${\cal PT}$ deformation of angular Calogero models

TL;DR

This paper explores ${ m PT}$-symmetric deformations of angular Calogero models, revealing how such transformations affect the potential, energy spectrum, and symmetries, especially for specific Coxeter systems.

Contribution

It introduces a ${ m PT}$-symmetric deformation method for angular Calogero models, showing how to modify potentials and analyze spectral and symmetry properties for various Coxeter systems.

Findings

${ m PT}$ deformation preserves energy eigenvalues.

Deformation can add unphysical states and double degeneracy.

For integral couplings, energy levels coincide, enabling nonlinear supersymmetry.

Abstract

The rational Calogero model based on an arbitrary rank-$n$ Coxeter root system is spherically reduced to a superintegrable angular model of a particle moving on $S^{n-1}$ subject to a very particular potential singular at the reflection hyperplanes. It is outlined how to find conserved charges and to construct intertwining operators. We deform these models in a ${\cal PT}$-symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The ${\cal PT}$ deformation does not change the energy eigenvalues but in some cases adds a previously unphysical tower of states. For integral couplings the new and old energy levels coincide, which roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry charge. We present the details for the generic rank-two ($A_2$, $G_2$) and all rank-three Coxeter systems ($AD_3$, $BC_3$ and $H_3$), including a reducible case ($A_1^{\otimes 3}$).

Equations

$$\sum _{i=1}^n(x^i{)}^2=:r^2.$$

$$H=\frac{1}{2}{L}^2+U\mathrm{with}{L}^2=-\sum _{i<j}(x^i{\partial }_j-x^j{\partial }_i{)}^2\mathrm{and}U=r^2\sum _{\alpha \in {\mathcal{R}}_{+}}\frac{g_{\alpha }(g_{\alpha }-1)\alpha \cdot \alpha }{2(\alpha \cdot x{)}^2}$$

$$H{v}_{\{\ell \}}=E_{\ell }{v}_{\{\ell \}}\mathrm{with}\{\ell \}=({\ell }_3,{\ell }_4,\dots ,{\ell }_{n+1})\mathrm{and}\ell =d_3{\ell }_3+d_4{\ell }_4+\cdots +d_{n+1}{\ell }_{n+1}$$

$$E_{\ell }=\frac{1}{2}q(q+n-2)\mathrm{with}q=\ell +\sum _{\alpha \in {\mathcal{R}}_{+}}{g}_{\alpha }.$$

$$v^{(g)}_{\{\ell\}}\ =\ r^{-q}\,{\Delta\vphantom{\big{|}}}^{g}\,h^{(g)}_{\{\ell\}}(x)\quad{\quad{\rm with}\quad}\quad{\Delta\vphantom{\big{|}}}^{g}=\prod_{\alpha\in{\cal R}_{+}}(\alpha\cdot x)^{g_{\alpha}}$$

$$h^{(g)}_{\{\ell\}}(x)\ =\ r^{n-2+2q}\,\biggl{(}\prod_{\mu=3}^{n+1}\sigma_{\mu}\bigl{(}\{\widetilde{{\cal D}_{i}}\}\bigr{)}^{\ell_{\mu}}\biggr{)}\,r^{2-n-2(q-\ell)}\ ,$$

$$\widetilde{{\cal D}}_{i}\ =\ \partial_{i}\ +\ \sum_{\alpha\in{\cal R}_{+}}\frac{g_{\alpha}\,\alpha_{i}}{\alpha\cdot x}\,(1-s_{\alpha})\ =\ {\Delta\vphantom{\big{|}}}^{-g}\,{\cal D}_{i}\,{\Delta\vphantom{\big{|}}}^{g}$$

$${\cal D}_{i}\ =\ {\Delta\vphantom{\big{|}}}^{g}\,\widetilde{{\cal D}}_{i}\,{\Delta\vphantom{\big{|}}}^{-g}\ =\ \partial_{i}\ -\ \sum_{\alpha\in{\cal R}_{+}}\frac{g_{\alpha}\,\alpha_{i}}{\alpha\cdot x}\,s_{\alpha}\ .$$

$$h_{\{0\}}^{(g)}\ =\ 1\qquad\Longrightarrow\qquad v_{\{0\}}^{(g)}\ =\ r^{-q}{\Delta\vphantom{\big{|}}}^{g}\ =\ \prod_{\alpha\in{\cal R}_{+}}\bigl{(}\alpha\cdot{\textstyle\frac{x}{r}}\bigr{)}^{g_{\alpha}}\ ,$$

$${\mathcal{L}}_{ij}=x^i{\mathcal{D}}_j-x^j{\mathcal{D}}_i,$$

$$-\frac{1}{2}\sum _{i<j}{\mathcal{L}}_{ij}^2=\mathcal{H}-\frac{1}{2}S(S+n-2)$$

$$\mathcal{H}=\frac{1}{2}{L}^2+r^2\sum _{\alpha \in {\mathcal{R}}_{+}}\frac{\alpha \cdot \alpha /2}{(\alpha \cdot x{)}^2}{g}_{\alpha }(g_{\alpha }-s_{\alpha })\mathrm{and}S=\sum _{\alpha }{g}_{\alpha }{s}_{\alpha }.$$

$$-{{\textstyle\frac{1}{2}}}\textrm{res}\Bigl{(}\sum_{i<j}{\cal L}_{ij}^{2}\Bigr{)}\ =\ \textrm{res}({\cal H})\ -\ {{\textstyle\frac{1}{2}}}\sum_{\alpha}g_{\alpha}\Bigl{(}\sum_{\alpha}g_{\alpha}+n{-}2\Bigr{)}\ =\ H\ -\ E_{0}\ .$$

$$\mathcal{C}\text{Weyl invariant}⟹[C,H]=0\mathrm{for}C=\text{res}(\mathcal{C}).$$

$${\mathcal{R}}_{+}={\mathcal{R}}^{\prime }\cup {\mathcal{R}}^{\prime \prime },$$

