age. Since
this is one of the few L9s where an independent age is known it is important
to clarify this discrepancy. Assuming the literature age and distance from the
primary, Loutrel et al. find a $T_{\rm eff}=1360_{1280}^{1410}\,K$, which is
an important constraint for the temperature at the the L/T boundary. If we
assume the lower age_flame_spec this will increase the temperature estimate at
this boundary.

HD 49197 B (D) has been studied extensively since its first discovery by
Metchev & Hillenbrand (2004) using high resolution observing techniques. It is
at a separation of 0.95″ from the primary. There are ongoing adaptive optics
projects to try to determine a binary solution (Bowler et al. 2020; Tokovinin
2014). With a magnitude difference of greater than 10 Gaia will not be able to
resolve the system. If we adopt the low end of the literature age range, HD
49197 B is a brown dwarf, if we adopt the high end - for example that
indicated by age_flame_spec \- the object becomes a star. Since there is also
a possibility of finding the mass of this companion either through high
resolution imaging or the detection of acceleration terms in the Gaia primary
solution (proper motion anomalies between the Hipparcos and Gaia results have
already been detected in Kervella et al. 2022), knowing its age will be
crucial for constraining the stellar-substellar boundary.

2MASS J12173646+1427119 (E) was first discovered in the Pan-STARRS survey as a
companion to HIP 59933 at 40″. The secondary is detected by Gaia (EDR3
3921177219942653696) but with only a two-parameter solution. The primary, EDR3
3921176983720146560, has a non-single star solution which indicates a
companion of $0.09{M}_{\odot}$ with a period of 1 yr and a corresponding
separation of 1 AU; given the small separation there must be a third component
in the system. Any age above 0.5 Ga would indicate that this is a stellar
object but very close to the stellar-substellar boundary as indicated by our
82 $M_{\jupiter}$.

HD 118865B (F) is a T5 in a system with an F4 spectral type first noted in
Burningham et al. (2013) where they find an age range of 1.5–4.9 Ga and mass
of 45–60 $M_{\jupiter}$. We find a primary age that is at the top end of their
range and hence a slightly larger mass. If confirmed it will provide a high
mass for this T5 compared to other similar type brown dwarfs.

2MASS J14165987+5006258 (G) is noted as a binary system in Faherty et al.
(2010) with a very large separation of $\sim$26 000 AU. The primary
age_flame_spec estimate is consistent with published values and also with blue
near-IR colors observed in Faherty et al. (2010) where they also re-evaluate
its spectral type from L5.5 to L4. The estimated mass indicates that this
object is of stellar and not sub-stellar type and further characterisation
will contribute to our understanding of very old borderline stellar objects.

ULAS J142320.79+011638.2 (H) is the coolest object in this sample in a system
with an early-G dwarf, HIP 70319. There are a significant number of age
estimates from very young to very old and the age_flame_spec is in agreement
with the median. This age is consistent with a low metallicity primary and
also with a broader Y-band peak and more depressed K-band peak than other T8s
(Kirkpatrick et al. 2021). This is an important benchmark for metal poor T
dwarfs.

Gl 564 B/C (I) is an L4+L4 binary in a triple system with Gl 564, a G2 V star.
The majority of the published age ranges are very young because Gl 564 is
chromospherically active with a high lithium abundance and fast rotation
(Potter et al. 2002). The space motion also puts the object in the Ursa Major
moving group from the Banyan $\Sigma$ tool (Gagné et al. 2018), which has an
age of around 500 Myr (King et al. 2003). This is in contrast to the high
age_flame which is difficult to reconcile given the high lithium abundance and
space motion. A possible explanation for this discrepancy is in the
limitations of the models that were used, for example, they do not include
rotation. If the system is in the first 0.5 Ga they will be contracting brown
dwarfs. The orbital period of the UCD binary system is around 10 yr (Potter et
al. 2002) and we will therefore soon have dynamical masses with Gaia. These
objects will provide a well-constrained calibration point for the theoretical
models describing low-mass, ultracool objects.

Gl779B (J) is an L4.5 UCD at 0.7″ from GJ 779, a G0 star. High levels of
chromospheric activity suggest a young age, lithium abundance indicates a
slightly older than the Hyades age but kinematics indicate an old disk star.
The age_flame_spec is consistent with the published estimates. The orbit is
such that it should be visible in the future Gaia observations which will lead
to a dynamical mass estimate (Crepp et al. 2014). A comparison of the
accelerations found from comparisons of Hipparcos and Gaia DR2 results
indicate a mass of around $0.07{M}_{\odot}$ (Brandt et al. 2019) and this is
therefore on the stellar-substellar boundary currently defining the end of the
main sequence, and in agreement with our estimated mass.

Eps Ind C (K) is the second closest brown dwarf binary T1+T6 system in a
triple system with the K5V star eps Ind. One of the brown dwarfs has a Gaia
solution (Gaia EDR3 6412596012146801152) which we assume is the T1. Later
releases should provide a dynamical solution for the component masses. There
is a significant history of publications for both the primary and the
secondary system. With a period of around 11 yr and an observed separation
that varies from 0.6 to 2″, it is a defining system for parameters of early T
dwarfs. The age_flame_spec is at the low end of the published age range for
the primary, and the masses of the secondaries from Dieterich et al. (2018)
also imply an inconsistency with such a young age. A dynamical mass
determination from the Gaia observations should resolve this inconsistency.

We have seen that the results of Gaia can be brought to bear on our
understanding of objects fainter than its magnitude limit. Indeed there will
probably be less than 1000 brown dwarfs brighter than the Gaia magnitude limit
(Smart et al. 2019) while we expect there to be tens of thousands in binary
systems or detected from astrometric and radial velocity perturbations.
Therefore, the contribution of Gaia to brown dwarf studies will be
predominantly due to indirectly detected objects rather than direct
detections.

