# Dunkl symplectic algebra in generalized Calogero models

Tigran Hakobyan<EMAIL_ADDRESS><EMAIL_ADDRESS>Yerevan State
University, 1 Alex Manoogian Street, Yerevan, 0025, Armenia  Alikhanyan
Natianal Laboratory, 2 Aliknanyan br. Street, Yerevan, 0036, Armenia

###### Abstract

We study the properties of the symplectic $sp(2N)$ algebra deformed by means
of the Dunkl operators, which describe the dynamical symmetry of the
generalized $N$-particle Calogero model. It contains a symmetry subalgebra
formed by the deformed unitary generators as well as the (nondeformed)
$sl(2,R)$ conformal subalgebra. An explicit relation among the deformed
symplectic generators is derived. Based on matching between the Casimir
elements of the conformal spin and Dunkl angular momentum algebras, the
wavefunctions of the both the standard and generalized Calogero models, being
expressed in terms of the deformed spherical harmonics, are classified
according to infinite-dimensional lowest-state $sl(2,R)$ multiplets.
Meanwhile, any polynomial integral of motion of the (generalized) Calogero-
Moser model generates a finite-dimensional highest-state conformal multiplet
with descendants expressed via the Weyl-ordered product in quantum field
theory.

##  I Introduction

Most of many-body systems with particle interaction do not possess an exact
solution both in classical and quantum mechanics and are amenable to numerical
simulations and analytical approximations only. However in one dimension,
there is a remarkable family of interacting $N$-particle systems based on the
Calogero model, which are integrable in the Liouville sense. The latter
describes one-dimensional identical particles with pairwise inverse-square
interaction [1, 2]:

$H=\frac{1}{2}\sum_{i=1}^{N}(p_{i}^{2}+x_{i}^{2})+\sum_{i<j}\frac{g(g\mp
1)}{(x_{i}-x_{j})^{2}}.$ (1)

The presence or absence of the confining harmonic potential (as well as any
other potential depending on the radial coordinate only) does not affect on
the integrability. The $\pm$ sign in the potential coupling constant is chosen
for the bosons/fermions, correspondingly. The particle mass, frequency, and
the Plank’s constant are set to unity in above Hamiltonian. More general
inverse-square potentials based on the finite reflection groups, trigonometric
functions [3], spin exchanges, etc. are also integrable. (See the reviews [4,
5, 6, 7] and references therein.)

The family of conservation laws in such systems is much more diverse and
wealthy. In particular, apart from the Liouville integrals, the Calogero
systems with rational potentials possess $N-1$ additional constants of motion,
i.e. are maximally superintegrable [8, 9, 10]. This property results in a huge
degeneracy of the energy levels and simplifies drastically the solution. The
(super)integrability is robust under certain external potentials. It persists
for the related angular models, as well as for systems defined on surfaces
with constant curvature [11, 12, 13, 14].

Another closely related and intriguing property of the described systems is
the elimination of the inverse-square potential by a nonlocal unitary
transformation, which makes them equivalent to the related noninteracting
counterparts [15, 16]. In quantum case, the equivalence becomes even more
transparent when the particle interaction term is packed into a nonlocal
covariant derivative with exchange operators introduced by Dunkl [17]. A
generalized interaction, obtained in this way, contains the particle exchanges
and is reduced to the original Calogero potential for the identical particles
[18, 19]. Moreover, the quantum systems with even more common particle
exchange interactions can be solved in the thermodynamic limit as has been
discovered recently [20].

The described correspondence between the noninteracting and interacting models
spreads further to their symmetries. In particular, the Dunkl deformations of
the momentum and angular momentum are the invariants of the unbound Calogero
Hamiltonian with particle exchanges, which argues the superintegrability at
the level of the Dunkl operator. The Dunkl angular momentum algebra describes
also the complete symmetries of the generalized angular Calogero model [14].
The same deformation applied to the symmetry generators of the $N$-dimensional
isotropic oscillator results in the Dunkl-deformed version of the related
unitary Lie algebra $u(N)$ which describes the symmetry of the generalized
Calogero model [14]. In case of the confining Coulomb potential, the Dunkl
deformation of the $so(N+1)$ algebra is obtained [21]. The latter gathers all
symmetries of the generalized Calogero-Coulomb model and consists of the Dunkl
analogs of the Runge-Lenz vector and angular momentum tensor [12].

The Dunkl deformation procedure retains or barely changes many important
properties of the original symmetries. Particularly, the commutation and
algebraic relations among the generators, the Casimir elements and the action
on the wavefunctions look quite close to their nondeformed analogs. Meanwhile,
the deformed generators do not form a Lie algebra based on the commutations,
but constitute a quadratic algebra together with the exchange operators.

The described mapping is continued further to the spectrum generating
algebras, or dynamical symmetries. For the $N$-dimensional isotropic
oscillator, the dynamical symmetry may be described by the semidirect product
Weyl and symplectic groups, $W_{N}\rtimes SP(2N)$, which are generated,
respectively, by the linear and bilinear combinations of the creation-
annihilation operators [22]. In the current article, we study the Dunkl
deformation of the symplectic Lie algebra paying special attention to the
derivation of explicit commutation relations among its generators. It consists
of the deformed unitary subalgebra which combines the symmetries of the
generalized Calogero Hamiltonian as well as the spectrum generating part,
which maps in-between different energy levels. The Dunkl $sp(2N)$ algebra
involves also the standard $sl(2,R)$ conformal subalgebra. Due to the actual
coincidence of the quadratic invariants of both the conformal spin and the
Dunkl angular momentum, the eigenfunctions of the generalized Calogero
Hamiltonian in spherical coordinates (expressed via the deformed spherical
harmonics) unite in the lowest-weight $sl(2,R)$ representations. Then the
second quantum number in the wavefunction varies within an individual
multiplet and describes the conformal spin’s projection while the remaining
ones parameterize a representation. Meanwhile, for the (generalized) Calogero-
Moser model, any integral of motion generates a (finite-dimensional) highest
weight $sl(2,R)$ multiplet with the descendants expressed in terms of the
Weyl-ordered product in quantum field theory. The highest states in a product
of two or more such multiplets produce additional integrals of motion which
are responsible for the superintegrability.

The article is organized as follows. In Sect. II, the extended version of the
Calogero model based on the Dunkl operators is we briefly reviewed. Sect. III
is devoted to the Dunkl analog of the symplectic group generators. Together
with the Dunkl creation-annihilation operators they generate a Dunkl
deformation of the $w_{N}\rtimes sp(2N)$ algebra. In Sect. IV the behaviour of
the eigenfunctions of the generalized Calogero system under the conformal
group is studied using the Dunkl spherical harmonics. Sect. V reveals the
conformal algebra structure of the Liouville integrals of motion of the
Calogero-Moser system. Using the momentum sum rules and Weyl ordering, the
additional integrals are constructed. The results are summarized in
Conclusion. For the completeness, we provide the definitions of the operator
ordering and deformed spherical harmonics and related details in Appendices.

##  II Generalized Calogero model

###  II.1 Hamiltonian and Dunkl operators

The solution and integrals of motion of the quantum Calogero system (1) may be
formulated in an elegant way after a slight but nontrivial modification. The
expanded version involves the particle exchanges in the potential [18, 19],

$H=\frac{1}{2}\sum_{i=1}^{N}(p_{i}^{2}+x_{i}^{2})+\sum_{i<j}\frac{g(g-s_{ij})}{(x_{i}-x_{j})^{2}}.$
(2)

Here $g>0$ is an attractive coupling constant, and the operator $s_{ij}$
permutes the $i$-th and $j$-th particles while resting the others untouched.
For identical bosons or fermions, the eigenfunctions are symmetric or
antisymmetric, correspondingly. Therefore, $s_{ij}=\pm 1$, and above
Hamiltonian is reduced to the original Calogero model (1).

Moreover, the inverse-square potential may be absorbed by a covariant
momentum, so that the above Hamiltonian can be rewritten as follows:

$H=\frac{1}{2}\sum_{i=1}^{N}(\pi_{i}^{2}+x_{i}^{2}),\qquad\pi_{i}=-\imath\nabla_{i}.$
(3)

The covariant momentum $\pi_{i}$ is not local and depends on the particle
permutations. It is defined by means of the Dunkl operator [17]

$\nabla_{i}=\partial_{i}-\sum_{j\neq i}\frac{g}{x_{i}-x_{j}}s_{ij}.$ (4)

The system (2), (3) is known as a generalized Calogero model often called also
a Dunkl oscillator.

The components of the Dunkl momentum are mutually commutative so that the
related connection is flat. Meanwhile, the commutations with coordinates are
more involved in comparison with the standard canonical commutation relation:

$[\nabla_{i},\nabla_{j}]=0,\qquad[\nabla_{i},x_{j}]=S_{ij}.$ (5)

The last equation contains a matrix composed from the particle-exchange
operators,

$S_{ij}=\begin{cases}-gs_{ij}&i\neq j,\\\ 1+g\sum_{k\neq
i}s_{ik}&i=j.\end{cases}$ (6)

In the free-particle limit ($g\to 0$), it is reduced to the Kronecker delta:
$S_{ij}\to\delta_{ij}$, so that the algebra (5), (6) is often referred in the
literature as a $S_{N}$-extended Heisenberg algebra.

###  II.2 Creatio-annihilation operators

The generalized Calogero Hamiltonian (3) is reminiscent of the isotropic
oscillator and, hence, is amenable to the simple but powerful tools for the
study of the spectrum, integrals of motion, etc. In particular, the
Hamiltonian may be expressed in terms of the Dunkl analogs of the energy
level’s lowering and rising operators [18, 19],

$\displaystyle H=\sum_{i=1}^{N}a_{i}^{+}a_{i}+\frac{1}{2}N-S,$ (7)
$\displaystyle a_{i}^{\pm}=\frac{1}{\sqrt{2}}(x_{i}\mp\nabla_{i}).$ (8)

They play the same role as the particle creation-annihilation operators in the
field-theoretical treatment. The last element in the Hamiltonian (7) is the
permutation-invariant sum over all pairwise exchanges [14]:

$S=\sum_{i<j}S_{ij},\qquad[S,S_{ij}]=0.$ (9)

It disappears in the free-particle limit.

The staircase operators obey a similar relarions as the coordinate and Dunkl
derivative:

$[a_{i},a_{j}^{+}]=S_{ij},\qquad[a_{j},a_{j}]=[a_{j}^{+},a_{j}^{+}]=0.$ (10)

They form a Dunkl version of the Weyl algebra $W_{N}$.

The standard spectrum generating commutation rules hold for the Calogero
Hamiltonian with particle exchanges:

$[H,a_{i}^{\pm}]=\pm a_{i}^{\pm}.$ (11)

The total Dunkl momentum yet does not contain a particle exchange and, hence,
reduces to the usual momentum. So, the center-of-mass creation-annihilation
operators obey the standard one-particle Weyl algebra commutation relation,

$A^{\pm}=\frac{1}{\sqrt{N}}\sum_{i}a_{i}^{\pm},\qquad[A,A^{+}]=1.$ (12)

###  II.3 Energy spectrum and eigenstates

Alike for the isotropic oscillator, the energy levels and eigenstates of the
generalized Calogero model are constructed in the algebraic way.