$$\textrm{res}\bigl{(}{\cal M}\,{\cal H}^{(g^{\prime},g^{\prime\prime})}\bigr{)}\ =\ M\,H^{(g^{\prime},g^{\prime\prime})}\qquad\textrm{but}\qquad\textrm{res}\bigl{(}{\cal H}^{(g^{\prime},g^{\prime\prime})}\,{\cal M}\bigr{)}\ =\ H^{(g^{\prime}+1,g^{\prime\prime})}\,M$$

$${\cal M}\quad\begin{cases}{\cal R}^{\prime}\ \textrm{antiinvariant}\\ {\cal R}^{\prime\prime}\ \textrm{invariant}\end{cases}\!\!\!\!\!\!\Bigg{\}}\qquad\Longrightarrow\qquad M\,H^{(g^{\prime},g^{\prime\prime})}\ =\ H^{(g^{\prime}+1,g^{\prime\prime})}\,M\ .$$

$$M{v}_{\{\ell \}}^{(g^{\prime },g^{\prime \prime })}=\sum _{\genfrac{}{}{0ex}{}{\{{\ell }^{\prime }\}}{{\ell }^{\prime }=\ell -|{\mathcal{R}}^{\prime }|}}{c}_{\{{\ell }^{\prime }\}}^{\{\ell \}}{v}_{\{{\ell }^{\prime }\}}^{(g^{\prime }+1,g^{\prime \prime })}$$

$$\widetilde{M}\,h_{\{\ell\}}^{(g^{\prime},g^{\prime\prime})}\ =\ \smash{\sum_{\{\ell^{\prime}\}\atop\ell^{\prime}=\ell-|{\cal R}^{\prime}|}}c^{\{\ell\}}_{\{\ell^{\prime}\}}\ {\Delta\vphantom{\big{|}}}\,h_{\{\ell^{\prime}\}}^{(g^{\prime}+1,g^{\prime\prime})}\quad{\quad{\rm with}\quad}\widetilde{M}\ =\ {\Delta\vphantom{\big{|}}}^{-g}M\,{\Delta\vphantom{\big{|}}}^{g}\ .$$

$$\Gamma (ϵ):{\mathbb{R}}^n\to {\mathbb{C}}^n\mathrm{with}\Gamma (0)=\text{id}$$

$$\Gamma (ϵ{)}^{\top }=\Gamma (ϵ{)}^{-1}.$$

$$\Gamma(\epsilon)\ =\ \exp\Bigl{\{}\sum_{i<j}\epsilon_{ij}G_{ij}\Bigr{\}}\quad{\quad{\rm with}\quad}\quad G_{ij}:x^{k}\ \mapsto\ {\mathrm{i}}\bigl{(}\delta^{kj}x^{i}-\delta^{ki}x^{j}\bigr{)}\ ,$$

$$\Gamma (ϵ{)}^{\ast }=\Gamma (ϵ{)}^{\top }=\Gamma (-ϵ).$$

$$(x^1,x^2,\dots ,x^n{)}^{\top }=x\mapsto \Gamma (ϵ)x=:x(ϵ),$$

$$\alpha \cdot x\mapsto \alpha \cdot \Gamma (ϵ)x=\Gamma (ϵ{)}^{\top }\alpha \cdot x,$$

$$v_{\{\ell\}\,\epsilon}^{(g)}\ =\ r^{-q}\,{\Delta\vphantom{\big{|}}}_{\epsilon}^{g}\,h_{\{\ell\}\,\epsilon}^{(g)}\quad{\quad{\rm with}\quad}\quad{\Delta\vphantom{\big{|}}}_{\epsilon}^{g}=\prod_{\alpha\in{\cal R}_{+}}\bigl{(}\alpha\cdot x(\epsilon)\bigr{)}^{g_{\alpha}}{\quad{\rm and}\quad}h_{\{\ell\}\,\epsilon}^{(g)}(x)=h_{\{\ell\}}^{(g)}\bigl{(}x(\epsilon)\bigr{)}\ .$$

$$\frac{\mathcal{P}\Gamma (ϵ)=s\Gamma (ϵ)s\text{buildrel}!}{=\Gamma (-ϵ)=\Gamma (ϵ{)}^{\ast }=\mathcal{T}\Gamma (ϵ).}$$

$$\{s,ϵ:G\}=0\iff s(ϵ:G)s=-ϵ:G\iff P_{\pm }(ϵ:G)P_{\pm }=0$$

$$P_{-}=\frac{1}{2}(1-s)\mathrm{and}{P}_{+}=\frac{1}{2}(1+s)$$

$$\textrm{rank}(\epsilon{:}G)\ =\ \textrm{min}\bigl{(}2\,\textrm{rank}(P_{-}),2\,\textrm{rank}(P_{+})\bigr{)}\ .$$

Full text

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${\cal PT}$** deformation of angular Calogero models **

Francisco Correa*+* and  Olaf Lechtenfeld*×*

+*Instituto de Ciencias Físicas y Matemáticas

Universidad Austral de Chile, Casilla 567, Valdivia, Chile

*Email: francisco.correa@uach.cl

×*Institut für Theoretische Physik and Riemann Center for Geometry and Physics

Leibniz Universität Hannover

Appelstraße 2, 30167 Hannover, Germany

*Email: olaf.lechtenfeld@itp.uni-hannover.de

The rational Calogero model based on an arbitrary rank-$n$ Coxeter root system is spherically reduced to a superintegrable angular model of a particle moving on $S^{n-1}$ subject to a very particular potential singular at the reflection hyperplanes. It is outlined how to find conserved charges and to construct intertwining operators. We deform these models in a ${\cal PT}$-symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The ${\cal PT}$ deformation does not change the energy eigenvalues but in some cases adds a previously unphysical tower of states. For integral couplings the new and old energy levels coincide, which roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry charge. We present the details for the generic rank-two ($A_{2}$, $G_{2}$) and all rank-three Coxeter systems ($AD_{3}$, $BC_{3}$ and $H_{3}$), including a reducible case ($A_{1}^{\otimes 3}$).