##  11 Conclusion

In this work we defined homogeneous samples of high quality astrophysical
parameters by exploiting many Gaia data products that appear in Gaia DR3,
while focusing on the sources and data products in the
astrophysical_parameters and the astrophysical_parameters_supp tables which
were produced by the Apsis software (Creevey et al. 2022; Delchambre et al.
2022; Fouesneau et al. 2022). We considered different regimes of stars all
across the HR diagram. In the first part of this work we considered large
samples of young massive disk OBA stars (Sect. 3), FGKM spectral type stars
(Sect. 4), and faint ultra-cool dwarfs (UCDs, Sect. 5). Then we focussed on
smaller samples of specific object types; carbon stars (Sect. 6), solar
analogues (Sect. 7), and the Spectro Photometric Standard stars (SPSS, Pancino
et al. 2021a, Sect. 8). Concerning the latter, this paper provides the first
homogeneous determination of the SPSS dataset to date. We validated each of
the samples using the Gaia data itself and external catalogues, and our
results are published in six tables that will appear alongside Gaia DR3, see
Sect. 9 and Table 7.

In Sect. 10, we demonstrated some use cases of these samples of stars. We used
a subset of the OBA sample to illustrate its usefulness to analyse the Milky
Way rotation curve (Sect. 10.1). We then used the properties of the FGKM stars
to analyse known exoplanet systems including the determination of planet radii
and masses (Sect. 10.2). We then predicted the colours of the Sun in various
passbands using the solar analogue sample (Sect. 10.3). Finally, we analysed
the ages of some unseen UCD-companions to the FGKM stars (Sect. 10.4).

The aim of this work was to highlight the science that can be done with Gaia
DR3. We focused on specific types of stars using strict quality criteria on
many of the data products, which sometimes included some ad hoc filtering
criteria tuned with a particular science case in mind. We emphasize that our
strict personal selections may not be applicable to a user’s specific science
case, and the user should acknowledge this before exploiting these samples. We
fully encourage all users to exploit all of the astrophysical parameters in
Gaia DR3 independent of our specific selection criteria highlighted in this
work. Indeed there are up to 470 million stars with stellar parameters derived
using the mean BP and RP spectra, up to 6 million stellar parameters and
abundances derived from the mean RVS spectra, along with up to 130 million
masses and ages, and many other new stellar products that were not the focus
of this work, such as DIB estimates, activity index of active stars, and
H$\alpha$ emission. As illustrated in this work, many science cases can be
explored with these data.

###### Acknowledgements.

We thank the referee for their constructive comments on the manuscript. This
work has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement. The full list of funding
agencies and grants is listed after the references.
This research has used NASA’s Astrophysics Data System, and the VizieR
catalogue access tool (CDS, Strasbourg, France).
The data processing and analysis made use of matplotlib (Hunter 2007), NumPy
(Harris et al. 2020), the IPython package (Pérez & Granger 2007), Vaex
(Breddels & Veljanoski 2018), TOPCAT (Taylor 2005), pyphot
(http://github.com/mfouesneau/pyphot), R (R Core Team 2013), Astropy (Astropy
Collaboration et al. 2013, 2018), CmdStanPy (https://github.com/stan-
dev/cmdstanpy), and ArviZ (Kumar et al. 2019).
In case of errors or omissions, please contact the Gaia Helpdesk. The full
list of acknowledgements can also be found in the official online
documentation for Gaia DR3.

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##  Gaia funding institutions

This work presents results from the European Space Agency (ESA) space mission
Gaia. Gaia data are processed by the Gaia Data Processing and Analysis
Consortium (DPAC). Funding for the DPAC is provided by national institutions,
in particular the institutions participating in the Gaia MultiLateral
Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia.
The Gaia archive website is https://archives.esac.esa.int/gaia.