The lowering operators annihilate the ground state: $a_{i}|0\rangle=0$. The
solution is provided by the following wavefunction (with omitted normalization
constant) [19]:

$|0\rangle=e^{-\frac{1}{2}r^{2}}\prod_{i<j}|x_{i}-x_{j}|^{g}$ (13)

with the radial coordinate defined by $r^{2}=\sum_{i}x_{i}^{2}$. The lowest
energy is given by the following expression:

$E_{0}=\frac{1}{2}gN(N-1)+\frac{1}{2}N.$ (14)

Apart from the standard oscillatory energy, which grants one-half per each
degree of freedom, is contains also the interaction part. The latter assigns
the value $g$ to each particle pair. According to Eq. (11), the excitation
above the lowest state are generated by the monomials in rising operators, so
that the spectrum is equidistant and highly degenerate alike in the oscillator
case:

$|{\bf n}\rangle=a_{1}^{+n_{1}}\dots a_{N}^{+n_{N}}|0\rangle,\qquad E_{\bf
n}=E_{0}+\sum_{l=1}^{N}n_{l}$ (15)

with omitted normalization constant. Note that the particle interaction just
shifts all levels by the same value given by the $g$-term in the ground-state
energy (14).

Generally, above states describe distinguishable particles since vary under
the particle exchanges. In order to infer the identical bosons (fermions) from
them, an additional (anti)symmetrization procedure is required. Then the
generalized Calogero model (2) is reduced to the original Hamiltonian (1).
Alternatively, one can apply (anti)symmetric polynomials in rising operators
to the ground state. For instance, by means of the power sums,

$A_{l}^{\pm}=\sum_{i=1}^{N}a^{\pm}_{i}\mbox{}^{l},$ (16)

the related (unnormalized) bosonic eigenstates are labeled by $N$ non-negative
integrals, ${\bf k}=\\{k_{1},\dots,k_{N}\\}$, and given by the formula [1, 13]

$\displaystyle|{\bf k}\rangle_{\text{sym}}=A_{1}^{+k_{1}}A_{2}^{+k_{2}}\dots
A_{N}^{+k_{N}}|0\rangle.$ (17)

The corresponding energy levels are highly degenerate and given by the
following expression:

$E_{\bf k}=E_{0}+m_{\bf k}\qquad\text{with}\qquad m_{\bf
k}=\sum_{l=1}^{N}lk_{l}$ (18)

being the order of the symmetric polynomial in $a_{i}^{+}$ operators. Note
that the above eigenfunctions may be obtained also using the quantum Lax
operator algebra [23].

The first quantum number $k_{1}$ describes the center-of-mass energy of the
system. The related symmetrized operator (16) is proportional to the Weyl
algebra generator (12): $A_{1}^{\pm}=\sqrt{N}A^{\pm}$. The annihilation
operator reduces the energy level by one, and its action on the energy
eigenstates can be easily calculated:

$A_{1}^{-}|{\bf k}\rangle_{\text{sym}}=\sum_{l=1}^{N}lk_{l}|{\bf k}-{\bf
e}_{l}+{\bf e}_{l-1}\rangle_{\text{sym}}.$ (19)

Here ${\bf e}_{l}$ is the $l$-th standard unit vector in $N$ dimensional space
with the components $\delta_{l}^{i}$ for $1\leq l\leq N$, and ${\bf e}_{0}=0$.
The states with a negative coefficient $k_{i}$ must be eliminated from the
above sum.

##  III Dunkl deformed symplectic algebra

###  III.1 Dunkl deformation of $sp(2N)$ generators

Let us supplement the staircase operators (8) with their bilinear
combinations,

$E_{ij}=a^{+}_{i}a_{j},\qquad F_{ij}=a_{i}a_{j},\qquad
F_{ij}^{+}=a_{i}^{+}a_{j}^{+}.$ (20)

Clearly, among them only $N(2N+1)$ are different. These elements form a Dunkl
deformation of the symplectic Lie algebra $sp(2N)$. Together with the
operators $a^{\pm}_{i}$ they form a Dunkl deformation of the dynamical
symmetry algebra of the generalized Calogeo model (2). The latter generalizes
the dynamical symmetry group $W_{N}\rtimes SP(2N)$ of the isotropic harmonic
oscillator [22].

When considering the original Calogero model (1) instead of the generalized
one (2), the symmetric polynomials in the elements $E_{ij}$ and $F^{\pm}_{ij}$
are taken since the first system usually deals with the identical particles
(bosons or fermions). Looking ahead, the spectrum generating algebra for the
first system is formed by the independent polynomials from the set
$\sum_{i=1}^{N}a_{i}^{+k}a_{i}^{l}$ (alike the Hamiltonian itself and the
conformal generators considered below) with rather complicated commutation
relations among them [24].

###  III.2 Deformed unitary symmetry subalgebra

The first set of generators in (20) is formed by the $N^{2}$ elements $E_{ij}$
and constitutes the symmetry algebra of the generalized Calogero Hamiltonian
[25, 14]. Each transposes its indexes under Hermitian conjugate,

$[H,E_{ij}]=0,\qquad E_{ij}^{+}=E_{ji}.$

They form the Dunkl-operator deformation of the standard $gl(N)$ basis and
obey the following commutation rules [14],

$[E_{ij},E_{kl}]=E_{il}S_{jk}-S_{il}E_{kj}+[S_{kl},E_{ij}].$ (21)

The symmetry operators are closed under commutation with the Dunkl staircase
operator. The explicit relation is given by the

$[E_{ij},a_{k}^{+}]=a_{i}^{+}S_{jk}.$ (22)

and its Hermitian conjugate.

They split into the antisymmetric and symmetric parts yielding the Dunkl
angular momentum and Fradkin tensors, correspondingly [9, 26, 14, 27]. Both
them are Hermitian and represent the Dunkl analog of the $U(N)$ generators,

$L_{ij}=\imath(E_{ij}-E_{ji}),\qquad I_{ij}=E_{ij}+E_{ji}.$ (23)

In the absence of the interaction part ($g=0$), they generate the unitary
symmetry of the isotropic oscillator.

The Dunkl angular momentum components are closed under the commutation with
coefficients depending on the particle exchanges:

$[L_{ij},L_{kl}]=\imath(L_{il}S_{jk}+L_{jk}S_{il}-L_{ik}S_{jl}+L_{jl}S_{ik}).$

The Casimir element, which commutes with above generators and modifies the
usual angular momentum square, yields the angular part of the Calogero model
(see also Eq. (45) below) [14],

$\displaystyle H_{\Omega}=L^{2}+S(S-N+2),\qquad L^{2}=\sum_{i<j}L_{ij}^{2},$
(24) $\displaystyle[H_{\Omega},L_{ij}]=0.$

Using the commutation relations among the coordinates and Dunkl operators, the
Dunkl angular momentum square may be expressed as follows [14]:

$H_{\Omega}=-r^{2}\nabla^{2}+r\partial_{r}(r\partial_{r}+N-2)$ (25)

where $r=\sqrt{x^{2}}$ is the radial coordinate. Notice that recently the
Dunkl angular momentum appears also as the symmetry of the Dunkl-operator
extended version of the free relativistic particle [28].

The Dunkl $u(N)$ algebra also possesses a Casimir element. Alike for the
isotropic oscillator, it is not independent but expressed via the Hamiltonian
in the following way:

$C=\sum_{i,j}E_{ij}(E_{ji}+S_{ji})=H^{2}-(S-{\textstyle\frac{1}{2}}N)^{2}.$

The above formula follows from a quadratic relation (crossing relation) among
the generators of the Dunkl unitary algebra [14],

$E_{ij}(E_{kl}+S_{kl})=E_{il}(E_{kj}+S_{kj}),$ (26)

which, in fact, implies the commutation relation (21).

###  III.3 Spectrum generating part

Finally, the remaining $N(N+1)$ operators from (20) form Dunkl analogs of the
staircase generators in the symplectic group $SP(2N)$. The corresponding
matrices are symmetric, $F^{\pm}_{ij}=F^{\pm}_{ji}$, and map between the next-
to-nearest neighboring energy levels of the generalized Calogero Hamiltonian
(2):

$[H,F^{\pm}_{ij}]=\pm 2F^{\pm}_{ij},\qquad F^{\pm}_{ij}=F^{\pm}_{ji}.$ (27)

The commutation relations with the Dunkl creation-annihilation operators are
similar to those for the symmetry generators (22). The nontrivial commutators
among them are easy to obtain using the algebra (10),

$\displaystyle=S_{ik}a_{j}+a_{i}S_{jk},$ (28)
$\displaystyle[F^{+}_{ij},a_{k}]$
$\displaystyle=-S_{ik}a^{+}_{j}-a^{+}_{i}S_{jk}.$

The remaining commutators among the $F$-s as well as between the $F$ and $E$
operators are a bit trickier. For the distinct values of the four indexes, the
Wick’s theorem holds for the string of creation and annihilation operators
with the contractor among the pairs $a_{i}$ and $a_{k}^{+}$ provided by the
exchange $S_{ik}$. The latter behaves alike a number since commutes with the
other operators in the monomial. Therefore, the only nontrivial commutators
are provided by the following relations,

$\displaystyle[F_{ij},F_{kl}^{+}]=E_{ki}S_{jl}+E_{kj}S_{li}+E_{li}S_{kj}+E_{lj}S_{ik},$
$\displaystyle[F_{ij},E_{kl}]=F_{jl}S_{ik}+F_{kl}S_{jk},$

and their conjugates (with distinct $i,j,k,l$).

For general values of indexes, the commutation relations among the generators
may be presented in the following form:

$\displaystyle[F_{ij},F_{kl}^{+}]=$
$\displaystyle\,S_{ik}(E_{lj}+S_{lj})+E_{ki}S_{lj}+a_{i}S_{jk}a_{l}^{+}$
$\displaystyle\qquad\qquad\qquad\qquad+a_{k}^{+}S_{il}a_{j},$ (29)
$\displaystyle[F_{ij},E_{kl}]=$ $\displaystyle\,S_{ik}F_{jl}+a_{i}S_{jk}a_{l}$
(30)

with the last equation supplemented by its Hermitian conjugate. The relation
with the Dunkl angular momentum (23) is inherited from Eq. (30) and its
conjugate,

$[F_{ij}^{\pm},L_{kl}]=\imath(S_{ik}F_{jl}^{\pm}+F_{il}^{\pm}S_{jk}-S_{il}F^{\pm}_{jk}-F_{ik}^{\pm}S_{jl}).$

In the formulas (29), (30), the terms which are bilinear in the Dunkl
creation-annihilation operators can be always re-expressed in terms of the $E$
or $F$ generators. This can be achieved by moving the permutation operators
$S_{ij}$ to the left or right. In particular, when the four indexes are split
into two equal pairs with values $i\neq j$, the relation (29) reduces as
follows:

$\displaystyle[F_{ii},F_{kk}^{+}]=$
$\displaystyle(E_{ki}+E_{ik}+E_{kk}+E_{ii}+S_{ii})S_{ik}+g^{2},$ (31)
$\displaystyle[F_{ik},F_{ik}^{+}]=$
$\displaystyle(2E_{ii}+S_{ii})S_{ik}+S_{ii}(E_{kk}+S_{kk})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+E_{ii}S_{kk}.$ (32)

Then the next expression (30) turns into the following couple:

$\displaystyle[F_{ii},E_{kk}]$ $\displaystyle=(F_{ik}+F_{ii})S_{ik},$ (33)
$\displaystyle[F_{ik},E_{ik}]$ $\displaystyle=F_{ii}S_{ik}+S_{ii}F_{kk}.$ (34)

In case when the four indexes coincide, the commutator contains many other
generators. Using the equation $S_{ii}=1-\sum_{k\neq i}S_{ik}$ implied by the
definition of the exchange matrix (6), one can obtain the following
expressions from the most general commutators (29), (30) [compare with Eqs.
(31) and (33)]:

$\displaystyle[F_{ii},F_{ii}^{+}]=\,$ $\displaystyle 4E_{ii}-\sum_{k\neq
i}(E_{ii}+E_{kk}+E_{ik}+E_{ki})S_{ik}$
$\displaystyle\qquad\qquad\qquad\qquad+S_{ii}^{2}-(N-1)g^{2},$ (35)
$\displaystyle[F_{ii},E_{ii}]=\,$ $\displaystyle 2F_{ii}-\sum_{k\neq
i}(F_{kk}+F_{ik})S_{ik}.$ (36)

Apart from the commutation relations, there are, of course, additional
algebraic relations among the considered operators. For example, the index
order is not essential in the products $F_{ij}F_{kl}$, $F_{ij}a_{k}$ as well
as the order of the last three indexes in the $E_{ij}F_{kl}$. Likewise, the
expression $F_{ij}(E_{kl}+S_{kl})$ remains invariant under the cyclic
permutation of the same indexes as the $E_{ij}(E_{kl}+S_{kl})$ does (26).