1 Introduction and summary

The rational Calogero model (for a review, see [1]) generalizes to any root system of a (finite-dimensional) Lie algebra or, better, to any Coxeter root system. Given such a system of rank $n$, it describes a conformal particle moving in $\mathbb{R}^{n}$ under the influence of a very special potential. Since this potential has a universal inverse-square radial dependence and otherwise depends only on the angular coordinates (of $S^{n-1}$), a spherical reduction to its angular subsystem, the angular Calogero model, is natural. Like the full model on $\mathbb{R}^{n}$, the reduced dynamics on $S^{n-1}$ is superintegrable, so that it enjoys $2n{-}3$ integrals of motion, which are however not in involution. Recently, the angular models have been analyzed in some detail, both classically and quantum mechanically [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

It has been known for a long time that hermiticity is not an essential feature of a Hamiltonian for its spectrum to be real. For instance, it suffices that the Hamiltonian commutes with an antilinear involution (one example is provided by the $\cal{PT}$ operator where ${\cal P}$ correspond to the parity operator and ${\cal T}$ the time reversal operator) which also leaves the eigenfunctions invariant (“unbroken $\cal{PT}$ symmetry”) [13]. Such a non-hermitian Hamiltonian is related to a hermitian one by a (non-unitary) similarity transformation, which may be impossibly complicated. Often, however, there exists a family $H_{\epsilon}$ of non-hermitian $\cal{PT}$-invariant Hamiltonians representing a smooth deformation of a hermitian $H_{0}$. In this case we speak of a “$\cal{PT}$ deformation”, with the parameter $\epsilon$ measuring the deviation from hermiticity. For rational Calogero models, a particularly nice set of $\cal{PT}$ deformations can be generated by a specific complex orthogonal deformation of the coordinates in the expression for the Hamiltonian. If such a $\cal{PT}$ deformation is in accordance with the Coxeter reflection symmetry of the system, integrability will be preserved. This kind of $\cal{PT}$ deformation has been applied to the full rational Calogero model about ten years ago by Fring and Znojil [14], and corresponding complex root systems were constructed by Fring and Smith thereafter [15, 16, 17]. For a review of $\cal{PT}$ deformations of integrable models, see [18].

It is worth recalling the relevant part (for this paper) of the Calogero model’s long history:

• •

1971  Calogero [19]:

Solution of the one-dim’l N-body problem with $\ldots$ inversely quadratic pair potentials

• •

1981  Olshanetsky & Perelomov [20, 21]:

Classical integrable finite-dimensional systems related to Lie algebras (1983: quantum)

• •

1983  Wojciechowski [22]:

Superintegrability of the Calogero–Moser system

• •

1989  Dunkl [23]:

Differential-difference operators associated to reflection groups

• •

1990  Chalykh & Veselov [24]:

Commutative rings of partial differential operators and Lie algebras, supercompleteness

• •

1991  Heckman [25]:

Elementary construction for commuting charges and intertwiners (shift operators)

• •

2003  M. Feigin [2]:

Intertwining relations for the spherical parts of generalized Calogero operators

• •

2008  A. Fring, M. Znojil [14]:

${\cal PT}$-symmetric deformations of Calogero models

• •

2008  Hakobyan, Nersessian, Yeghikyan [3]:

The cuboctahedric Higgs oscillator from the rational Calogero model (classical)

• •

2010  A. Fring, M. Smith [15, 16, 17]:

Complex root systems in the Calogero model

• •

2013  M. Feigin, Lechtenfeld, Polychronakos [9]:

The quantum angular Calogero–Moser model (spectra, eigenstates)

• •

2013  Correa, Lechtenfeld, Plyushchay [10]:

Nonlinear supersymmetry in the quantum Calogero model

• •

2014  M. Feigin, Hakobyan [11]:

On the algebra of Dunkl angular momentum operators

• •

2015  Correa, Lechtenfeld [12]:

The tetrahexahedric angular Calogero model

The present paper describes the superintegrable spherical reduction of the rational quantum Calogero model for any Coxeter root system (Section 2) and some of its complex ${\cal PT}$ deformations (Section 3). The emphasis is on the Weyl-singlet energy spectrum including degeneracy and eigenstates, and on the conserved charges and intertwiners, in particular for a coupling strength $g(g{-}1)$ with $g\in\mathbb{Z}$. We discuss all features in some detail for the rank-two cases of $A_{2}$ and $G_{2}$ (Sections 4 and 5) and for all rank-three cases, i.e. $AD_{3}$, $BC_{3}$ and $H_{3}$ (Sections 6, 7 and 9), as well as for $A_{1}^{3}$ as a reducible example (Section 8). Tables of low-lying states are collected in the Appendix.

Our results generalize those of [12] to general Coxeter root systems, in particular to the non-simply-laced case, where two independent couplings wrongly suggest the existence of long-root and short-root intertwiners. Instead, we find that all intertwiners respecting the reflection symmetry either shift both couplings or only one of them, so not all states with integral couplings can be connected. We identify a geometric condition for complex orthogonal coordinate transformations to yield a ${\cal PT}$ deformation (with ${\cal P}$ given by a Coxeter element) and display the simplest solutions. It turns out that such deformations reduce the singularities of the angular Calogero potential from codimension one to codimension two. We also present a nonlinear ${\cal PT}$ deformation which may completely remove those singularities (it does so for rank three). In such a situation, the non-normalizable eigenstates (formally given by sending $g\mapsto 1{-}g$ for $g\in\mathbb{N}$) become normalizable and have to be added to the spectrum. Not only does this roughly double the state degeneracy, but it also gives rise to new ‘odd’ conserved charges, which connect the old and the new states. We display these effects for the generic rank-two and all rank-three Coxeter systems.