The Gaia mission and data processing have financially been supported by, in
alphabetical order by country:
– the Algerian Centre de Recherche en Astronomie, Astrophysique et Géophysique
of Bouzareah Observatory;
– the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (FWF)
Hertha Firnberg Programme through grants T359, P20046, and P23737;
– the BELgian federal Science Policy Office (BELSPO) through various PROgramme
de Développement d’Expériences scientifiques (PRODEX) grants, the Research
Foundation Flanders (Fonds Wetenschappelijk Onderzoek) through grant
VS.091.16N, the Fonds de la Recherche Scientifique (FNRS), and the Research
Council of Katholieke Universiteit (KU) Leuven through grant C16/18/005
(Pushing AsteRoseismology to the next level with TESS, GaiA, and the Sloan
DIgital Sky SurvEy – PARADISE);
– the Brazil-France exchange programmes Fundação de Amparo à Pesquisa do
Estado de São Paulo (FAPESP) and Coordenação de Aperfeicoamento de Pessoal de
Nível Superior (CAPES) - Comité Français d’Evaluation de la Coopération
Universitaire et Scientifique avec le Brésil (COFECUB);
– the Chilean Agencia Nacional de Investigación y Desarrollo (ANID) through
Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Regular
Project 1210992 (L. Chemin);
– the National Natural Science Foundation of China (NSFC) through grants
11573054, 11703065, and 12173069, the China Scholarship Council through grant
201806040200, and the Natural Science Foundation of Shanghai through grant
21ZR1474100;
– the Tenure Track Pilot Programme of the Croatian Science Foundation and the
École Polytechnique Fédérale de Lausanne and the project TTP-2018-07-1171
‘Mining the Variable Sky’, with the funds of the Croatian-Swiss Research
Programme;
– the Czech-Republic Ministry of Education, Youth, and Sports through grant LG
15010 and INTER-EXCELLENCE grant LTAUSA18093, and the Czech Space Office
through ESA PECS contract 98058;
– the Danish Ministry of Science;
– the Estonian Ministry of Education and Research through grant IUT40-1;
– the European Commission’s Sixth Framework Programme through the European
Leadership in Space Astrometry (ELSA) Marie Curie Research Training Network
(MRTN-CT-2006-033481), through Marie Curie project PIOF-GA-2009-255267 (Space
AsteroSeismology & RR Lyrae stars, SAS-RRL), and through a Marie Curie
Transfer-of-Knowledge (ToK) fellowship (MTKD-CT-2004-014188); the European
Commission’s Seventh Framework Programme through grant FP7-606740
(FP7-SPACE-2013-1) for the Gaia European Network for Improved data User
Services (GENIUS) and through grant 264895 for the Gaia Research for European
Astronomy Training (GREAT-ITN) network;
– the European Cooperation in Science and Technology (COST) through COST
Action CA18104 ‘Revealing the Milky Way with Gaia (MW-Gaia)’;
– the European Research Council (ERC) through grants 320360, 647208, and
834148 and through the European Union’s Horizon 2020 research and innovation
and excellent science programmes through Marie Skłodowska-Curie grant 745617
(Our Galaxy at full HD – Gal-HD) and 895174 (The build-up and fate of self-
gravitating systems in the Universe) as well as grants 687378 (Small Bodies:
Near and Far), 682115 (Using the Magellanic Clouds to Understand the
Interaction of Galaxies), 695099 (A sub-percent distance scale from binaries
and Cepheids – CepBin), 716155 (Structured ACCREtion Disks – SACCRED), 951549
(Sub-percent calibration of the extragalactic distance scale in the era of big
surveys – UniverScale), and 101004214 (Innovative Scientific Data Exploration
and Exploitation Applications for Space Sciences – EXPLORE);
– the European Science Foundation (ESF), in the framework of the Gaia Research
for European Astronomy Training Research Network Programme (GREAT-ESF);
– the European Space Agency (ESA) in the framework of the Gaia project,
through the Plan for European Cooperating States (PECS) programme through
contracts C98090 and 4000106398/12/NL/KML for Hungary, through contract
4000115263/15/NL/IB for Germany, and through PROgramme de Développement
d’Expériences scientifiques (PRODEX) grant 4000127986 for Slovenia;
– the Academy of Finland through grants 299543, 307157, 325805, 328654,
336546, and 345115 and the Magnus Ehrnrooth Foundation;
– the French Centre National d’Études Spatiales (CNES), the Agence Nationale
de la Recherche (ANR) through grant ANR-10-IDEX-0001-02 for the
‘Investissements d’avenir’ programme, through grant ANR-15-CE31-0007 for
project ‘Modelling the Milky Way in the Gaia era’ (MOD4Gaia), through grant
ANR-14-CE33-0014-01 for project ‘The Milky Way disc formation in the Gaia era’
(ARCHEOGAL), through grant ANR-15-CE31-0012-01 for project ‘Unlocking the
potential of Cepheids as primary distance calibrators’ (UnlockCepheids),
through grant ANR-19-CE31-0017 for project ‘Secular evolution of galxies’
(SEGAL), and through grant ANR-18-CE31-0006 for project ‘Galactic Dark Matter’
(GaDaMa), the Centre National de la Recherche Scientifique (CNRS) and its SNO
Gaia of the Institut des Sciences de l’Univers (INSU), its Programmes
Nationaux: Cosmologie et Galaxies (PNCG), Gravitation Références Astronomie
Métrologie (PNGRAM), Planétologie (PNP), Physique et Chimie du Milieu
Interstellaire (PCMI), and Physique Stellaire (PNPS), the ‘Action Fédératrice
Gaia’ of the Observatoire de Paris, the Région de Franche-Comté, the Institut
National Polytechnique (INP) and the Institut National de Physique nucléaire
et de Physique des Particules (IN2P3) co-funded by CNES;
– the German Aerospace Agency (Deutsches Zentrum für Luft- und Raumfahrt e.V.,
DLR) through grants 50QG0501, 50QG0601, 50QG0602, 50QG0701, 50QG0901,
50QG1001, 50QG1101, 50QG1401, 50QG1402, 50QG1403, 50QG1404, 50QG1904,
50QG2101, 50QG2102, and 50QG2202, and the Centre for Information Services and
High Performance Computing (ZIH) at the Technische Universität Dresden for
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of the Las Cumbres Observatory Global Telescope Network.


# New Objects in Scattering Theory with Symmetries

Andrey S. Losev  Wu Wen-Tsun Key Lab of Mathematics, Chinese Academy of
Sciences  National Research University Higher School of Economics, Moscow,
Russia<EMAIL_ADDRESS>and  Tim V. Sulimov  Petersburg Department of the
Mathematical Institute of the Russian Academy of Science, St. Petersburg
<EMAIL_ADDRESS>
###### Abstract.

We consider 1D quantum scattering problem for a Hamiltonian with symmetries.
We show that the proper treatment of symmetries in the spirit of homological
algebra leads to new objects, generalizing the well known T- and K-matrices.
Homological treatment implies that old objects and new ones are be combined in
a differential. This differential arises from homotopy transfer of induced
interaction and symmetries on solutions of free equations of motion.
Therefore, old and new objects satisfy remarkable quadratic equations. We
construct an explicit example in SUSY QM on $S^{1}$ demonstrating
nontriviality of the above relation.