###  III.4 Conformal subalgebra

The Dunkl deformed symplectic algebra contains a conventional (nondeformed)
conformal subalgebra $sl(2,R)$ formed by the following elements,

$\displaystyle
K_{\pm}=\frac{1}{2}\sum_{i}F^{\pm}_{ii}=\frac{1}{2}\sum_{i}a_{i}^{\pm 2},$
(37) $\displaystyle
K_{3}=\frac{1}{2}H=\frac{1}{2}\sum_{i}E^{\pm}_{ii}-\frac{1}{2}S+\frac{1}{4}N.$

They obey the standard commutation relation

$[K_{-},K_{+}]=2K_{3},\qquad[K_{3},K_{\pm}]=\pm K_{\pm},$ (38)

which follows from the algebra (10). Alternatively, it may be obtained from
Eqs. (31), (33), (III.3), (36), and the relation

$\sum_{i}S_{ii}^{2}=N-2S.$

The above generators may be regarded as analogs of ones constructed long time
ago for the original Calogero model without the exchange operators [29]. It
differs also from the conformal subalgebra of the Virasoro algebra’s Dunkl
deformation [30]. The operators $K_{\pm}$ with excluded center of mass have
been considered in Ref. [31].

Instead of the lowering-rising generators, one can use the elements $K_{1}$
and $K_{2}$ defined in a standard way:

$K_{\pm}=K_{1}\pm K_{2}.$ (39)

Then the defining relations (38) can be written in a covariant form, which
reveals the equivalence with the three dimensional Lorentz algebra $so(1,2)$:

$[K_{\alpha},K_{\beta}]=-\epsilon_{\alpha\beta\gamma}K^{\gamma}.$ (40)

Here by the $\epsilon$, the Levi-Civita tensor is denoted, and the index is
risen by the Minkowski metrics $\gamma^{\alpha\beta}=\text{diag}(1,-1,-1)$.
The second generator is anti-Hermitian while the remaining two are Hermitian:

$K_{2}^{+}=-K_{2},\qquad K_{1,3}^{+}=K_{1,3}.$

The first two generators are expressed via the Hamiltonian and radial
coordinate:

$\displaystyle K_{1}=\frac{1}{2}(r^{2}-H),\qquad
K_{2}=-\frac{1}{2}r\partial_{r}-\frac{N}{4}.$ (41)

Therefore, the conformal algebra commutes with the Dunkl angular momentum:

$[K_{\alpha},L_{ij}]=0.$

Recall that the formal replacement of $K_{1}$ by $\imath K_{1}$ in the
equations (40) reproduces the familiar relations of the $su(2)$ angular
momentum algebra in quantum mechanics. The Casimir element of the conformal
algebra (40) is the square in Minkowski space:

$K^{2}=K^{\alpha}K_{\alpha}=K_{1}^{2}-K_{2}^{2}-K_{3}^{2}.$ (42)

Up to a constant, it coincides with the angular Calogero Hamiltonian (24), as
can be easily verified:

$H_{\Omega}=-4K^{2}-\left({\textstyle\frac{1}{4}}N-1\right)N.$ (43)

Finally note that the conformal (37) and Weyl (12) generators form together a
closed algebra. The nontrivial mixed commutation relations are given by

$\displaystyle[A^{\mp},K^{\pm}]=\pm
A^{\pm},\qquad[K_{3},A^{\pm}]=\pm\frac{1}{2}A^{\pm}.$ (44)

##  IV Conformal group structure of Calogero wavefunctions

###  IV.1 Deformed harmonic polynomials

The unbound generalized Calogero model (Calogero-Moser model with particle
exchanges) may be expressed via the angular Hamiltonian and radial coordinates
by inverting Eq. (25):

$H_{0}=-\frac{1}{2}\nabla^{2}=-\frac{1}{2}\partial_{r}^{2}-\frac{N-1}{2r}\partial_{r}+\frac{H_{\Omega}}{2r^{2}}.$
(45)

Let us use an equivalent Hamiltonian which is more suitable for our purpose in
the current section [32]. It is obtained from the primary one by applying the
similarity map with respect to the $g$-th power of the Vandermonde
polynomial’s absolute value,

$\phi=\phi(x)=\prod_{i<j}|x_{i}-x_{j}|^{g},$ (46)

which participates in the ground state’s expression (13):

$H^{\prime}_{0}=\phi^{-1}H_{0}\phi=-\frac{1}{2}\nabla^{\prime}\mbox{}^{2}=-\frac{1}{2}\partial^{2}\\\
-g\sum_{i<j}\left[\frac{1}{x_{i}-x_{j}}(\partial_{i}-\partial_{j})-\frac{1-s_{ij}}{(x_{i}-x_{j})^{2}}\right].$
(47)

Under such transform, the Dunkl operator, in its turn, undergoes the following
shift:

$\nabla^{\prime}_{i}=\phi^{-1}\nabla_{i}\phi=\partial_{i}+\sum_{j\neq
i}\frac{g}{x_{i}-x_{j}}(1-s_{ij}).$ (48)

Evidently, the applied similarity map is not unitary. As a consequence, both
$H^{\prime}_{0}$ and $\pi^{\prime}_{i}=-\imath\nabla^{\prime}_{i}$ are not
Hermitian.

An advantage of the shifted Dunkl operator in front of the initial one (4) is
the reduction to the conventional derivative when applying to a symmetric
function. Moreover, the deformed Laplace equation

$\nabla^{\prime 2}h(x)=0$ (49)

has polynomial solutions, which are Dunkl deformations of the usual harmonic
polynomials [17, 32]. The wavefunctions of the angular momentum square are
expressed via such polynomials. In fact, it is easy to see that the relation
(25) implies that any deformed homogeneous harmonic polynomial is an
eigenfunction of the shifted angular Calogero Hamiltonian.

A suitable set among the deformed harmonic polynomials is given by the
following compact expression [33]:

$h_{\bf n}(x)=r^{2(E_{0}+m_{\bf n}-1)}\nabla^{\prime n_{1}}_{1}\nabla^{\prime
n_{2}}_{2}\dots\nabla^{\prime n_{N}}_{N}r^{2(1-E_{0})}$ (50)

where the polynomial’s order takes the value

$m_{\bf n}=\sum_{i=1}^{N}n_{i},\qquad r\partial_{r}h_{\bf n}(x)=m_{\bf
n}h_{\bf n}(x).$ (51)

For the completeness, the proof that the functions (50) fulfill the deformed
Laplace equation is given in Apendix B.

The symmetrization procedure over all coordinates reduces the deformed
harmonic polynomials (50) to the expression [13]

$\displaystyle h_{\bf k}^{\text{sym}}(x)$ $\displaystyle=r^{2(E_{0}+m_{\bf
k}-1)}\mathcal{D}_{1}^{k_{1}}\mathcal{D}_{3}^{k_{3}}\dots\mathcal{D}_{N}^{k_{N}}r^{2(1-E_{0})}.$
(52)

Here we have used a shorten notations for the Newton powers in shifted Dunkl
operators,

$\mathcal{D}_{l}=\sum_{i=1}^{N}\nabla^{\prime}_{i}\mbox{}^{l}.$

The integer $m_{\bf k}$ also is equal to the polynomial’s order and defined by
Eq. (18) above,

$r\partial_{r}h_{\bf k}^{\text{sym}}(x)=m_{\bf k}h_{\bf
k}^{\text{sym}}(x)\quad\text{with}\quad k_{2}=0.$ (53)

The last condition is due to the absence of the Dunkl Laplacian
$\mathcal{D}_{2}=\nabla^{\prime 2}$ in the expression (52), since it acts on a
deformed harmonic function:

$\nabla^{\prime 2}r^{-2(E_{0}-1)}=0.$ (54)

Note that due to above condition, the deformed harmonic polynomials (50)depend
linearly and do not constitute a basis [33]:

$\sum_{i=1}^{N}h_{{\bf n}+2{\bf e}_{i}}(x)=0$ (55)

with the standard notation for the unit vectors (19).

###  IV.2 Wavefunctions in deformed harmonic polynomials

First, let us consider the Dunkl angular momentum square which coincides with
the angular part of the generalized Calogero model (24). The related
eigenfunctions are the product of the deformed harmonic polynomial (50) and
shift function (46) as was explained above. The energy levels are obtained
immediately using the expression for the angular Hamiltonian (25):

$\displaystyle H_{\Omega}\,\phi(x)h_{\bf
n}(x)=\mathcal{E}_{\beta}\phi(x)h_{\bf n}(x),$ (56) $\displaystyle{\cal
E}_{\beta}=\beta^{2}-({\textstyle\frac{1}{2}}N-1)^{2}.$ (57)

They depend on a single quantum number expressed via the order of the
polynomial (51):

$\beta=E_{0}+m-1\quad\text{with}\quad m=\sum_{l=1}^{N}n_{l}$ (58)

where $E_{0}$ is the lowest energy of the Calogero model (18). This property
leads to the highest possible level of degeneracy and maximal
superintegrability of the angular system [13, 11]. Note that the eigenstates
are not independent but subjected to a constraint inherited from the condition
(55).

Since we deal with the angular system, the wavefunctions have to depend only
on the angular coordinates $u_{i}=\cos\theta_{i}=x_{i}/r$ located on unit
$d=N-1$ sphere: $u^{2}=1$. Indeed, the radial coordinate may be eliminated
after canceling out the common factor $r^{\beta-\frac{1}{2}N+1}$ from both
sides of Eq. (56).

Using the wavefunctions of the angular Hamiltonian, the energy eigenfunctions
of the (generalized) Calogero model may be constructed in spherical
coordinates. This fact reflects a common feature of superintegrable
Hamiltonians to allow separation of variables in few coordinate systems. The
simplest and most conventional solution of the generalized Calogero model is
given above in terms of the Cartesian coordinated (15).

First note that the standard similarity transformation with respect to the
function $r^{\frac{1}{2}(N-1)}$, applied to the generalized Calogero
(Calogero-Moser) system (45), absorbs the term which is linear in the radial
derivative. As a result, Hamiltonian acquires the following simple form:

$\displaystyle r^{\frac{1}{2}(N-1)}Hr^{-\frac{1}{2}(N-1)}$
$\displaystyle=-\frac{1}{2}\partial_{r}^{2}+\frac{1}{2}r^{2}+\frac{H_{\Omega}+\varepsilon^{\prime}}{2r^{2}},$
(59) $\displaystyle\varepsilon^{\prime}$
$\displaystyle={\textstyle\frac{1}{4}}(N-1)(N-3).$

The deformed spherical harmonics (56), (57) are eigenfunctions of the shifted
Dunkl angular momentum square operator, $H_{\Omega}+\varepsilon^{\prime}$,
with the eigenvalue

$\varepsilon_{\beta}+\varepsilon^{\prime}=\beta^{2}-{\textstyle\frac{1}{4}}.$

So, the angular and radial coordinates of the transformed Hamiltonian (59) are
separated, and the radial part describes the one-dimensional singular
oscillator with the spectrum and eigenfunctions given, respectively, by

$E_{k}=\beta+2k+1,\qquad
e^{-\frac{r^{2}}{2}}r^{\beta+\frac{1}{2}}L_{k}^{\beta}(r^{2})$ (60)

with $k=0,1,2,\dots$ and

$L_{k}^{\beta}(z)=\frac{1}{k!}z^{-\beta}e^{z}\frac{d^{k}}{dz^{k}}\left(e^{-z}z^{k+\beta}\right)$

being the $k$-th order associated Laguerre polynomial [4].