2 The angular rational Calogero model

The well known rational Calogero model describing $n$ interacting identical particles moving on $\mathbb{R}$ can be formulated for any finite reflection group $W$, with the multi-particle potential encoded in the associated Coxeter root system ${\cal R}\subset\mathbb{R}^{n}$. Since this interaction is not translation invariant111 The $A_{n}$ model decribes the relative coordinates of $n{+}1$ particles after decoupling the center of mass. it is more natural to view such systems as a single particle moving in $\mathbb{R}^{n}$ under the influence of a rather particular external potential determined by $\cal R$. As the Hamiltonian is homogeneous under a common coordinate rescaling (the couplings are dimensionless) the model may be reduced over the $(n{-}1)$-sphere. The result is what we have named the angular Calogero model, since it describes a particle moving on $S^{n-1}$, parametrized by angular coordinates $\vec{\theta}$ only. Because hyperspherical coordinates are rather unwieldy however, we prefer to employ the homogeneous $\mathbb{R}^{n}$ coordinates $x=(x^{i})$ with $i=1,\ldots,n$ and define

[TABLE]

In terms of the latter, the angular Calogero Hamiltonian takes the form

[TABLE]

where ${\cal R}_{+}$ is the positive half of $\cal R$, $g_{\alpha}\in\mathbb{R}$ are the couplings, and $\cdot$ is the standard scalar product in $\mathbb{R}^{n}$. Due to the invariance of the Hamiltonian under $g_{\alpha}{+}1\leftrightarrow-g_{\alpha}$, it suffices to consider $g_{\alpha}\geq\frac{1}{2}$, but we shall not impose this restriction because intermediate results do not reflect this symmetry. Each positive root $\alpha$ contributes a term of the form $\cos^{-2}\phi_{\alpha}$, where $\phi_{\alpha}$ is the geodesic distance to $\hat{\alpha}$. This so-called Higgs oscillator potential [26, 27] is singular on a great $S^{n-2}$, where the hyperplane orthogonal to $\alpha$ cuts our $(n{-}1)$-sphere into two hemispheres. Taken together, these singular loci of codimension one tessalate the $(n{-}1)$-sphere, and our particle is confined to a given Weyl chamber, with its wave function vanishing at the walls (except for $g{=}0$ and $g{=}1$). The potential breaks the SO($n$) invariance of $L^{2}$ to its discrete subgroup $W$, so the energy eigenstates fall into $W$ representations. Motivated by the physical interpretation, we admit only singlet states, i.e. wave functions are either totally symmetric or totally antisymmetric under Coxeter reflections.

The Weyl-invariant spectrum of $H$ has been derived in [9] (see also the appendices of [19]),

[TABLE]

where $d_{2}{=}2$, $d_{3},\ldots,d_{n+1}$ are the degrees of the basic homogeneous $W$-invariant polynomials $\sigma_{2}=\sum_{i}(x^{i})^{2}$, $\sigma_{3},\ldots,\sigma_{n+1}$ and the quantum numbers $\ell_{3},\ell_{4},\ldots,\ell_{n+1}$ are nonnegative integers.222 The unconventional labelling is chosen to match with the standard choice for the $A_{1}\oplus A_{n}$ model. Note that $\sigma_{2}$ does not contribute because $\ell_{2}$ labels the radial excitations. The energy depends only on the ‘deformed angular momentum’ $q$,

[TABLE]

For vanishing couplings, $H={\textstyle\frac{1}{2}}L^{2}$, and $q=\ell$ is the familiar total angular momentum for a free particle on $S^{n-1}$. Nevertheless, the degeneracy of $E_{\ell}$ is greatly reduced by $W$-invariance to the number of partitions of $\ell$ into integers from the set $\{d_{3},\ldots,d_{n+1}\}$.

The angular wave function $v^{(g)}_{\{\ell\}}$ for couplings $g=\{g_{\alpha}\}$ can be constructed in the following way [9]. First, we split off a suitable power of $r$ and a ‘Vandermonde factor’,

[TABLE]

and obtain a homogenous polynomial $h^{(g)}_{\{\ell\}}$ of degree $\ell$ in $x$. Second, the latter is a $W$-invariant Dunkl-deformed harmonic function given by

[TABLE]

where

[TABLE]

denotes the Dunkl differential-reflection operator [23, 28], which involves the Coxeter reflections $s_{\alpha}$ about the hyperplane $\alpha\cdot x=0$. The tilde signifies the so-called potential-free frame, which is related to the ‘potential frame’ by a similarity transformation with {\Delta\vphantom{\big{|}}}^{g},

[TABLE]

In particular, for the ground state one has

[TABLE]

and hence the full ground-state wave function is totally symmetric (antisymmetric) under Coxeter reflections for even (odd) integer values of $g_{\alpha}$. Since all other ingredients besides {\Delta\vphantom{\big{|}}}^{g} in (2.5) are completely symmetric, this symmetry property of the integer-$g_{\alpha}$ ground state extends to all excited states above it. The degeneracy of the energy levels decreases with growing values of $g_{\alpha}$. Furthermore, the reflection symmetry $g_{\alpha}{+}1\leftrightarrow{-}g_{\alpha}$ of the Hamiltonian (2.2) is broken since one tower of states is Weyl symmetric while the other one is antisymmetric. However, due to singularities at $\alpha\cdot x=0$ coming from the Vandermonde factor in (2.5), for $g_{\alpha}<0$ the formal eigenstates are not normalizable (i.e. not in $L_{2}(S^{n-1})$) and thus unphysical. In other words, the singularities in the potential $U$ enforce boundary conditions, which admit only one of the two symmetry types. The free case is an exception, because then those boundary conditions are absent, and so both values $g_{\alpha}=0$ and $g_{\alpha}=1$ contribute to the same spectrum, leading to a rough doubling of the states.