##  1\. Introduction and Results

Consider nonrelativistic 1D quantum mechanics for a particle that has internal
degrees of freedom, e.g. spin or isospin. Its space of states is described by
a tensor product of functions on a circle or a line and a finite dimensional
vector space $I$ corresponding to internal degrees of freedom. The Hamiltonian
is taken to be $H=H_{0}+V$, where $H_{0}$ is the _free Hamiltonian_ that acts
by $-\partial_{x}^{2}$ and does not rotate the internal space $I$, and the
_interaction_ $V$ that in general rotates the space $I$. We shall use the
perturbative approach, i.e., start with a solution to $H_{0}$ and consider the
corresponding solution of $H$ as a series in interaction $V$. Since we will be
interested in solutions of $H_{0}$ with fixed energy $E$, it is convenient to
define a shifted Hamiltonian $H_{0,E}=H_{0}-E$.

Consider the problem on a line and assume that interaction falls off fast
enough at space infinities. Define the _perturbative T-matrix_ as follows. Its
matrix element between two states of the continuous spectrum
$|\varphi_{\alpha}\rangle,|\varphi_{\beta}\rangle\in\ker{H_{0,E}}$ is equal to

(1.1) $\displaystyle
T_{\beta,\alpha}=\langle\varphi_{\beta}|T|\varphi_{\alpha}\rangle=\langle\varphi_{\beta}|V+VG_{\text{caus}}V+VG_{\text{caus}}VG_{\text{caus}}V+\ldots|\varphi_{\alpha}\rangle,$

where $G_{\text{caus}}$ is the _causal Green’s function_

(1.2) $G_{\text{caus}}=\lim_{\varepsilon\to 0+}(E-H_{0}+{\mathrm{i}\mkern
1.0mu}\varepsilon)^{-1}.$

It is known that $T$ is the perturbative solution to the Lippman-Schwinger
equation for the (nonperturbative) T-matrix $T^{\text{np}}$:

(1.3) $T^{\text{np}}=V+VG_{\text{caus}}T^{\text{np}}.$

It is also well-known that all questions about scattering of particles may be
expressed through $T^{\text{np}}$ (see, for example, [5]).

Let $\mathfrak{g}$ be a symmetry (super)algebra of $H$ with generators
$S_{a}$:

(1.4) $[S_{a},H]=0,\qquad\\{S_{a},S_{b}\\}=f_{ab}^{d}S_{d},$

where $\\{,\\}$ denotes the supercommutator. Note, that Hamiltonian itself is
also a symmetry, because $[H,H]=0$. One is prompted to ask if the symmetries
of the Hamiltonian are reflected in any kind of symmetries of $T^{\text{np}}$,
or at least in symmetries of $T$.

In this paper we will attempt to answer this question. Surprisingly, the
answer is very nontrivial. In order to formulate it we have to introduce _new
objects_ that (to the best of our knowledge) have not been introduced in
scattering theory before. We will also show that they are related by quadratic
equations. In our current understanding these objects may be seen as coming
from the problem on a circle of asymptotically big radius. However, we believe
that they could be defined directly in the theory on the line, we just do not
know of such a definition yet and we are working on it.

To formulate the answer we will use _ghosts_ for symmetries. The ghosts
$c^{a}$ are coordinates on $\mathfrak{g}$ with inverted parity. In simple
words it means that if $S_{a}$ is even then $c^{a}$ is odd and vice versa. The
ghost for the Hamiltonian will be denoted by $c$ and will be odd while the
rest of the ghosts will be denoted by $\tilde{c}^{a}$.

It turns out that $T$ is just a particular component of the differential
$Q_{\text{ind}}$ acting on the space
$\ker{H_{0,E}}\otimes\mathbb{C}[[c^{a},R^{-1}]]$. Here $R^{-1}$ is an even
variable. We call it $R^{-1}$ because it is related to the problem on the
circle of radius $R$ which is discussed in subsections 3.2 and 3.3.

_Appearance of the differential $Q_{\text{ind}}$ is novel in the context of
scattering theory, but is natural in the context of homological algebra._

Quadratic equations are just

(1.5) $Q_{\text{ind}}^{2}=0.$

Now we will proceed to details of construction of $Q$ out of symmetries. First
we assume that like Hamiltonian symmetries $S_{a}$ are split into two pieces:

(1.6) $S_{a}=S_{0,a}+S_{1,a}$

where $S_{0,a}$ is a symmetry of the free Hamiltonian and $S_{1,a}$ (like
interaction of Hamiltonian) vanish at space infinities.

Then,

(1.7)
$Q_{\text{ind}}=\tfrac{1}{2}f_{ab}^{d}c^{a}c^{b}\partial_{c^{d}}+c^{a}S_{a}(R^{-1})+c^{a}c^{b}S_{ab}(R^{-1})+\ldots$

In the zeroth order in $R^{-1}$ expansion the second term in equation above
equals to the symmetry of free Hamiltonian acting on solutions to free problem

(1.8)
$Q_{\text{ind}}^{(0)}=\tfrac{1}{2}f_{ab}^{d}c^{a}c^{b}\partial_{c^{d}}+c^{a}S_{0,a}\big{|}_{\ker{H_{0,E}}}.$

In the first order in $R^{-1}$ expansion we have

(1.9)
$Q_{\text{ind}}^{(1)}=c^{a}S_{a}^{(1)}+c^{a}c^{b}S_{ab}^{(1)}+\ldots=cK+\tilde{c}^{a}S_{a}^{(1)}+c^{a}c^{b}S_{ab}^{(1)}+\ldots$

It is remarkable that the coefficient in front of the distinguished ghost $c$
is the well known K-matrix (see, for instance, [4]), which is related to
T-matrix by

(1.10) $K=T+\frac{{\mathrm{i}\mkern 1.0mu}}{2\sqrt{E}}TK.$

######  Example.

Consider $V(x)=\lambda\delta(x)$. The T-matrix can be easily acquired through
any of the formulae above giving

(1.11) $T(p,q)=\frac{\lambda}{1-\frac{\lambda}{2{\mathrm{i}\mkern
1.0mu}\kappa}},$

which is independent of the incoming or outgoing momenta. Using (1.10) we get

(1.12) $K(p,q)=\lambda.$

Coincidently, $K$ turned out to be equal to $V$. One can easily see that this
will not be the case in general.