Hence, the eigenstates of the generalized Calogero model in terms of the
spherical harmonics are provided by the following functions:

$\displaystyle\Psi_{{\bf n},k}(x)$
$\displaystyle=e^{-\frac{r^{2}}{2}}r^{\beta-\frac{1}{2}N+1}L_{k}^{\beta}(r^{2})\phi(u)h_{\bf{n}}(u)$
(61)
$\displaystyle=e^{-\frac{r^{2}}{2}}L_{k}^{\beta}(r^{2})\phi(x)h_{\bf{n}}(x),$
(62)

where $u_{i}$ are above defined cosine functions in angular coordinates. The
corresponding energies are obtained from Eqs. (58) and (60):

$E_{{\bf n},k}=E_{0}+m+2k=E_{0}+\mathop{\sum_{l=1}^{N}}_{l\neq
2}lk_{l}+2(k_{2}+k).$ (63)

We see that the second quantum number participating in the energy expression
comes both from the radial and spherical parts in the wavefunction.

Choosing the symmetric harmonic polynomials (52) instead of the common ones,
one obtains the states for indistinguishable bosons, which also are
eigenfunctions of the conventional Calogero model,

$\Psi^{\text{sym}}_{k_{1}k_{2}k_{3}\dots
k_{N}}(x)=e^{-\frac{r^{2}}{2}}L_{k_{2}}^{\alpha}(r^{2})\phi(x)h^{\text{sym}}_{k_{1}0\,k_{3}\dots
k_{N}}(x).$ (64)

Now the lower index of the Laguerre polynomial defines the second quantum
number describing the state. The parameter $\alpha$ is similar to $\beta$ but
with the degree $m$ of the symmetric homogeneous polynomial (52), (53):

$\alpha=E_{0}+m-1,\qquad m=k_{1}+\sum_{l=3}^{N}lk_{l}.$ (65)

The energy spectrum is provided by the same formula as in case of the
Cartesian coordinates (18),

$E_{\bf k}=E_{0}+\sum_{l=1}^{N}lk_{l}=\alpha+2k_{2}+1.$ (66)

However, generally, the quantum numbers $k_{l}$, characterizing the energy
eigenstates in the Cartesian (17) and spherical (64) coordinate systems,
differ mutually. The cause of this discrepancy is a degeneracy of energy
levels which reflects the superintegrability of the Calogero model.

In comparison with the spectrum of the generalized Calogero mode (63), the
second quantum number in the energy here (66) is spawned exclusively by the
radial part in the wavefunction given by the associated Laguerre polynomial
(64). Below it will be shown to be controlled by the conformal group.

Finally we mention that the detailed study of the wavefunctions expressed in
terms of the deformed spherical harmonics has been carried out recently for
the generalized Calogero based the reflection groups $B_{2}$ and $H_{3}$ which
describe the symmetries of the square and icosahedron correspondingly [34].

###  IV.3 Conformal group action on wavefunctions in spherical coordinates

The symmetric wavefunctions of the Calogero model in spherical coordinates
(64) with fixed values of all quantum numbers $k_{l}$ but the second one (with
$l=2$) are grouped into the infinite-dimensional lowest-weight representation
of the conformal algebra. Applying the $SL(2,R)$ generators to them, one
obtains the following expressions:

$\displaystyle K_{-}\Psi^{\text{sym}}_{\dots k_{2}\dots}$
$\displaystyle=-(\alpha+k_{2})\Psi^{\text{sym}}_{\dots k_{2}-1\dots},$ (67)
$\displaystyle K_{+}\Psi^{\text{sym}}_{\dots k_{2}-1\dots}$
$\displaystyle=-k_{2}\Psi^{\text{sym}}_{\dots k_{2}\dots},$ (68)
$\displaystyle K_{3}\Psi^{\text{sym}}_{\dots k_{2}\dots}$
$\displaystyle={\textstyle\frac{1}{2}}E_{\bf k}\Psi^{\text{sym}}_{\dots
k_{2}\dots},$ (69)

where the condition $\Psi=0$ is imposed for the negative values of $k_{2}$.
Here the dots mark the remaining quantum numbers ($k_{l}$ with $l\neq 2$)
which are not affected by the conformal group. According to the definition of
conformal algebra (37), the eigenvalue of the diagonal element equals a half
the state’s energy (66). Since the normalization constants in front of the
wavefunctions are neglected, the matrices of the operators $K_{\pm}$ and
$K_{3}$ have no conventional forms used in quantum mechanics.

The relations (68) and (67) follow from the radial representation of the
conformal generators (39), (41), Eq. (66), and the following recurrence
relation among the associated Laguerre polynomials:

$\displaystyle(z\partial_{z}-k)L_{k}^{\alpha}(z)$
$\displaystyle=-(\alpha+k)L_{k-1}^{\alpha}(z),$
$\displaystyle(z\partial_{z}-z+\alpha+k)L_{k-1}^{\alpha}(z)$
$\displaystyle=kL_{k}^{\alpha}(z)$

holding for any integer $k\geq 1$.

The infinite-dimensional $SL(2,R)$ multiplet (67)–(69) is generated by the
wavefunction with $k_{2}=0$,

$\Psi^{\text{sym}}_{k_{1}0k_{3}\dots}(x)=e^{-\frac{r^{2}}{2}}\phi(x)h^{\text{sym}}_{k_{1}0\,k_{3}\dots
k_{N}}(x),$

obeying the lowest-state condition

$K_{-}\Psi^{\text{sym}}_{k_{1}0k_{3}\dots}=0,\qquad
K_{3}\Psi^{\text{sym}}_{k_{1}0k_{3}\dots}=s\Psi_{k_{1}0k_{3}\dots}$

with the conformal spin $s$ depending on all quantum numbers except from the
second one, $k_{2}$,

$s={\textstyle\frac{1}{2}}(\alpha+1).$ (70)

The Casimir element of the conformal algebra (42) takes on the current
representation the constant value:

$K^{2}=-{\textstyle\frac{1}{4}}(\alpha^{2}-1)=-s(s-1).$ (71)

The relations (43) between the Casimir elements of the conformal algebra and
Dunkl angular momentum on the discussed wavefunctions leads to to an explicit
relation between the corresponding constants given by Eq. (57) with the
$\beta$ replaced by the $\alpha$.

Resuming the above consideration, the radial quantum number and second quantum
number $k_{2}$ in the Dunkl spherical harmonics care about the conformal group
in the Calogero model. In contrast, in Cartesian coordinates, the conformal
transformation can alter other quantum numbers too. More explicitly, the
Cartesian basis (17) behaves alike the spherical one (64) under the rising and
diagonal generators:

$\displaystyle K_{+}|{\bf k}\rangle_{\text{sym}}$
$\displaystyle={\textstyle\frac{1}{2}}|{\bf k}+{\bf
e}_{2}\rangle_{\text{sym}},$ $\displaystyle K_{3}|{\bf k}\rangle_{\text{sym}}$
$\displaystyle={\textstyle\frac{1}{2}}E_{\bf k}|{\bf k}\rangle_{\text{sym}}$

while the lowering generator changes all quantum numbers. For example, if
$k_{l}=0$ for all $l\geq 3$, one gets:

$\displaystyle
K_{-}|k_{1},k_{2}\rangle_{\text{sym}}=2k_{2}(\alpha+k_{2})|k_{1},k_{2}-1\rangle_{\text{sym}}$
$\displaystyle+(k_{1}-1)|k_{1}-2,k_{2}\rangle_{\text{sym}},$

where the vanishing quantum numbers are omitted within the ket states. In the
absence of the last term, the above representation becomes equivalent to the
previous one (67)–(69) with a similarity map:

$|k_{1},k_{2}\rangle_{\text{sym}}\to(-1)^{k_{2}}k_{2}!\Psi^{\text{sym}}_{\dots
k_{2}\dots}.$

Of course, using the Jacobi coordinates, one can separate the center-of-mass
and its quantum number $k_{1}$ but in the higher-level quantum numbers
starting from $k_{3}$ still will remain coupled for the Cartesian
wavefunctions.

Similar type of $SL(2,R)$ representations appears for the Calogero model with
particle exchanges (2), which has more common, nonsymmetric eigenstates
$\Psi_{{\bf n},k}$ (62). The corresponding multiplets are generated from the
ground state $\Psi_{{\bf n},0}(x)=e^{-\frac{r^{2}}{2}}\phi(x)h_{\bf n}(x)$ and
characterized by the conformal spin given by the formula (70) with the
parameter $\alpha$ substituted by $\beta$ (58):

$s={\textstyle\frac{1}{2}}(\beta+1).$ (72)

An eigenvalue associated with the diagonal generator equals half the state’s
energy given now by Eq. (63):

$\displaystyle K_{-}\Psi_{{\bf n},k}$ $\displaystyle=-(\beta+k)\Psi_{{\bf
n},k-1},$ (73) $\displaystyle K_{+}\Psi_{{\bf n},k-1}$
$\displaystyle=-k\Psi_{{\bf n},k},$ $\displaystyle K_{3}\Psi_{{\bf n},k}$
$\displaystyle={\textstyle\frac{1}{2}}E_{{\bf n},k}\Psi_{{\bf n},k}.$

It depends on the radial quantum number $k$ which also distinguishes the
individual states within a single multiplet. The quantum numbers $n_{i}$,
characterizing the spherical harmonics, remain untouched under the conformal
transformation. They define the value of the conformal spin (79), see Eq.
(58).

##  V Conformal group structure of Calogero-Moser integrals

###  V.1 Rotated $sl(2,R)$ basis

In the previous section, the behavior of the spectrum and wavefunctions of the
Calogero model with and without particle exchanges under the action of the
conformal group has been revealed. The latter is a part of more general
dynamical symmetry formed by the Dunkl symplectic group. In the current
section, we study the conformal multiplets spawned by the Liouville integrals
model by adapting the method used in Ref. [35] to the exchange-operator
approach. The additional constants of motion then are recovered from the
tensor-product decomposition which is equivalent to the usual angular momentum
sum rule in quantum mechanics.

Since both the second and third generators of the conformal algebra have a
space-like signature (42), a rotation in this plane results in an equivalent
algebra (40). In particular, the $\pi/2$ rotation along the first conformal
component maps the initial generators to their modified counterparts:

$K^{\prime}_{\alpha}=e^{\frac{\pi}{2}K_{1}}K_{\alpha}e^{-\frac{\pi}{2}K_{1}}.$
(74)

Clearly, the above similarity transformation does not touch the generator
$K_{1}$ while exchanges the two other components as follows:

$K^{\prime}_{2}=-K_{3},\qquad K_{3}^{\prime}=K_{2}.$ (75)

As a result, the rotated lowering and rising $SL(2,R)$ generators (39) acquire
the following explicit form:

$\displaystyle K^{\prime}_{+}$
$\displaystyle=K_{1}-K_{3}={\textstyle\frac{1}{2}}\nabla^{2}=-H_{0},$ (76)
$\displaystyle K^{\prime}_{-}$
$\displaystyle=K_{1}+K_{3}={\textstyle\frac{1}{2}}r^{2}.$

Virtually, the conformal algebra representation via the Dunkl operators
appeared in above form at first in Ref. [36]. Recall that the observable
$H_{0}=-{\textstyle\frac{1}{2}}I_{2}$ is the generalized Calogero-Moser
Hamiltonian (45).