Our Hamiltonian and other conserved quantities are conveniently constructed from the algebra of Dunkl-deformed angular momenta,

[TABLE]

which yields

[TABLE]

with

[TABLE]

The restriction ‘res’ to $W$-symmetric functions provides the Hamiltonian,

[TABLE]

As was shown in [2], the center of the algebra generated by $\{{\cal L}_{ij}\}$ is spanned by ${\cal H}$ and the constants. Therefore, any polynomial ${\cal C}$ built from the ${\cal L}_{ij}$ will commute with ${\cal H}$. If such a polynomial is Weyl invariant, then its restriction yields a conserved quantity,

[TABLE]

It is not clear whether some combinations of these are in involution or how to classify them.

It is actually more fruitful to investigate Weyl antiinvariant polynomials in ${\cal L}_{ij}$, since they give rise to intertwiners (shift operators) which connect Hamiltonians and eigenspaces differing by unit values in the couplings. To be more precise, let us split the set of positive roots into Weyl orbits,

[TABLE]

where one of the following four situations occurs:

[TABLE]

Because all couplings $g_{\alpha}$ in a given Weyl orbit must coincide, we can have at most two different values, $g^{\prime}$ and $g^{\prime\prime}$. The objects of interest are polynomials ${\cal M}$ in ${\cal L}_{ij}$ which are Weyl antiinvariant under ${\cal R}^{\prime}$ reflections but Weyl invariant under ${\cal R}^{\prime\prime}$ reflections. Because the structure of (2.12) implies that

[TABLE]

the commutation of ${\cal M}$ and ${\cal H}$ qualifies $M=\textrm{res}({\cal M})$ as an intertwiner,

[TABLE]

Note that ${\cal M}$ and $M$ depend on $(g^{\prime},g^{\prime\prime})$, which we have suppressed. This operator relation may be applied to $W$-noninvariant states. Hence, $M$ maps $H^{(g^{\prime},g^{\prime\prime})}$ eigenstates of energy $E_{\ell}^{(g^{\prime},g^{\prime\prime})}$ to $H^{(g^{\prime}+1,g^{\prime\prime})}$ eigenstates of (the same) energy $E_{\ell^{\prime}}^{(g^{\prime}+1,g^{\prime\prime})}$ with $\ell^{\prime}=\ell-|{\cal R}^{\prime}|$ (see (2.4)). In particular,

[TABLE]

with some coefficients $c^{\{\ell\}}_{\{\ell^{\prime}\}}\in\mathbb{R}$. Generically, such a map $M$ has a nonempty kernel. The action on the deformed harmonic polynomials $h_{\{\ell\}}^{(g)}$ is obtained by passing to the potential-free frame,

[TABLE]

It is a nontrivial problem for a given Coxeter group $W$ to identify a complete set of intertwiners, their algebra and its generators. We remark that case D does not shift any coupling and describes the constants of motion $C$ mentioned above, while case A pertains to the simply-laced Coxeter groups. When both couplings $g^{\prime}$ and $g^{\prime\prime}$ are integer, repeated intertwining may relate all quantities with their analogs in the free theory, which allows one to generate analytic expressions for all wave functions.

3 ${\cal PT}$-symmetric complex coordinate deformations

We implement a complex deformation of the (angular) coordinates $\vec{\theta}$ through a family of complex linear maps

[TABLE]

which respect the standard scalar product of $\mathbb{R}^{n}$, so

[TABLE]

Hence, $\Gamma(\epsilon)\in\textrm{SO}(n,\mathbb{C})$, but because real coordinate rotations are inessential our family is parametrized by the coset $\textrm{SO}(n,\mathbb{C})/\textrm{SO}(n,\mathbb{R})$ of real dimension $\frac{1}{2}n(n{-}1)$,

[TABLE]

and thus we also have

[TABLE]

A coordinate change effected by $\Gamma(\epsilon)$,

[TABLE]

leaves $r^{2}$ and the kinetic term ${{\textstyle\frac{1}{2}}}L^{2}$ invariant but generates a complex deformation $U\mapsto U(\epsilon)$ of the angular potential (2.2), via

[TABLE]

which may also be interpreted as a complex (dual) deformation of the roots $\alpha$. Formally, the deformed Hamiltonian $H(\epsilon)$ is isospectral to $H=H(0)$, and its $W$-invariant eigenfunctions are simply given by

[TABLE]

Our Hamiltonian is ${\cal PT}$ symmetric if there exist two involutions, one linear (${\cal P}$) and one antilinear (${\cal T}$), under whose combined action it is invariant. For ${\cal T}$ we take the conventional choice of complex conjugation. In the context of Calogero models, a natural ${\cal P}$ transformation is provided by some element $s$ of order 2 in the Coxeter group $W$. The kinetic term ${{\textstyle\frac{1}{2}}}L^{2}$ is separately invariant under ${\cal P}$ and ${\cal T}$ but, in order for $U(\epsilon)$ to be ${\cal PT}$ invariant, the action of the involutive Coxeter element $s$ on the deformed coordinate $x(\epsilon)$ has to be undone by complex conjugation, implying

[TABLE]

On the Lie-algebra level this condition reads

[TABLE]

with $\epsilon{:}G=\sum_{i<j}\epsilon_{ij}G_{ij}$ and projectors

[TABLE]

on the $-1$ and $+1$ eigenspaces of $s$, repectively. It means that $\epsilon{:}G$ intertwines between those two eigenspaces, and so

[TABLE]

If $s$ is just a Coxeter reflection $s_{\gamma}$ pertaining to some (positive) root $\gamma$, then we can say a bit more. Since in this case $P_{-}$ is of rank one, it follows that $\epsilon{:}G$ is of rank two only and parallel to $\gamma$,

[TABLE]

for some real vector $\eta$, carrying $n{-}1$ parameters. The hats denote unit vectors, and the overall scale has been absorbed into a single parameter $\epsilon$. For this situation, the infinitesimal transformation can be integrated explicitly to

[TABLE]

with the help of projectors onto the plane spanned by $\gamma$ and $\eta$ and orthogonal to it. This is just a complex rotation (boost) in the plane determined by $\gamma$ and $\eta$. A similar analysis applies in the co-rank-one case, i.e. when $P_{+}$ is of rank one. In adapted coordinates,

[TABLE]