The perturbative expansion for $K$ can be acquired from (1.1) by replacing the
causal Green’s function with the standing wave one.

_Other terms in the expansion of $Q_{\text{ind}}$ are novel objects in
scattering theory, related to standard objects by $Q_{\text{ind}}^{2}=0$._

##  2\. Theory

###  2.1. Homotopy Transfer

In this paper we shall illustrate ideas of section 1 using the simplest
nontrivial example of $\mathscr{N}=1$ supersymmetric quantum mechanics on
$S^{1}$ of radius $R$. Moreover, for simplicity we consider an algebra
consisting of the Hamiltonian together with a single odd symmetry that squares
to zero:

(2.1) $[S,H]=0,\qquad S^{2}=0.$

Thus, all the structure constants $f_{ab}^{d}$ are zero, so the first term in
(1.7) is also zero.

The corresponding ghosts are $c$ and $\tilde{c}$ with the following
properties:

  1. (1)

$c$ is odd while $\tilde{c}$ is even;

  2. (2)

$c$ anti-commutes with $S$.

These ghosts allow us to construct differentials

(2.2) $Q_{0}=cH_{0,E}+\tilde{c}S_{0},\quad Q_{1}=cV+\tilde{c}S_{1},\quad
Q=Q_{0}+Q_{1},$

and a _homotopy_

(2.3) $h=G_{\text{st}}\partial_{c},\quad
G_{\text{st}}|\varphi\rangle=\begin{cases}(E-H_{0})^{-1}|\varphi\rangle,&|\varphi\rangle\notin\ker{H_{0,E}},\\\
0,&|\varphi\rangle\in\ker{H_{0,E}},\end{cases}$

where $\partial_{c}=\frac{\partial}{\partial{c}}$. In physics $G_{\text{st}}$
is known as the standing wave Green’s function, but in the spirit of
homological algebra we will refer to it as the _homotopical_ Green’s function.

From (2.1) and the properties of the ghosts it follows that

(2.4) $Q_{0}^{2}=Q^{2}=0.$

Let $\mathscr{H}$ denote the original Hilbert space. Define
$U=\mathscr{H}\otimes\mathbb{C}[c,\tilde{c}]$, then the notation is as follows

(2.5) $U=\ker{H_{0,E}}\oplus U_{\text{ac}},\quad\pi:U\to\ker{H_{0,E}},\quad
i:\ker{H_{0,E}}\to U,$

where $\pi$ and $i$ are the projection and inclusion operators. Composing them
we get the projection on $\ker{H_{0,E}}$ that can be written using the
homotopy

(2.6) $i\pi=\Pi_{\ker{H_{0,E}}}=1+\\{h,Q_{0}\\}.$

We can now write down the central object of homotopy transfer — the _induced
differential_ (see, for example, [3] and [1])

(2.7) $Q_{\text{ind}}=\pi Q_{0}i+\pi Q_{1}i+\pi Q_{1}hQ_{1}i+\pi
Q_{1}hQ_{1}hQ_{1}i+\ldots.$

A brief calculation gives

(2.8)
$Q_{\text{ind}}=\tilde{c}S_{0}+cV_{\text{ind}}+c\partial_{c}\tilde{c}S_{\text{R}}+\partial_{c}c\tilde{c}S_{\text{L}}-\partial_{c}\tilde{c}^{2}S_{\text{LR}},$

where $S_{0}$ is actually $S_{0}\big{|}_{\ker{H_{0,E}}}$, but since
$[S_{0},H_{0}]=0$ we simplify the notation. Additionally,

(2.9)
$V_{\text{ind}}=\pi\left(V+VG_{\text{st}}V+VG_{\text{st}}VG_{\text{st}}V+\ldots\right)i,$
(2.10)
$S_{\text{R}}=\pi\left(S+VG_{\text{st}}S+VG_{\text{st}}VG_{\text{st}}S+\ldots\right)i,$
(2.11)
$S_{\text{L}}=\pi\left(S+SG_{\text{st}}V+SG_{\text{st}}VG_{\text{st}}V+\ldots\right)i,$
(2.12)
$S_{\text{LR}}=\pi\left(SG_{\text{st}}S+SG_{\text{st}}VG_{\text{st}}S+SG_{\text{st}}VG_{\text{st}}VG_{\text{st}}S+\ldots\right)i.$

We can now use the main result of homotopy transfer. Since $Q^{2}=0$, by
Kadeishvili’s theorem we have

(2.13) $Q_{\text{ind}}^{2}=0.$

This equation can be expanded as

(2.14)
$\begin{split}Q_{\text{ind}}^{2}&=c\tilde{c}[V_{\text{ind}},S_{0}]+c\partial_{c}\tilde{c}^{2}\\{S_{0},S_{\text{R}}\\}+\partial_{c}c\tilde{c}^{2}\\{S_{0},S_{\text{L}}\\}+\partial_{c}\tilde{c}^{3}[S_{0},S_{\text{LR}}]\\\
&\quad+c\tilde{c}\left(V_{\text{ind}}S_{\text{L}}-S_{\text{R}}V_{\text{ind}}\right)\\\
&\quad+\tilde{c}^{2}\Big{(}c\partial_{c}(S_{\text{R}}^{2}-V_{\text{ind}}S_{\text{LR}})+\partial_{c}c(S_{\text{L}}^{2}-S_{\text{LR}}V_{\text{ind}})\Big{)},\end{split}$

which gives

(2.18)

##  3\. Example of New Objects and Their Relations to Scattering

###  3.1. SUSY QM in More Detail

Supersymmetric quantum mechanics can be formulated as follows

(3.1) $Q_{+}=\mathop{}\\!\mathrm{d}+\mathop{}\\!\mathrm{d}{W},\quad
Q_{-}=(\mathop{}\\!\mathrm{d}-\mathop{}\\!\mathrm{d}{W})^{*},\quad
H=\\{Q_{+},Q_{-}\\},$

where $\mathop{}\\!\mathrm{d}=\psi\partial_{x}$ is the de Rham differential,
$\psi^{*}=-\partial_{\psi}$ and $\psi$ is the Grassmann variable associated
with supersymmetry (for a broad review see [2]).