Note that the transformation (74) is not unitary so that both
$K^{\prime}_{\pm}$ are Hermitian while $K_{\pm}$ are conjugate of each other.
In fact, the same rotation relates the generators of the deformed Weyl and
Dunkl algebras,

$a_{i}\to\nabla_{i},\qquad a_{i}^{+}\to x_{i}.$

As a result, the generalized Calogero Hamiltonian (7) transforms as

$H\to-2K^{\prime}_{3}=r\partial_{r}+\frac{1}{2}N$

while the ground state maps to $\phi$ (46) which disappears under the action
of the Dunkl operator: $\nabla_{i}\phi(x)=0$.

###  V.2 Conformal multiplets generated by Liouville integrals

The Liouville integrals of motion of the unbound Calogero (Calogero-Moser)
model are given by symmetric polynomials in Dunkl operators, usually, the
power sums [18]:

$I_{n}=\sum_{i=1}^{N}\nabla_{i}^{n}.$ (77)

In fact, the valid quantities must be Hermitian and expressed via the Dunkl
momentum, $\sum_{i}\pi_{i}^{n}=(-\imath)^{n}I_{n}$. Here we ignore that
requirement in order to avoid imaginary unit in formulas below.

The adjoint action of the conformal algebra on the above integral has quite
simple form in the newly defined basis. Indeed, the third generator is
diagonal while the rising one just kills it:

$\hat{K}^{\prime}_{+}I_{n}=0,\qquad\hat{K}^{\prime}_{3}I_{n}={\textstyle\frac{1}{2}}nI_{n},$
(78)

where the hat here means the commutator,

$\hat{X}f=[X,f].$

It is easy to see that the integral (77) generates the highest weight
irreducible representation of the $SL(2,R)$ group with the conformal spin

$s^{\prime}={\textstyle\frac{1}{2}}n$ (79)

composed from the states $I_{n,l}$ with $l=0,1,\dots,n$, and the
$I_{n,0}=I_{n}$ is the highest one:

$I_{n,l}=(\hat{K}^{\prime}_{-})^{l}I_{n},\qquad\hat{K}^{\prime}_{3}I_{n,l}=(s^{\prime}-l)I_{n,l}.$
(80)

The rising generator acts on these states in a standard way:

$\hat{K}^{\prime}_{+}I_{n,l}=-\hat{H}_{0}I_{n,l}=-l(n-l+1)I_{n,l-1}.$ (81)

In particular, the first Liouville integral $I_{1}=\sum_{i}\partial_{i}$,
which is proportional to the total momentum, generates a doublet
($s^{\prime}=1/2$) with the lowest function proportional to the center-of-mass
coordinate $I_{1,1}=-\sum_{i}x_{i}$. The second integral generates a triplet
($s^{\prime}=1$) consisting of the following members:

$\begin{gathered}I_{2}=2K^{\prime}_{+},\qquad
I_{2,1}=4K^{\prime}_{3}=-2r\partial_{r}-N,\\\
I_{2,2}=4K^{\prime}_{-}=2r^{2}.\end{gathered}$ (82)

The square of the conformal spin (42) takes (80) a constant value on the
common multiplet:

$\hat{K}^{2}=-{\textstyle\frac{1}{4}}n(n+1)=-s^{\prime}(s^{\prime}+1).$ (83)

Note that the conformal spin’s square in the infinite dimensional lowest (71)
and highest weight (83) representations have different shapes but map to each
other under the substitution $s^{\prime}\to-s$.

###  V.3 Descendants of Liouville integral in Weyl-ordered form.

The descendants of the Liouville integral of motion are expressed via the
Weyl-ordered operator product of the coordinate and Dunkl operator in the
following way:

$I_{n,l}=(-1)^{l}\frac{n!}{(n-l)!}\sum_{i=1}^{N}\left(x_{i}^{l}\nabla_{i}^{n-l}\right)_{W}.$
(84)

Appendix A provides the definition and brief explanation of the Weyl order.

The above relation follows immediately from the following commutation
relations:

$\hat{K}^{\prime}_{-}\nabla_{i}=-x_{i},\qquad\hat{K}^{\prime}_{-}x_{i}=0.$

Note that the factorials in Eq. (84) contract with the binomial factor in
front of the Weyl product (88) which results in the overall $l!$ factor.

In particular, the first descendant from the entire family has the following
explicit form:

$\displaystyle I_{n,1}$
$\displaystyle=-\sum_{i=1}^{N}\sum_{k=0}^{n-1}\nabla_{i}^{k}x_{i}\nabla_{i}^{n-k-1}.$
(85)

Meanwhile, the last one reduces to the center of mass:

$I_{n,n}=(-1)^{n}n!\sum_{i}x_{i}.$

Note that a generating polynomial gathering all the descendants (84) may be
built as a Newton’s power sum in an appropriate superposition of the Dunkl
operator with the related coordinate (89):

$I_{n}(v):=\sum_{l=0}^{n}(-v)^{l}I_{n,l}=\sum_{i=1}^{N}(\nabla_{i}-vx_{i})^{n}.$

###  V.4 Additional integrals from product multiplets

As was mentioned in Introduction, the Calogero-Moser model is superintegrable.
Apart from the Liouville integrals $I_{n}$ with $n=1,\dots,N$, it possesses
additional constants of motion, which constitute together a set of $2N-1$
independent quantities commuting with the Hamiltonian [8, 9, 10]. In fact, the
superintegrability is closely related with the dynamical $SL(2,R)$ symmetry
which can be used in order to retrieve the additional integrals from the
Liuoville ones [10, 37, 35].

Alternatively, they can be recovered as the highest-weight functions in the
product representation [35]. Indeed, a symmetric bilinear combination of two
Liouville integrals of motion,

$\\{I_{n_{1}},I_{n_{2}}\\}=I_{n_{1}}I_{n_{2}}+I_{n_{2}}I_{n_{1}},$ (86)

possesses a structure of the tensor product representation with respect to the
$sl(2,R)$ algebra’s adjoint action. The symmetrized invariant (86) decomposes
according to the usual momentum sum rule in quantum mechanics:
$(s_{1})\otimes(s_{2})=(s_{1}+s_{2})\oplus(s_{1}+s_{2}-1)\oplus\dots\oplus(|s_{1}-s_{2}|)$.
Apparently, the highest-weight states are integrals of motion:

$\displaystyle I^{n_{1},n_{2}}_{n_{1}+n_{2}-2k}=$
$\displaystyle\sum_{l=0}^{k}\frac{(-1)^{l}}{2}\binom{n_{1}-k+l}{l}\binom{n_{2}-l}{k-l}$
$\displaystyle\qquad\qquad\qquad\quad\quad\times\\{I_{n_{1},k-l},I_{n_{2},l}\\},$
$\displaystyle\hat{K}^{\prime}_{+}I^{n_{1},n_{2}}_{n_{1}+n_{2}-2k}=$
$\displaystyle\big{[}H_{0},I^{n_{1},n_{2}}_{n_{1}+n_{2}-2k}\big{]}=0$

where $0\leq k\leq|n_{1}-n_{2}|$. Such integral is parameterized by twice the
spin value of the conformal multiplet it generates:

$\hat{K}^{\prime}_{3}I^{2s_{1},2s_{2}}_{2s}=sI^{2s_{1},2s_{2}}_{2s}\quad\text{with}\quad
s=s_{1}+s_{2}-k.$

Here for simplicity, we omit the prime symbol in the spin’s notation (79).

The first nontrivial integral within this family corresponds to the $k=1$
case:

$I^{n_{1},n_{2}}_{n_{1}+n_{2}-2}=s_{2}\\{I_{n_{1},1},I_{n_{2}}\\}-s_{1}\\{I_{n_{1}},I_{n_{2},1}\\}.$
(87)

In case when the second multiplet is the triplet defined in Eq. (82), the
above integral of motion may be expressed also as a commutator with the
conformal spin’s square, or generalized angular Calogero Hamiltonian:

$\displaystyle I^{n,2}_{n}=$
$\displaystyle\,\\{I_{n,1},K^{\prime}_{+}\\}-2n\\{I_{n},K^{\prime}_{3}\\}$
$\displaystyle=$
$\displaystyle\,2[K^{2},I_{n}]=-\frac{1}{2}[H_{\Omega},I_{n}].$

These equations are derived easily from the expression
$K^{2}=\frac{1}{2}\\{K^{\prime}_{+},K^{\prime}_{-}\\}-K^{\prime 2}_{3}$ for
the conformal spin square and the relation (43).

###  V.5 Generalized Calogero-Moser’s integrals

The described formation of additional integrals of motion based on the
dynamical $SL(2,R)$ symmetry is even more transparent for the Calogero-Moser
model with particle exchanges. In this case, the Liouville integrals form just
by the Dunkl momentum components $\pi_{i}=-\imath\nabla_{i}$ without any
symmetrization procedure. Each component of the Dunkl operator generates a
conformal doublet ($s^{\prime}=1/2$) with

$I_{1}=\nabla_{i}\qquad\text{and}\qquad I_{1,1}=-x_{i}.$

Then according to the momentum sum rule, a pair of such multiplets generated
by two different components, $I_{1}=\nabla_{i}$ and
$I^{\prime}_{1}=\nabla_{j}$ produce a singlet ($s^{\prime}=0$). The latter
corresponds to a conserved quantity, which is nothing but the Dunkl angular
momentum component (23) as is easy to see:

$I^{1,1}_{0}=\frac{1}{2}\\{I_{1,1},I^{\prime}_{1}\\}-\frac{1}{2}\\{I_{1},I^{\prime}_{1,1}\\}=-\imath
L_{ij}.$

So, one can resume that the Dunkl angular momentum symmetry is generated from
the Dunkl momentum and the dynamical conformal symmetries.

##  VI Conclusion

It is well known that the Weyl algebra $w(N)$ generated by the creation-
annihilation operators and the symplectic algebra $sp(2N)$ formed by their
bilinear combinations generate a largest possible finite-dimensional group
which generates the spectrum of the $N$-dimensional isotropic oscillator. In
the current article, the spectrum generating algebra was deformed in case of
the presence of additional inverse square (Calogero) potential. The
construction was based on the exchange operator formalism, in which the
derivatives in observables are replaced by the Dunkl operators. In particular,
the commutation relations among the deformed generators are derived explicitly
and presented in compact form.

The Dunkl analog of the symplectic algebra contains of the deformed unitary
subalgebra which combines the symmetries of the generalized Calogero
Hamiltonian [14]. The remaining, spectrum generating part maps in-between the
different energy levels and involves the standard $sl(2,R)$ conformal
subalgebra. A simple relation is obtained between the Casimir elements of both
the conformal spin and Dunkl angular momentum, given by (modified) squares in
the corresponding generators. This correspondence has suggested to analyze of
the conformal structure of the Calogero wavefunctions in the spherical
coordinates based on the Dunkl analog of the spherical harmonics. It turned
out that, in fact, the second quantum number is driven by the conformal group,
and each eigenstate, in which it vanishes, generates the infinite dimensional
lowest-weight $sl(2,R)$ multiplet.

Next, the $\pi/2$ rotation has been applied to the conformal generators in
order to demonstrate that the $n$-th Liouville integral of motion of the
(generalized) Calogero-Moser model generates the finite-dimensional (non-
unitary) highest weight spin-$\frac{n}{2}$ $sl(2,R)$ multiplet. The
descendants of this representation have been expressed in terms of the Weyl-
ordered product in quantum field theory. The highest states in a product of
two or more such multiplets produce additional integrals of motion.