The complex deformation greatly improves the singularities of $U$ by generically increasing their codimension from one to two. The singularity relation $\alpha\cdot\Gamma(\epsilon)\,x=0$ decomposes into a real and imaginary part giving two conditions,

[TABLE]

leaving an $S^{n-3}$ plus its antipode as the singular locus for each positive root $\alpha$ contributing to $U$. Specializing to ${\cal PT}$-symmetric deformations (3.9), the second condition may be empty if $\alpha$ lies in the kernel of $\epsilon{:}G$. However, such a situation can be avoided by a slight change in the parameters $\epsilon_{ij}$. For the case of $s=s_{\gamma}$, the singular loci appear at

[TABLE]

The second condition gets lost if $\alpha$ lies in the kernel of $P_{\gamma\wedge\eta}$, i.e. if

[TABLE]

However, by a suitable (generic) choice of $\eta$ one can tilt the plane spanned by $\gamma$ and $\eta$ such as to avoid any roots and so evade this degenerate situation.

The deformation also ameliorates the singularities in the unphysical wave functions for negative values of the couplings. From the form of (3.7) it is clear that {\Delta\vphantom{\big{|}}}_{\epsilon} vanishes at antipodal pairs $(x_{\alpha},-x_{\alpha})$ obeying (3.16), for each $\alpha\in{\cal R}_{+}$. Hence, on a collection of $(n{-}3)$-spheres in $S^{n-1}$ our wave functions have nodes for positive values of $g_{\alpha}$, but they still blow up for negative couplings when $n>2$. Hence, for rank 3 and larger, the formal energy eigenstates at $g_{\alpha}<0$ remain non-normalizable under the linear deformation (3.3). Passing to the deformed metric under which $H$ becomes hermitian unfortunately does not change this, and so the ${\cal PT}$ deformation in general does not enlarge the degeneracy of the energy spectrum. An exception occurs for $n{=}2$, which will be outlined below.

The conserved quantities and intertwiners naturally carry over to the deformed situation,

[TABLE]

built from ‘doubly deformed’ angular momenta ${\cal L}_{ij}^{\epsilon}$ made from $x(\epsilon)$ and

[TABLE]

in the case of a linear deformation. Therefore, the superintegrability of the model is unchanged.

One may consider also nonlinear complex deformations of the coordinates. A particular one consists in a complex shift of each angle in a hyperspherical parametrization,

[TABLE]

Such a deformation will (for $n{>}2$) also modify the kinetic term ${{\textstyle\frac{1}{2}}}L^{2}$. The obvious choice for ${\cal P}$ is

[TABLE]

The correspondingly deformed Hamiltonian is ${\cal PT}$ invariant if this transformation is a symmetry of the root system, i.e. if it is contained in the Coxeter group extended by the outer automorphisms (symmetries of the Dynkin diagram).

The advantage of such a deformation is that the singular locus of the potential $U(\epsilon)$ and thus the zero set of the Vandermonde {\Delta\vphantom{\big{|}}}_{\epsilon} may be empty. This renders the formal energy eigenstates for $g_{\alpha}<0$ normalizable and, hence, produces new towers of physical states for negative couplings. Due to $H(-g_{\alpha})=H(g_{\alpha}{+}1)$, these new states enlarge the state space for $g_{\alpha}>1$. For integral $g_{\alpha}$ we can connect the two towers by a string of intertwiners.333 This is also possible for odd half-integral couplings but does not yield an independent charge. In the enlarged state space then acts an additional, ‘odd’ conserved charge,

[TABLE]

which intertwines between the $g^{\prime}>0$ and $g^{\prime}\leq 0$ towers. In the potential-free frame,

[TABLE]

relates the two Dunkl- and ${\cal PT}$-deformed harmonic polynomials to each other. Note that in contrast to $Q_{\epsilon}^{(g^{\prime},g^{\prime\prime})}$, the potential-free intertwiner $\widetilde{Q}_{\epsilon}^{(g^{\prime},g^{\prime\prime})}$ is not conserved. The new odd charge squares to a polynomial in the conserved ‘even’ charges $C$ and extends the algebra of conserved quantities to a nonlinear supersymmetric one. Due to the ${\cal PT}$ regularization of the negative-coupling states, $Q_{\epsilon}^{(g^{\prime},g^{\prime\prime})}$ now has a regular action in the state space. In general there exist more than one intertwiner, giving rise to various such odd charges.

4 $A_{2}$ model

The simplest case to consider is the $A_{2}$ model, which is based on the roots

[TABLE]

yielding the Coxeter reflections

[TABLE]

Its spherical reduction yields the Pöschl-Teller model, which describes a particle on $S^{1}$ in the potential

[TABLE]

where we introduced a complex homogeneous $\mathbb{R}^{2}$ coordinate $w$ and polar coordinates $(r,\phi)$,

[TABLE]

Since $A_{2}$ is simply-laced, all couplings must coincide, $g_{\alpha}=g$. The two basic homogeneous polynomials invariant under $W=S_{3}$ are

[TABLE]

Hence, $d_{3}=3$, $\{\ell\}=\ell_{3}$ and $\ell=3\ell_{3}$, and we have the $S_{3}$-invariant spectrum

[TABLE]

For $g>0$ this implies $E_{\textrm{min}}=\frac{9}{2}g^{2}$, but for $g<0$ the spectrum goes down to zero energy. The Vandermonde factor takes the simple form

[TABLE]

and the Dunkl operator in the potential-free frame reads ($\rho={\mathrm{e}}^{2\pi{\mathrm{i}}/3}$)

[TABLE]

with the Coxeter reflections

[TABLE]

Thus, the $S_{3}$-invariant wave functions in the potential-free frame are (with $r^{0}\to\ln r$)

[TABLE]

expressed in terms of the hypergeometric function ${}_{2}F_{1}$ or the Jacobi polynomials $P_{n}^{(\alpha,\beta)}$. The gamma-function prefactors are irrelevant for $g>0$ but are chosen such as to enable an analytic continuation to $g<0$, which will become relevant in a while. A table of states for small values of $\ell$ can be found in Appendix A.1.