Let us write down the Hamiltonian and its symmetries explicitly:

(3.2) $H=-\partial_{x}^{2}+W^{\prime
2}-W^{\prime\prime}(1-2\psi\partial_{\psi}),$ (3.3)
$Q_{+}=\psi(\partial_{x}+W^{\prime}),\qquad
Q_{-}=\partial_{\psi}(-\partial_{x}+W^{\prime}).$

######  Remark.

Both $Q_{+}$ and $Q_{-}$ are symmetries of the full Hamiltonian but not of any
of its subparts.

Let us choose $S=Q_{+}$. As we can see, $S_{\text{LR}}\sim\psi^{2}=0$, so
equations (LABEL:odd_S_2)-(2.18) are of no use in this case. On the other
hand, equation (LABEL:odd_S_1) presents an interesting symmetry of
$V_{\text{ind}}$ which, as we have shown earlier, is the K-matrix.

######  Remark.

If we were to consider $S=Q_{+}+Q_{-}$, $S_{\text{LR}}$ would surely appear.

We shall proceed to studying this equation for a particular potential on the
circle of radius $R$ and then acquire the results for the real line by taking
the limit $R\to\infty$.

###  3.2. Double $\delta$-potential on $S^{1}$

The reason for studying the problem on a circle rather than on a line will
become apparent later. For now we say that this case is easier to understand
due to its discrete spectrum. With a continuous spectrum the projection on
$\ker{H_{0,E}}$, represented by two points on the momentum line brings about
unnecessary complications.

As for why we are considering two $\delta$-potentials instead of one, this is
because a single $\delta$-potential can not be produced by a periodic
$W^{\prime}$.

On a circle of radius $R$ we use the coordinate $x\in[-\pi R,\pi R)$. We
consider

(3.4) $W^{\prime}(x)=\frac{\lambda}{2}\operatorname{sgn}(a^{2}-x^{2}),$

where $\lambda$ and $0<a<\pi R$ are fixed parameters. According to (3.2) this
produces the potential

(3.5) $\frac{\lambda^{2}}{4}+(-1)^{F}\lambda(\delta(x-a)-\delta(x+a)),$

where $F$ is the fermionic number, so $(-1)^{F}=1-2\psi\partial_{\psi}$.

Clearly, as it is now, this potential is inconvenient for a scattering problem
because of the constant shift. We shall send $\frac{\lambda^{2}}{4}$ to the
¡¡free part¿¿ of the problem so that the perturbation has a finite support:

(3.6) $H_{0,E}=-\partial_{x}^{2}-E-\frac{\lambda^{2}}{4},\quad
V(x)=(-1)^{F}\lambda(\delta(x-a)-\delta(x+a)).$

Similarly, we do the same for the symmetry:

(3.7) $S_{0}=\psi(\partial_{x}-\tfrac{\lambda}{2}),\quad
S_{1}=\psi\lambda\theta(a^{2}-x^{2}).$

The spectrum of the free problem is discrete due to periodicity of the wave
functions:

(3.8) $E_{n}=k_{n}^{2}-\frac{\lambda^{2}}{4},\qquad k_{n}=\frac{n}{R},\quad
n\in\mathbb{Z}.$

We fix the ratio $\kappa=\frac{n_{0}}{R}>0$ for the energy.

######  Remark.

This means that in what follows $R$ will not be an arbitrary positive number,
but an arbitrary multiple of $\frac{1}{\kappa}$.

Any level with positive $n$ is double-degenerate, so $\ker{H_{0,E}}$ has
dimension four.

######  Remark.

We should not forget to count bosons and fermions separately. As with our
notation this is done through the Grassmann variable $\psi$, our matrices will
be $2\times 2$, not $4\times 4$.

The normalized wave functions are

(3.9) $\varphi_{n}(x)=\langle x|n\rangle=\frac{{\mathrm{e}}^{{\mathrm{i}\mkern
1.0mu}k_{n}x}}{\sqrt{2\pi R}}.$

The homological Green’s function is simply

(3.10) $G_{\text{st}}=\sum_{|n|\neq n_{0}}\frac{|n\rangle\langle
n|}{E_{n_{0}}-E_{n}}=\sum_{|n|\neq n_{0}}\frac{|n\rangle\langle
n|}{\kappa^{2}-k_{n}^{2}}.$

The matrix elements of operators we need are

(3.11) $\displaystyle V_{n,m}$ $\displaystyle=\frac{{\mathrm{i}\mkern
1.0mu}(-1)^{F}\lambda}{\pi R}\sin{a(k_{m}-k_{n})},$ $\displaystyle
G_{\text{h},n,m}$ $\displaystyle=\frac{\delta_{n,m}}{\kappa^{2}-k_{n}^{2}},$
(3.12) $\displaystyle S_{0,n,m}$ $\displaystyle=\psi({\mathrm{i}\mkern
1.0mu}k_{n}-\tfrac{\lambda}{2})\delta_{n,m},$ $\displaystyle S_{1,n,m}$
$\displaystyle=\psi\frac{\lambda}{\pi
R}\frac{\sin{a(k_{m}-k_{n})}}{k_{m}-k_{n}}.$

Restriction on-shell is trivial: for some operator $A$ we construct a $2\times
2$ matrix

(3.13) $A\big{|}_{\ker{H_{0,E}}}=\pi
Ai=\begin{pmatrix}A_{n_{0},n_{0}}&A_{n_{0},-n_{0}}\\\
A_{-n_{0},n_{0}}&A_{-n_{0},-n_{0}}\end{pmatrix}.$

######  Remark.

Keep in mind that the matrix elements of $A\big{|}_{\ker{H_{0,E}}}$ may depend
on $\psi$ and $\partial_{\psi}$, as is the case for $V$ and $S$.