###### Acknowledgements.

The author is grateful to M. Feigin for preliminary discussions. The work was
supported by the Armenian Science Committee Grants No. 20TTWS-1C035, No.
20TTAT-QTa009, and No. 21AG-1C047.

##  Appendix A Weyl ordering

Usually in quantum field theory, the the normal and chronological orderings
are used in order to arrange the operators in a way suitable for calculations.
However, sometimes the other order types appear to be more suitable. In
particular, the Weyl ordering just symmetrizes over all possible products of
two operators with given powers. Such operators may be represented by a
particle creation/annihilation, coordinate/momentum, or any other couple of
noncommuting variables $a,b$.

For example, some simple Weyl-ordered monomials are defined as follows:

$\displaystyle\left(ab\right)_{W}={\textstyle\frac{1}{2}}(ab+ba),\quad\left(a^{2}b\right)_{W}={\textstyle\frac{1}{3}}(a^{2}b+aba+ba^{2}),$
$\displaystyle\left(a^{2}b^{2}\right)_{W}={\textstyle\frac{1}{6}}(a^{2}b^{2}+abab+baba+ab^{2}a+ba^{2}b+b^{2}a^{2}),$

with numerical factors reflecting the number of terms in the sum. It is easy
to see that in the most common case, this factor is given by an inverse
binomial coefficient:

$(a^{l}b^{k})_{W}={\textstyle\binom{k+l}{l}}^{\scriptscriptstyle-1}(a^{l}b^{k}+a^{l-1}bab^{k-1}+\dots+b^{k}a^{l}).$
(88)

The Weyl-ordered polynomials appear in the Newton’s binomial formula for
noncommutative variables $a,b$ and numerical parameter $v$:

$(a+vb)^{n}=\sum_{l=0}^{n}\binom{n}{l}v^{l}\left(a^{n-l}b^{l}\right)_{W}.$
(89)

The above formula may be considered also as a definition of Weyl order via
generating function in $v$.

Notice that the symmetrized product (88) was appeared in the Lax and Dunkl
staircase operators [23, 38].

##  Appendix B Dunkl deformed harmonic functions

In the current Appendix, the derivation of the expression (52) for the Dunkl
analog of the harmonic polynomials is briefly outlined.

First note that the radial function

$1/r^{a}\quad\text{with}\quad a=2E_{0}-2,$ (90)

where $E_{0}$ is the ground state energy of the Calogero model (14), obeys a
Dunkl harmonic equation (49), (48). In three dimensions and the absence of
particle interaction, it reduces to the usual Coulomb potential $1/r$. Next,
consider the last equation in the sequence (47) and apply its both sides to
the function (90). Using the identities

$\partialderivative{x_{i}}\frac{1}{r^{a}}=-\frac{ax_{i}}{r^{a+2}},\qquad\partial^{2}\frac{1}{r^{a}}=\frac{a(a-N+2)}{r^{a+2}},$

it is easy to calculate the deformed Laplacian’s action:

$\nabla^{\prime
2}\frac{1}{r^{a}}=\frac{a\left(a-gN(N-1)-N+2\right)}{r^{a+2}}.$

Finally note that the only nontrivial value of $a$ for which the right hand
side of above equation vanishes is given by Eq. (90).

Now fix that value of the parameter and consider the functions

$h_{\bf n}(x)=r^{a+2m}\nabla^{\prime n_{1}}_{1}\nabla^{\prime
n_{2}}_{2}\dots\nabla^{\prime n_{N}}_{N}r^{-a}$ (91)

with $m=\sum_{l}n_{l}$ as was already defined (58).

In common, the Dunkl derivative does not obey the standard Leibniz rule.
Instead, the deformed analog of the Leibniz rule takes place as is easy to
verify. Namely, any two wavefunctions $\phi(x)$ and $\psi(x)$ must satisfy the
following identity:

$\nabla^{\prime}_{i}(\phi\psi)=\phi\nabla^{\prime}_{i}\psi+\psi\nabla^{\prime}_{i}\phi\\\
-g\sum_{k\neq i}\frac{(\phi-s_{ik}\phi)(\psi-s_{ik}\psi)}{x_{i}-x_{k}},$ (92)

where the permutation operator, as usually, exchanges the corresponding
coordinates in the function argument: $s_{ij}\phi(x)=\phi(s_{ij}x)$. Notice
that the sophisticated third term vanishes if just one two functions is
symmetric, say $\phi$: $s_{ij}\phi=\phi$. In that case we arrive at the
standard Leibniz rule.

Let us choose now both functions as $\phi=r^{a+2m}$ and
$\psi=\nabla^{\prime}_{\bf n}r^{-a}$. The first one is symmetric. In order to
simplify the notations, the multiple Dunkl derivatives are abbreviated in the
second function, namely:

$\nabla^{\prime}_{\bf n}=\nabla^{\prime n_{1}}_{1}\dots\nabla^{\prime
n_{N}}_{N}.$

Then the standard Leibniz rule remains valid for the product $\phi\psi$, which
implies the following identity:

$\displaystyle\nabla^{\prime}_{i}h_{\bf
n}=(a+2m)r^{a+2m-2}x_{i}\nabla^{\prime}_{\bf
n}r^{-a}+r^{a+2m}\nabla^{\prime}_{i}\nabla^{\prime}_{\bf n}r^{-a}.$

Apply now the operator $\nabla^{\prime}_{i}$ again and sum over the index $i$.
Then employ the Leibniz rule again for the Dunkl operator, taking the power of
the radial coordinate on the left as the first function, and take into account
the identity (54). Combining together similar terms, we arrive at the
following result:

$\nabla^{\prime 2}h_{\bf n}(x)=(a+2m)r^{a+2m-2}\big{(}a+2m-2\\\ +{\bf
x}\cdot\nabla^{\prime}+\nabla^{\prime}\cdot{\bf x}\big{)}\nabla^{\prime}_{\bf
n}r^{-a}\\\ =(2m+a)(2E_{0}-2-a)r^{a+2m-2}\nabla^{\prime}_{\bf n}r^{-a},$ (93)

where the operator order is essential. In the last step, an explicit value for
the anticommutator has been applied:

$\frac{1}{2}\sum_{i}\\{x_{i},\nabla^{\prime}_{i}\\}=r\partial_{r}+E_{0}.$

The above equation is a consequence of the Dunkl-operator definitions (4),
(48) and the canonical commutation relations (5), (6).

For the proper choice of the parameter (90), the last expression in Eqs. (93)
vanishes for any set of integers $n_{i}$.

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# On the Spectra of Threshold Hypergraphs

Anirban Banerjee111Department of Mathematics and Statistics, IISER Kolkata,
Kolkata, India (anirban.banerjee@iiserkol.ac.in). Rajiv Mishra222Department of
Mathematics and Statistics, IISER Kolkata, Kolkata, India
(rm20rs017@iiserkol.ac.in). Samiron Parui333Department of Mathematics and
Statistics, IISER Kolkata, Kolkata, India<EMAIL_ADDRESS>
###### Abstract

Starting with an isolated vertex, here we construct a threshold hypergraph by
repeatedly adding an isolated vertex or a $k$-dominating vertex set. We
represent a threshold hypergraph by a string of non-negative integers and find
the Laplacian spectrum of threshold hypergraphs from their string
representation. We also compute the complete Laplacian spectrum of certain
threshold hypergraphs from the Ferrer’s diagram of their degree sequences. We
show that the Laplacian spectra of threshold hypergraphs are $r$-integral,
i.e., integral multiple of $r$, for some $r\in\mathbb{Q}$. We also construct
another class of hypergraphs whose Laplacian spectra are $r$-integral.

##  1 Introduction

In $1977$, Chvàtal and Hammer [8] introduced threshold graphs to investigate
the set packing problems. Henderson and Zalcstein [10] studied the same graphs
with a different name in $1977$ to explore the process synchronization
problems. Threshold graphs attracted many researchers as they have various
applications in psychology, computer science, scheduling theory, etc. [12].
The spectra of different connectivity matrices associated with threshold
graphs have been studied by many researchers [19, 11, 5, 2]. There are many
ways to produce threshold graphs [9, 12]. In one of the constructions, a
threshold graph can be created from an isolated vertex by repeatedly adding an
isolated vertex or a dominating vertex (a vertex that is connected to all
other vertices). Here, we have generalized this construction to define
threshold hypergraphs. Researchers have attempted to generalize other methods
of creating threshold graphs in order to introduce threshold hypergraphs [16,
9]. The threshold hypergraphs constructed by using their method are different
from ours.

A hypergraph $G$ is an ordered pair $(V,E)$ with a nonempty set $V$, called
the vertex set and a hyperedge set $E$, such that
$E\subseteq\mathcal{P}(V)\setminus\emptyset$. A hypergraph $G$ is called
$m$-uniform hypergraph if $|e|=m$ for all $e\in E$. For a vertex $v\in V$, we
define $E_{v}=\\{e\in E:v\in e\\}$ and the degree $d(v)=|E_{v}|$. For any
$k\in\mathbb{N}$, a set $S\subseteq V$ is called $k$-dominating set if $|S|=k$
and for every $v\in V\setminus S$, $S\cup\\{v\\}\in E$. Thus adding a
$k$-dominating set $S$ to a hypergraph $G(V,E)$ is to generate a new
hypergraph $G_{1}(V_{1},E_{1})$ with the vertex set $V_{1}=V\cup S$ and the
edge set $E_{1}=E\cup E_{S}$ where $E_{S}=\\{S\cup\\{v\\}:v\in V\\}$.

######  Definition 1.1.

A hypergraph $G$ is called a threshold hypergraph if it is obtained from a
single vertex by repeatedly applying any of the following steps an arbitrary
number of times:

  1. $(1)$

Add an isolated vertex: Add a new vertex to the set of vertices without
modifying the edge set.

  2. $(2)$

Apply $k$-domination: Add a $k$-dominating set to the existing hypergraph for
some $k\in\mathbb{N}$.

A threshold graph on $n$ vertices can be uniquely represented by a binary
string $s_{1}s_{2}\dots s_{n}$ where $s_{i}=1$ if at the $i$-th step we add a
dominating vertex, and $s_{i}=0$ if at the $i$-th step we add an isolated
vertex [2]. Since we start the construction with an isolated vertex, $s_{1}$
always takes the value zero. Our threshold hypergraph can be represented by a
by a string $s_{1}\,s_{2}\,\ldots\,s_{l}$ of nonnegative integers, where

$s_{j}=\begin{cases}k&\text{~{}if~{}}k\text{-domination is applied in
}j\text{-th step,}\\\ 0&\text{~{}otherwise}.\end{cases}$

Here the first term $s_{1}$ is also zero since we start with an isolated
vertex to construct a threshold hypergraph. Now we rewrite our string
concisely. We assume $m_{i}$ is the number of isolated vertices added between
the $(i-1)$-th and $i$-th domination, $k_{i}$-dominating set is added at the
$i$-th domination, and $F_{i}$ is the set of all the hyperedges added in the
$i$-th domination. Therefore, we can represent a connected threshold
hypergraph by the following string

$0^{m_{1}}\,k_{1}\,0^{m_{2}}\,k_{2}\,\ldots\,0^{m_{d}}\,k_{d}$

where $0^{m}$ denotes $m$ consecutive zeros in the string, and $d$ denotes the
total number of dominations applied to construct the threshold hypergraph.
Clearly, $m_{1}\geq 1$. For a threshold graph, the Laplacian spectrum can be
computed directly from its binary string representation [2]. In this work, one
of our main results is to show that the complete Laplacian spectrum of a
threshold hypergraph can be obtained from its string
$0^{m_{1}}\,k_{1}\,0^{m_{2}}\,k_{2}\,\ldots\,0^{m_{d}}\,k_{d}$. This is
addressed in 1.2.