The Dunklized angular momentum is given by

[TABLE]

with ${\cal D}_{w}=\widetilde{{\cal D}}_{w}-\frac{3\,g\,w^{2}}{w^{3}+{\bar{w}}^{3}}$. From this we can build only one algebraically independent $S_{3}$-symmetric polynomial (case D),

[TABLE]

whose restriction $C_{2}$ to $S_{3}$-symmetric functions provides the Pöschl-Teller Hamiltonian minus its ground-state energy. The single basic $S_{3}$-antiinvariant polynomial (case A) is ${\cal L}$ itself, from which we get

[TABLE]

which obeys

[TABLE]

Because ${\cal M}_{1}$ is linear in ${\cal L}$, in this case it is also true that

[TABLE]

which exceptionally does not depend on $g$. The ladder relation for the deformed harmonic polynomials (remember deg$(E_{\ell})=1$),

[TABLE]

may for positive integer $g$ be iterated to generate them from the free ($g{=}0$) ones,

[TABLE]

which reproduces the analytic expression (4.10). Eventually, the iteration hits the kernel of $\widetilde{M}_{1}$, i.e. $h_{0}^{(g)}=1$ corresponding to the ground state, where it ceases.

The $g<0$ states can as well be obtained directly from (4.14), which also implies that

[TABLE]

Its iteration for negative integer $g$ produces

[TABLE]

which may be checked to reproduce the analytic continuation of (4.10) to $g<0$. However, without ${\cal PT}$ deformation the full wave functions $v_{\ell}^{(g<0)}$ are not normalizable. For illustration, in Appendix A.1 we display the polynomials $h_{\ell}^{(g)}$ for $g=-2,-1,0,1,2$ and $q\leq 12$.

Let us take a look at the possible ${\cal PT}$ involutions and the compatible complex deformations for the Pöschl-Teller model. The only order-2 elements in $S_{3}$ are the Coxeter reflections about the lines perpendicular to the roots, so without loss of generality we may fix ${\cal P}$ as the action of $s_{0}$, which belongs to the root $\gamma=\sqrt{2}e_{1}$ and is a reflection on the $x^{2}$-axis,

[TABLE]

Obviously, $P_{-}$ and $P_{+}$ project onto the $x^{1}$ axis and the $x^{2}$ axis, respectively. As usual, ${\cal T}$ is complex conjugation, but please be aware that this does not swap $w$ with ${\bar{w}}$ because the complex linear combination of the real coordinates $x^{1}$ and $x^{2}$ is unaffected by ${\cal T}$.

The coset SO$(2,\mathbb{C})/$SO$(2,\mathbb{R})$ is one-dimensional and parametrized as

[TABLE]

Since there is just one plane, necessarily $\hat{\eta}=e_{2}$ and $P_{\gamma\wedge\eta}=\mathbbm{1}$. Clearly, $s_{0}$ and $G$ anticommute, and so all such complex deformations

[TABLE]

are ${\cal PT}$ symmetric. In polar coordinates, this deformation takes a particularly simple form,

[TABLE]

but for the complex combinations $(w,{\bar{w}})$ one has to keep in mind that ${\cal T}$ does not conjugate

[TABLE]

but only flips the sign of $\epsilon$. For any root $\alpha$ contributing to the potential, the singular locus of $U(\epsilon)$ for $\epsilon{\neq}0$ lies at

[TABLE]

since ${\mathrm{i}}G$ is a $\pi/2$ rotation in our plane. Hence, the deformed potential

[TABLE]

as well as the deformed wave functions (see (4.10))

[TABLE]

for $g<0$ are free of singularities because

[TABLE]

is regular everywhere. Because the complex deformation is merely a constant shift of the polar angle, the angular momentum and the potential-free intertwiner exceptionally remain undeformed,

[TABLE]

Our intertwiner $\widetilde{M}_{1}$ has a simple kernel. Since

[TABLE]

$\widetilde{M}_{1}$ at any fixed $g$ annihilates this one state but no other one. Our ${\cal PT}$ deformation leads to a rough doubling of the energy eigenstates, because the spectrum of $H^{(g)}$ now has to be joined with that of $H^{(1-g)}$. So, for a given $g{>}\frac{1}{2}$, we encounter two towers of states with $E={{\textstyle\frac{1}{2}}}q^{2}$, for

[TABLE]

where the second tower yields negative $q$ for $\ell<3(g{-}1)$. When $g$ is integral or half-integral, the two towers meet, so the degeneracy doubles. However, it turns out that flipping the sign of $q$ yields the same state again, and so for positive integral $g$ the level degeneracy becomes

[TABLE]

The new states are again given by (4.10), where in the limit of negative integral $g$ the zeros of the Jacobi polynomial are cancelled by poles of the prefactor, so a careful limit has to be taken. Such a state structure is common for systems possessing a hidden supersymmetric structure [29], which is indeed the case here and revealed by the additional ‘odd’ conserved charge

[TABLE]

In the potential-free frame, it simplifies to

[TABLE]

and clearly obeys the intertwining relation

[TABLE]

relating the deformed harmonic polynomials at couplings $1{-}g$ and $g$. Since the transition from $h^{(g)}_{\ell}$ to $v^{(g)}_{\ell}$ involves the ($g$-dependent) factor of {\Delta\vphantom{\big{|}}}^{g} and $\widetilde{H}^{(1-g)}\neq\widetilde{H}^{(g)}$, only in the potential frame this intertwining relation becomes a commutation relation,

[TABLE]

The $g$ singlet states (for $q<3g$) are annihilated by $Q^{(g)}_{\epsilon}$,

[TABLE]

at energies

[TABLE]

For all other states, $Q^{(g)}_{\epsilon}$ maps the doublet partners to each other. The square of $Q^{(g)}_{\epsilon}$ is a polynomial in the Hamiltonian,

[TABLE]

which also reveals the properties of the combined spectrum.