These expressions allow us to calculate $V_{\text{ind}}$, $S_{\text{L}}$ and
$S_{\text{R}}$ perturbatively in $\lambda$. Using a computer algebra system we
were able to compute the series up to $\lambda^{3}$. The results are given in
the appendix.

An important thing to note is that the $n$-th term in the series turns out to
be a polynomial in $R^{-1}$ of degree $n$. Since the supersymmetry holds for
any radius $R$, we have _several_ equations for each power of $\lambda$. We
shall denote terms proportional to $\lambda^{\alpha}R^{-\beta}$ with
$(\alpha,\beta)$. Equation (LABEL:odd_S_1) can now be checked perturbatively
for the first few terms:

(3.14) $\displaystyle(1,1):$
$\displaystyle[V_{\text{ind}}^{(1,1)},S_{0}^{(0,0)}]=0,$ (3.15)
$\displaystyle(2,1):$
$\displaystyle[V_{\text{ind}}^{(2,1)},S_{0}^{(0,0)}]+[V_{\text{ind}}^{(1,1)},S_{0}^{(1,0)}]=0,$
(3.16) $\displaystyle(2,2):$
$\displaystyle[V_{\text{ind}}^{(2,2)},S_{0}^{(0,0)}]+V_{\text{ind}}^{(1,1)}S_{\text{L}}^{(1,1)}-S_{\text{R}}^{(1,1)}V_{\text{ind}}^{(1,1)}=0,$
(3.17) $\displaystyle(3,1):$
$\displaystyle[V_{\text{ind}}^{(3,1)},S_{0}^{(0,0)}]+[V_{\text{ind}}^{(2,1)},S_{0}^{(1,0)}]=0,$
$\displaystyle(3,2):$
$\displaystyle[V_{\text{ind}}^{(3,2)},S_{0}^{(0,0)}]+[V_{\text{ind}}^{(2,2)},S_{0}^{(1,0)}]$
(3.18)
$\displaystyle+V_{\text{ind}}^{(1,1)}S_{\text{L}}^{(2,1)}-S_{\text{R}}^{(2,1)}V_{\text{ind}}^{(1,1)}$
$\displaystyle+V_{\text{ind}}^{(2,1)}S_{\text{L}}^{(1,1)}-S_{\text{R}}^{(1,1)}V_{\text{ind}}^{(2,1)}=0,$
$\displaystyle(3,3):$ $\displaystyle[V_{\text{ind}}^{(3,3)},S_{0}^{(0,0)}]$
(3.19)
$\displaystyle+V_{\text{ind}}^{(1,1)}S_{\text{L}}^{(2,2)}-S_{\text{R}}^{(2,2)}V_{\text{ind}}^{(1,1)}$
$\displaystyle+V_{\text{ind}}^{(2,2)}S_{\text{L}}^{(1,1)}-S_{\text{R}}^{(1,1)}V_{\text{ind}}^{(2,2)}=0,$
$\displaystyle(4,1):$ $\displaystyle\ldots$ $\displaystyle\ldots$

###  3.3. $R\to\infty$ Limit

To get more understanding of the equations above it is beneficial to consider
the infinite radius limit. As $R\to\infty$, the discretization becomes
negligible and the problem coincides with that on a line. To recover the
$\delta$-normalization of states we have to multiply each $|n\rangle$ by
$\sqrt{2\pi R}$, which imposes the following relation between matrix elements
of operators:

(3.20) $A^{\mathbb{R}}(k_{n},k_{m})=\lim_{R\to\infty}2\pi RA^{S^{1}}_{n,m}.$

Expression (3.10) becomes the principal value integral and

(3.21) $\displaystyle V_{\text{ind}}^{\mathbb{R}}$ $\displaystyle=2\pi
R(V_{\text{ind}}^{(1,1)}+V_{\text{ind}}^{(2,1)}+V_{\text{ind}}^{(3,1)}+\ldots),$
(3.22) $\displaystyle S_{\text{L}}^{\mathbb{R}}$ $\displaystyle=2\pi
R(S_{\text{L}}^{(1,1)}+S_{\text{L}}^{(2,1)}+S_{\text{L}}^{(3,1)}+\ldots),$
(3.23) $\displaystyle S_{\text{R}}^{\mathbb{R}}$ $\displaystyle=2\pi
R(S_{\text{R}}^{(1,1)}+S_{\text{R}}^{(2,1)}+S_{\text{R}}^{(3,1)}+\ldots).$

The terms given in the appendix agree with the analytical results for
$V_{\text{ind}}^{\mathbb{R}}$, $S_{\text{L}}^{\mathbb{R}}$ and
$S_{\text{R}}^{\mathbb{R}}$ on a line, which we were able to acquire. These
formulae are also given in the appendix.

However, when it comes to $S_{0}$, there is no limit when using rule (3.20).
This is expected, as trying to evaluate

(3.24) $S_{0}(p,q)=({\mathrm{i}\mkern 1.0mu}p-\tfrac{\lambda}{2})\delta(p-q)$

on-shell results in a singularity, and it is well known that $\delta(0)\sim R$
when finite box regularization is used. As a result, there is no simple way to
write, say, equation (3.16) on a line, since the first term is inderteminate
of type ¡¡$0\times\infty$¿¿. This result is the main reason we chose to start
with the problem on $S^{1}$ rather than on $\mathbb{R}$.