######  Theorem 1.2.

Let $G=0^{m_{1}}\,k_{1}\,0^{m_{2}}\,k_{2}\,\ldots\,0^{m_{d}}\,k_{d}$ be a
threshold hypergraph. Then the eigenvalues of $L_{G}$ are as follows

  1. $(i)$

$0$ with multiplicity $1$;

  2. $(ii)$

$d-i+1$ with multiplicity $m_{i}$, for each $i=2,\ldots,d$;

  3. $(iii)$

$d$ with multiplicity $m_{1}-1$;

  4. $(iv)$

$\frac{1}{k_{i}}\left(\sum\limits_{j=1}^{i}m_{j}+\sum\limits_{j=1}^{i}k_{j}\right)+d-i$
with multiplicity $1$, for each $i=1,\ldots,d$;

  5. $(v)$

$\frac{k_{i}+1}{k_{i}}\left(\sum\limits_{j=1}^{i}m_{j}+\sum\limits_{j=1}^{i-1}k_{j}\right)+d-i$
with multiplicity $k_{i}-1$, for each $i=1,\ldots,d$.

We prove 1.2 in Section 2.

The Laplacian spectrum of a threshold graph can also be computed from the
Ferrer’s diagram of its degree sequence [4].

######  Definition 1.3 (Ferrer’s diagram).

[4, p. $148$] Suppose that $S=\\{a_{1},a_{2},\ldots,a_{n}\\}$ is a finite
collection of non-negative integers such that $a_{i}\geq a_{i+1}$ for all
$i=1,\dots,n-1$. The Ferrer’s diagram of $S$ is a diagram consisting of $n$
rows of boxes such that

  1. $(1)$

the rows are left-justified, that is, start from the left, and

  2. $(2)$

the $i$-th row $($from above$)$ contains $a_{i}$ number of boxes.

In Ferrer’s diagram, a block is a collection of all the rows with the same
number of boxes. The number of rows within a block is called the height of the
block. The width of a block is the number of boxes in a row in that block (see
Figure 1).

######  Example 1.4.

Let $G$ be a threshold hypergraph whose hyperedge set $E$ is as follows

$\displaystyle E$ $\displaystyle=$
$\displaystyle\\{\\{1,2,3,4\\},\\{1,6,7,8\\},\\{2,6,7,8\\},\\{3,6,7,8\\},\\{4,6,7,8\\},\\{5,6,7,8\\},$
$\displaystyle\\{1,11,12,13\\},\\{2,11,12,13\\},\\{3,11,12,13\\},\\{4,11,12,13\\},\\{5,11,12,13\\},\\{6,11,12,13\\},$
$\displaystyle\\{7,11,12,13\\},\\{8,11,12,13\\},\\{9,11,12,13\\},\\{10,11,12,13\\},\\{1,14,15,16\\},\\{2,14,15,16\\},$
$\displaystyle\\{3,14,15,16\\},\\{4,14,15,16\\},\\{5,14,15,16\\},\\{6,14,15,16\\},\\{7,14,15,16\\},\\{8,14,15,16\\},$
$\displaystyle\\{9,14,15,16\\},\\{10,14,15,16\\},\\{11,14,15,16\\},\\{12,14,15,16\\},\\{13,14,15,16\\}\\};$

After arranging the degree sequence in decreasing order, we have

$\\{13,13,13,11,11,11,7,7,7,4,4,4,4,3,2,2\\}$

The Ferrer’s diagram of the degree sequence is shown in Figure 1.

$\\}$$k$$D_{1}=13$$D_{2}=11$$D_{3}=7$$D_{4}=4$16161413999666633

Figure 1: Ferrer’s diagram of the degree sequence
$\\{13,13,13,11,11,11,7,7,7,4,4,4,4,3,2,2\\}$ of a uniform $4$-threshold
hypergraph, where $D_{i}$ is the width of $i$-th block from the top.

We also find the complete Laplacian spectrum of certain threshold hypergraphs
from the Ferrer’s diagram of their degree sequences. A threshold graph is a
threshold hypergraph where each domination is $1$-domination. A threshold
hypergraph is said to be a $k$-threshold hypergraph if each domination applied
is a $k$-domination. Another focus of our work is to find the complete
Laplacian spectrum of a $k$-threshold hypergraph from the Ferrer’s diagram of
its degree sequence. This is described in the following theorem which is
proved in Section 2.

######  Theorem 1.5.

Let $d_{1}\geq d_{2}\geq\ldots\geq d_{n}$ be the degree sequence of a
$k$-threshold hypergraph $G$. Then the complete list of eigenvalues of $L_{G}$
is as follows.

  1. $(i)$

$0$ with multiplicity $1$;

  2. $(ii)$

$\frac{1}{k}C_{D_{i}}$ with multiplicity $D_{i}-D_{i+1}-k$ for all
$i=1,\ldots,d-1$;

  3. $(iii)$

$\frac{1}{k}C_{D_{d}}$ with multiplicity $D_{d}-D_{d+1}$ if $D_{d}\neq d$;

  4. $(iv)$

$\frac{1}{k}C_{i}$ with multiplicity multiplicity $1$, for all $i=1,\ldots,d$;

  5. $(v)$

$\frac{1}{k}[(k+1)D_{i}-i+1]$ with multiplicity $k-1$ for all $i=1,\ldots d$.

Here $D_{i}$ and $C_{i}$ are the width of $i$-th block from above and the
$i$-th column sum from the left, respectively, in the Ferrer’s diagram of the
degree sequence, and $d=\min\\{s\in\mathbb{N}:D_{s}<s\\}-1$.

A quest in spectral graph theory is: what are the graphs for which all the
eigenvalues of the associated Laplacian are integers, and what can we say
about the structure of those graphs? Till now, this is partially answered (see
[14, 15, 20] for some progress in this direction). This question is also open
for hypergraph. Here, we show that the Laplacian spectra of threshold
hypergraphs are integral multiple of $r$, for some $r\in\mathbb{Q}$. We also
derive another class of hypergraphs whose Laplacian spectra have the same
property. Section 3 is dedicated to this.

##  2 Spectrum of Threshold Hypergraphs

Let $G(V,E)$ be a finite hypergraph, that is, $V$ is finite and let $|e|\geq
2$ for all $e\in E$. Let $\mathbb{R}^{V}$ denote the set of all real-valued
functions on $V$. For $G$, the Laplacian
$L_{G}:\mathbb{R}^{V}\to\mathbb{R}^{V}$ is defined as

$(L_{G}f)(v)=\sum\limits_{e\in E_{v}}\frac{1}{|e|-1}\sum\limits_{u\in
e}(f(v)-f(u))$

for all $v\in V$. The eigenvalues of $L_{G}$ are real and we order the
eigenvalues of $L_{G}$ as
$0=\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{|V|}$, thus
$\lambda_{2}>0$ if the hypergraph is connected [1]. Now to prove 1.2, we prove
the following lemma.

######  Lemma 2.1.

Let $G(V,E)$ be a hypergraph and
$\lambda_{1}=0,\lambda_{2},\ldots,\lambda_{|V|}$ be the eigenvalues of
$L_{G}$. Suppose $G_{1}(V_{1},E_{1})$ is a hypergraph obtained by applying
$k$-domination on $G$ for some $k\in\mathbb{N}$. Then the eigenvalues of
$L_{G_{1}}$ are given by

  1. $(i)$

$0$ with multiplicity $1$;

  2. $(ii)$

$\lambda_{i}+1$ with multiplicity $1$ for $i=2,\ldots,|V|$;

  3. $(iii)$

$\frac{1}{k}(|V|+k)$ with multiplicity $1$;

  4. $(iv)$

$\frac{k+1}{k}|V|$ with multiplicity $k-1$.

###### Proof.

It is clear that $0$ is an eigenvalue of $L_{G_{1}}$ with eigenvector
$\chi_{|V_{1}|}$. Suppose $x_{i}\in\mathbb{R}^{|V|}$ is an eigenvector
corresponding to $\lambda_{i}$ of $L_{G}$ for $i=2,\ldots,|V|$. Now for
$i=2,3,\ldots,|V|$, define $\bar{x}_{i}\in\mathbb{R}^{|V_{1}|}$ as,

$\bar{x}_{i}(v)=\begin{cases}x_{i}(v)&\text{ if }v\in V\\\ 0&\text{
otherwise}\end{cases}$

Now note that $V_{1}=V\cup S$, where $S$ is the set of vertices added in the
domination, $|S|=k$ and $E_{1}=E\cup E_{S}$, where $E_{S}=\\{S\cup\\{u\\}:u\in
V\\}$.

Now for $v\in V$, $E_{v}=E_{v}\cap E\cup\\{S\cup\\{v\\}\\}$, thus

$\displaystyle(L_{G_{1}}\bar{x_{i}})(v)$ $\displaystyle=$
$\displaystyle\sum\limits_{e\in E_{v}\cap E}\frac{1}{|e|-1}\sum\limits_{u\in
e}\big{(}\bar{x}_{i}(v)-\bar{x}_{i}(u)\big{)}+\frac{1}{k}\sum\limits_{u\in
S}\big{(}\bar{x}_{i}(v)-\bar{x}_{i}(u)\big{)}$ $\displaystyle=$
$\displaystyle\lambda_{i}\bar{x}_{i}(v)+\frac{1}{k}k\bar{x}_{i}(v)$
$\displaystyle=$ $\displaystyle\big{(}\lambda_{i}+1\big{)}\bar{x}_{i}(v).$

Now for $v\in S$, $E_{v}=E_{S}$, thus

$\displaystyle(L_{G_{1}}\bar{x_{i}})(v)$ $\displaystyle=$
$\displaystyle\sum\limits_{e\in E_{S}}\frac{1}{|e|-1}\sum\limits_{u\in
e}\big{(}\bar{x}_{i}(v)-\bar{x}_{i}(u)\big{)}$ $\displaystyle=$
$\displaystyle-\frac{1}{k}\sum\limits_{u\in V}\bar{x}_{i}(u)$ $\displaystyle=$
$\displaystyle 0$

Therefore $\lambda_{i}+1$ is an eigenvalue of $L_{G_{1}}$ for
$i=2,\ldots,|V|$.

Now suppose $S=\\{v_{1},\ldots,v_{k}\\}$, and for $i=2,\ldots,k$, consider the
vectors $y_{i}\in\mathbb{R}^{|V_{1}|}$ defined by

$y_{i}(v)=\begin{cases}1&\text{ if }v=v_{1}\\\ -1&\text{ if }v=v_{i}\\\
0&\text{ otherwise.}\end{cases}$

Now for $v\in V$,

$\big{(}L_{G_{1}}y_{i}\big{)}(v)=\sum\limits_{e\in E_{v}\cap
E}\frac{1}{|e|-1}\sum\limits_{u\in
e}\big{(}y_{i}(v)-y_{i}(u)\big{)}+\frac{1}{k}\sum\limits_{u\in
S}\big{(}y_{i}(v)-y_{i}(u)\big{)}=0$

and for $v\in S$

$\displaystyle(L_{G_{1}}y_{i})(v)$ $\displaystyle=$
$\displaystyle\sum\limits_{e\in E_{S}}\frac{1}{|e|-1}\sum\limits_{u\in
e}\big{(}y_{i}(v)-y_{i}(u)\big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{k}\sum\limits_{e\in E_{S}}\sum\limits_{u\in
e}y_{i}(v)-y_{i}(u)$ $\displaystyle=$ $\displaystyle\frac{k+1}{k}|V|y_{i}(v)$

Therefore $\frac{k+1}{k}|V|$ is an eigenvalue of $L_{G_{1}}$ with multiplicity
$k-1$.