###  3.4. Concluding Conjecture

We conjecture that it is possible to acquire $(\cdot,\beta)$ terms in
$Q_{\text{ind}}$ for $\beta\geq 2$ without referring to the finite radius
circle, at least perturbatively in $\lambda$, as in (2.9).

## Appendix: Perturbation Series

Here we present the terms needed to verify equation (LABEL:odd_S_1) up to
$\lambda^{3}$ for the double $\delta$-potential on $S^{1}$.

As in the main text, terms proportional to $\lambda^{\alpha}R^{-\beta}$ are
denoted with $(\alpha,\beta)$.

###  $V_{\text{ind}}$ up to $\lambda^{3}$

(3.25)
$V_{\text{ind}}^{(1,1)}=\begin{pmatrix}V_{n_{0},n_{0}}&V_{n_{0},-n_{0}}\\\
V_{-n_{0},n_{0}}&V_{-n_{0},-n_{0}}\end{pmatrix}=(-1)^{F}\frac{\lambda\sin{2a\kappa}}{\pi
R}\sigma_{2}.$

(3.26) $V_{\text{ind}}^{(2,1)}=-\frac{\lambda^{2}\sin{2a\kappa}}{2\pi
R\kappa}(\cos{2a\kappa}+\sigma_{1}).$

(3.27)
$V_{\text{ind}}^{(2,2)}=\frac{\lambda^{2}a\sin{2a\kappa}}{\pi^{2}R^{2}\kappa}\left(\cos{2a\kappa}-\frac{\sin{2a\kappa}}{4a\kappa}+\sigma_{1}\right).$

(3.28)
$V_{\text{ind}}^{(3,1)}=-(-1)^{F}\frac{\lambda^{3}\sin^{3}{2a\kappa}}{4\pi
R\kappa^{2}}\sigma_{2}.$

(3.29)
$V_{\text{ind}}^{(3,2)}=(-1)^{F}\frac{\lambda^{3}(\cos{2a\kappa}+4a\kappa\sin{2a\kappa})\sin^{2}{2a\kappa}}{4\pi^{2}R^{2}\kappa^{3}}\sigma_{2}.$

(3.30)
$V_{\text{ind}}^{(3,3)}=-(-1)^{F}\frac{\lambda^{3}\big{(}8a\kappa\cos{2a\kappa}+(16a^{2}\kappa^{2}-1)\sin{2a\kappa}\big{)}\sin^{2}{2a\kappa}}{16\pi^{3}R^{3}\kappa^{4}}\sigma_{2}.$

###  $S_{\text{L}}$ up to $\lambda^{2}$

(3.31) $S_{\text{L}}^{(1,1)}=\psi\frac{a\lambda}{\pi
R}\left(1+\frac{\sin{2a\kappa}}{2a\kappa}\sigma_{1}\right).$

(3.32) $S_{\text{L}}^{(2,1)}=\psi\frac{{\mathrm{i}\mkern
1.0mu}\lambda^{2}a}{2\pi
R\kappa}\Big{(}(1-\tfrac{\sin{4a\kappa}}{4a\kappa})\sigma_{3}+{\mathrm{i}\mkern
1.0mu}(\cos{2a\kappa}-\tfrac{\sin{2a\kappa}}{2a\kappa})\sigma_{2}\Big{)}.$

(3.33) $\begin{split}S_{\text{L}}^{(2,2)}&=\psi\frac{{\mathrm{i}\mkern
1.0mu}\lambda^{2}}{8\pi^{2}R^{2}\kappa^{3}}\Big{(}\cos{4a\kappa}+2a\kappa\sin{4a\kappa}-4a^{2}\kappa^{2}-1\\\
&\qquad\qquad\qquad\qquad~{}+{\mathrm{i}\mkern
1.0mu}4a\kappa(\sin{2a\kappa}-a\kappa\cos{2a\kappa})\sigma_{2}\Big{)}.\end{split}$

###  $S_{\text{R}}$ up to $\lambda^{2}$

(3.34) $S_{\text{R}}^{(1,1)}=S_{\text{L}}^{(1,1)},\qquad
S_{\text{R}}^{(2,1)}=\left(S_{\text{L}}^{(2,1)}\right)^{\dagger},\qquad
S_{\text{R}}^{(2,2)}=S_{\text{L}}^{(2,2)}.$

### Analytical Results on $\mathbb{R}$

(3.35)
$V_{\text{ind}}^{\mathbb{R}}=-\frac{4\kappa\lambda\sin{2a\kappa}}{4\kappa^{2}+\lambda^{2}\sin^{2}{2a\kappa}}(\lambda\cos{2a\kappa}+\lambda\sigma_{1}+2\kappa\sigma_{2}).$

(3.36)
$\begin{split}S_{\text{L}}^{\mathbb{R}}&=\psi\frac{4a\kappa\lambda}{4\kappa^{2}+\lambda^{2}\sin^{2}{2a\kappa}}\Bigg{(}2\kappa-{\mathrm{i}\mkern
1.0mu}\lambda\left(1-\frac{\sin{4a\kappa}}{4a\kappa}\right)\sigma_{3}\\\
&\qquad\qquad+\frac{\sin{2a\kappa}}{a}\sigma_{1}+\lambda\left(\cos{2a\kappa}-\frac{\sin{2a\kappa}}{2a\kappa}\right)\sigma_{2}\Bigg{)}.\end{split}$

(3.37)
$S_{\text{R}}^{\mathbb{R}}=\left(S_{\text{L}}^{\mathbb{R}}\right)^{\dagger}.$

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