Again define $y\in\mathbb{R}^{|V_{1}|}$ by

$y(v)=\begin{cases}-\frac{k}{|V|}&\text{ if }v\in V\\\ 1&\text{ if }v\in
S\end{cases}$

Now for $v\in V$,

$\displaystyle\big{(}L_{G_{1}}y\big{)}(v)$ $\displaystyle=$
$\displaystyle\sum\limits_{e\in E_{v}\cap E}\frac{1}{|e|-1}\sum\limits_{u\in
e}\big{(}y(v)-y(u)\big{)}+\frac{1}{k}\sum\limits_{u\in
S}\big{(}y(v)-y(u)\big{)}$ $\displaystyle=$ $\displaystyle
0+\frac{1}{k}k\Big{(}-\frac{k}{|V|}-1\Big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{k}(|V|+k)y(v)$

and for $v\in S$

$\displaystyle(L_{G_{1}}y)(v)$ $\displaystyle=$
$\displaystyle\sum\limits_{e\in E_{S}}\frac{1}{|e|-1}\sum\limits_{u\in
e}\big{(}y(v)-y(u)\big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{k}\sum\limits_{e\in
E_{S}}\Big{(}1+\frac{k}{|V|}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{k}(|V|+k)y(v)$

Therefore $\frac{1}{k}(|V|+k)$ is an eigenvalue of $L_{G_{1}}$ with
multiplicity $1$. ∎

Now we are ready to prove 1.2.

Proof of the 1.2

###### Proof.

We use induction on the number of dominations, $d$ to prove the result. The
case $d=1$ follows from 2.1. We assume that the statement is true for $d=n$,
as the induction hypothesis. Suppose that, for $d=n$, the threshold hypergraph
is $G_{n}$. Let $H_{n}$ be the hypergraph obtained after adding all the
isolated vertices just after $d=n$ and before $d=n+1$. The Laplacian spectrum
of $H_{n}$ is the same as the Laplacian spectrum of $G_{n}$ but with an
additional $m_{n+1}$ number of zeros. Now apply $k_{n+1}$-domination on
$H_{n}$ and from 2.1, we find the statement is also true for $d=n+1$. ∎

In a $k$-threshold hypergraph, $k_{1}=k_{2}=\ldots k_{d}=k$. Thus a connected
$k$-threshold hypergraph is a $(k+1)$-uniform hypergraph. By taking $k_{i}=k$
for each $i=1,2,\ldots,d$ in 1.2, we can also obtain the complete spectra of
$k$-threshold hypergraph.

######  Corollary 2.2.

Let $G=0^{m_{1}}\,k\,0^{m_{2}}\,k\,\ldots\,0^{m_{d}}\,k$ be a $k$-threshold
hypergraph. Then the Laplacian spectrum of $G$ is given by

  1. $(i)$

0 with multiplicity 1;

  2. $(ii)$

$d-i+1$ with multiplicity $m_{i}$, for each $i=2,3,\ldots,d$;

  3. $(iii)$

$d$ with multiplicity $m_{1}-1$;

  4. $(iv)$

$\frac{1}{k}\sum\limits_{j=1}^{i}m_{j}+d$ with multiplicity 1, for each
$i=1,2,\ldots,d$;

  5. $(v)$

$\frac{k+1}{k}\sum\limits_{j=1}^{i}m_{j}+k(i-1)+d-1$ with multiplicity $k-1$,
for each $i=1,2,\ldots,d$.

As we have mentioned before the above spectrum can be directly computed only
from the degree sequence of the $k$-threshold hypergraph by using the Ferrer’s
diagram. Now we show the same in the proof of the 1.5.

Proof of the 1.5

###### Proof.

Suppose that $0^{m_{1}}\,k\,0^{m_{2}}\,k\,\ldots\,0^{m_{d}}\,k$ be a
$k$-threshold hypergraph and $d_{1}\geq d_{2}\geq\ldots\geq d_{n}$ be its
degree sequence. Let $D_{i}$ denote the width of $i$-th block from the top in
Ferrer’s diagram of the degree sequence. The total number of dominations, $d$,
is given by

$d=\min\\{s\in\mathbb{N}:D_{s}<s\\}-1$

since $D_{d}\geq d$ and $D_{d+i}<d+1,i\in\mathbb{N}$. If $u$ be any vertex
added in $i$-th domination then the degree of $u$ is given by,

$d(u)=\sum\limits_{j=1}^{i}m_{j}+k(i-1)+(d-i).$

Therefore

$D_{i}=\sum\limits_{j=1}^{d-i+1}m_{j}+k(d-i)+i-1\text{ for all
}i=1,2,\ldots,d.$

Moreover, we have

$m_{d-i+1}=D_{i}-D_{i+1}-k+1\text{ for all }i=1,2,\ldots d-1,$

and

$m_{1}=\begin{cases}1&\text{ if }D_{d}=d,\\\ D_{d}-D_{d+1}+1&\text{
otherwise.}\end{cases}$

Now as there is $D_{1}$ number of columns in the Ferrer’s diagram. Let $C_{i}$
denote the sum of all the boxes in the $i$-th column, for
$i=1,2,\ldots,D_{1}$. Thus

$C_{i}=\sum\limits_{j=1}^{d-i+1}m_{j}+kd\text{ for all }i=1,2,\ldots,d;$

and

$C_{D_{d-i+1}}=k(d-i+1)\text{ for all }i=2,3,\ldots,d$

moreover

$C_{D_{d}}=kd\text{ if }m_{1}>1.$

Therefore the proof follows from the 2.2. ∎

######  Example 2.3.

Let $G$ be a $3$-threshold hypergraph described in 1.4. Using 1.5, the
complete spectrum of $L_{G}$ is given by $\\{\frac{1}{3}x:x\in X\\}$ where

$X=\\{0,6,6,9,13,13,13,14,16,16,26,26,43,43,52,52\\}.$

Note that $1$-threshold hypergraph becomes a threshold graph, and thus the
result obtained for a $k$-threshold hypergraph holds for a threshold graph.

Now, we illustrate the computation of the result of 1.5 in the Algorithm 1.

Algorithm 1 Algorithm for computing the Laplacian spectrum of a $k$-threshold
hypergraph $G$ when $d\geq 2$.

$d_{1}\geq d_{2}\geq\ldots\geq d_{n}$ $\triangleright$ Degree sequence in non-
increasing order

$\triangleright$ Draw the Ferrer’s Diagram

$k\leftarrow\text{height of the first block }$.

Set $s=$ Total number of blocks.

while $D_{s}<s$ do

$s\leftarrow s-1$

end while

$d\leftarrow s$$\triangleright$ $d$ is the number of dominations

$D_{j}\leftarrow\text{width of }j^{th}\text{ block from the top }$
$\triangleright$ $j=1,2,\ldots,d.$

$C_{j}\leftarrow j^{th}\text{ column sum}$ $\triangleright$
$j=1,2,\ldots,d_{1}.$

for $j=1:d$ do

$\Gamma_{j}\leftarrow\frac{1}{k}((k+1)D_{j}-j+1);$

$\Gamma_{j}$ is an eigenvalue with multiplicity $k-1$.

end for

for $j=1:d$ do

$\frac{1}{k}\,C_{j}$ is an eigenvalue with multiplicity 1.

end for

for $j=1:d-1$ do

$\frac{1}{k}\,C_{D_{j}}$ is an eigenvalue with multiplicity $D_{j}-D_{j+1}-k$.

end for

if $D_{d}\neq d$ then

$\frac{1}{k}C_{D_{d}}$ is an eigenvalue with multiplicity $D_{d}-d$.

end if

##  3 Integral Hypergraphs

The Laplacian $L_{G}$ of a (hyper)graph $G$ is said to be $r$-integral if
there exists $r\in\mathbb{R}$ such that all the eigenvalues of $L_{G}$ are the
integral-multiple of $r$, that is, all the eigenvalues are of the form $rp$
where $p\in\mathbb{Z}$. The graphs for which the Laplacian is $1$-integral
(Laplacian integral graphs) have been well studied in the past [14, 15, 20].
Here, we derive some classes of hypergraphs for which the associated
Laplacians are $r$-integral. If for a hypergraph, the associated Laplacian is
$r$-integral then we say that the hypergraph is a Laplacian $r$-integral
hypergraph.

Let $G(V,E)$ be an $m$-uniform hypergraph on $n$ vertices and let
$\kappa_{m}^{n}$ denote the set of all $m$ element subsets of $V$. The
complement of $G$, denoted by $\overline{G}$, is the hypergraph with the same
vertex set $V$ and the edge set $\kappa_{m}^{n}\setminus E$. We can check that
(see [1]) if $G$ is an $m$-uniform hypergraph on $n$ vertices and
$0=\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}$ are the eigenvalues of
$L_{G}$ then the eigenvalues of $L_{\overline{G}}$ are
$0=\lambda_{1}\leq\phi_{m}(n)-\lambda_{n}\leq\ldots\leq\phi_{m}(n)-\lambda_{2}$,
where $\phi_{m}(n)=\frac{n}{n-1}\binom{n-1}{m-1}.$

######  Proposition 3.1.

Every threshold hypergraph is Laplacian $r$-integral for some
$r\in\mathbb{Q}$.

###### Proof.

Proof directly follows from the 1.2. ∎

Now we define another class of hypergraphs where each hypergraph in that class
is Laplacian $r$-integral for some $r\in\mathbb{Q}$.

######  Definition 3.2 (Co-$k$-threshold hypergraph).

A hypergraph is called a co-$k$-threshold hypergraph if it is constructed
using the following rules recursively:

  * $(i)$

Any $k$-threshold hypergraph is a co-$k$-threshold hypergraph.

  * $(ii)$

The complement of a co-$k$-threshold hypergraph is a co-$k$-threshold
hypergraph.

  * $(iii)$

The union of two vertex-disjoint co-$k$-threshold hypergraphs is a
co-$k$-threshold hypergraph.

The converse of the point $(i)$ in the 3.2 is not true. As an example, for any
two $k$-threshold hypergraphs $G_{1}$ and $G_{2}$, the hypergraph
$\overline{G_{1}\cup G_{2}}$ is a co-$k$-threshold hypergraph but not a
$k$-threshold hypergraph.

######  Theorem 3.3.

Any Co-$k$-threshold hypergraph is Laplacian $r$-integral for some
$r\in\mathbb{Q}$.

###### Proof.

From 3.1, we can say that any $k$-threshold hypergraph is $r$-integral for
some rational $r$. Note that we can assume that $r=\frac{1}{q}$ for some
$q\in\mathbb{N}$. Also, if $G$ be any $r$-integral hypergraph on $n$ vertices,
where $r=\frac{1}{q}$ for some $q\in\mathbb{N}$, then $\overline{G}$ is
$\frac{r}{n-1}$ -integral. Also, if $G_{1}$ be Laplacian
$\frac{1}{q_{1}}$-integral and $G_{2}$ be Laplacian $\frac{1}{q_{2}}$-integral
hypergraphs, then their union is Laplacian $\frac{1}{q_{1}q_{2}}$-integral.
Thus the result follows. ∎

## Acknowledgement

The work of Samiron Parui is supported by University Grant Commission, India
[Beneficiary Code: BININ00965055 A]. The work of Rajiv Mishra is supported by
the Council of Scientific$\And$Industrial Research, India[File number:
09/921(0347)/2021-EMR-I].

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