5 $G_{2}$ model

The $A_{2}$ model is the first of an infinite list of dihedral $I_{2}(p)$ models, with

[TABLE]

and where for odd $p$ all couplings must coincide while for even $p$ the root system decomposes into two $I_{2}(\frac{p}{2})$ subsystems with two couplings $g_{S}$ and $g_{L}$. Let us illustrate the latter situation on the $G_{2}$ example, since it can be obtained by a superposition of two $A_{2}$ systems (with a $\pi/2$ rotation),

[TABLE]

presented in the previous section. The corresponding Coxeter reflections read

[TABLE]

The potential is easily derived,

[TABLE]

and exhibits the two subsystems. The Coxeter group is the dihedral group ${\cal D}_{6}$ with 12 elements, which maps short roots to short roots and long roots to long roots. The two basic ${\cal D}_{6}$-invariant homogeneous polynomials are

[TABLE]

Hence, $d_{3}=6$, $\{\ell\}=\ell_{3}$ and $\ell=6\ell_{3}$, and we have the ${\cal D}_{6}$-invariant spectrum

[TABLE]

For $g_{L}{=}0$ or $g_{S}{=}0$, we fall back to the Pöschl-Teller model, but only its ‘even’ states survive the more restrictive Weyl invariance requirement, as $\ell$ must be a multiple of 6 now. Compared to the Pöschl-Teller model, the density of energy eigenstates is cut in half. The Vandermonde factorizes,

[TABLE]

and the (potential-free) Dunkl operator reads

[TABLE]

with the additional Coxeter reflections

[TABLE]

With these ingredients, the wave functions in the potential-free frame can be constructed,

[TABLE]

Since only even powers of $w$ or ${\bar{w}}$ occur, its form is a bit simpler than (4.10). Some low-lying wave functions are given explicitly in Appendix A.2.

The Dunklized angular momentum is given by

[TABLE]

and essentially squares to the Hamiltonian,

[TABLE]

via $H=\textrm{res}({\cal H})$. Again, for generic $g$ this is the only conserved charge (case D). Like before, ${\cal L}$ is Weyl antiinvariant (case A), thus providing the basic intertwiner

[TABLE]

The intertwining relations read

[TABLE]

and again the potential-free intertwiner trivializes,

[TABLE]

The corresponding ladder relations for the wave functions are

[TABLE]

with special relations for the vanishing of one of the couplings,

[TABLE]

where we intermediately allow Weyl ‘half-invariant’ states at $\ell=3,9,12,\ldots$. For integral couplings the above relations may be iterated for the alternative wave function reconstruction

[TABLE]

[TABLE]

and similarly for the four other domains of $(g_{S},g_{L})$, starting from

[TABLE]

When $g_{S}$ and $g_{L}$ are non-negative, the wave functions are normalizable. For integral couplings, the ${\cal D}_{6}$-invariant energy spectrum $E_{\ell}=\frac{1}{2}q^{2}$ is non-empty only for

[TABLE]

When a coupling turns negative, the zeros of the corresponding Vandermonde factor render the full wave function $v_{\ell}^{(g_{S},g_{L})}$ non-normalizable. In order to make these states physical, we turn to the ${\cal PT}$ deformation.

The order-2 elements in ${\cal D}_{6}$ are precisely the 6 root reflections, so there are only two inequivalent cases, corresponding to \hat{\gamma}=\bigl{(}\begin{smallmatrix}1\\ 0\end{smallmatrix}\bigr{)} and to \hat{\gamma}=\bigl{(}\begin{smallmatrix}0\\ 1\end{smallmatrix}\bigr{)},

[TABLE]

However, since all linear complex coordinate deformations are admissible in two dimensions, the discussion is identical to the $A_{2}$ case, and the potential and wave functions lose all their singularities,

[TABLE]

[TABLE]

The ${\cal PT}$ deformation now leads to an approximate quadrupling of the eigenstates because

[TABLE]

tells us to join four towers of states. Let us look at positive integral couplings $(g_{S},g_{L})$. Then, (5.6) implies that the first and fourth tower from (5.25) coincide, and likewise do the second and third tower. Depending on whether $g_{S}{+}g_{L}$ is even or odd, one pair of towers sits at $q=0,6,12,\ldots$ and the other one at $q=3,9,15,\ldots$. Therefore, the density of energy eigenstates is about the same as in the $A_{2}$ model. Like in the latter though, some states are missing for small values of $q$, since the towers do not reach all the way down to zero (see (5.21)).

When $g_{S}$ and $g_{L}$ are positive integers, we can write down an additional ‘odd’ conserved charge

[TABLE]

whose explicit form reads

[TABLE]

where the product order must be assumed from right to left due to noncommuting action of the intertwining operators. The potential-free form reads

[TABLE]

The form (5.27) or (5.28) represents an action $(g_{S},g_{L})\mapsto(1{-}g_{S},1{-}g_{L})$ on the couplings. Analogously to (4.35), $\widetilde{Q}^{(g_{S},g_{L})}_{\epsilon}$ obeys an intertwining relation, while ${Q}^{(g_{S},g_{L})}_{\epsilon}$ commutes with the potential-frame Hamiltonian as in (4.36). There exist other admissible actions like $(1{-}g_{S},g_{L})\mapsto(g_{S},1{-}g_{L})$ which only produce different factorizations of the same operator (5.27) but no new conserved charges, see Fig. 5. For $g_{S}\geq g_{L}$, $Q^{(g_{S},g_{L})}_{\epsilon}$ annihilates the singlet states with energies

[TABLE]

For $g_{L}\geq g_{S}$, the roles of $g_{L}$ and $g_{S}$ are reversed. In analogy with the $A_{2}$ case, (5.27) squares to a polynomial in the Hamiltonian [30],

[TABLE]

The structure presented in the last two sections are easily generalized to all dihedral $I_{2}(p)$ models. Essentially, $w^{3}$ is replaced by $w^{p}$ or $w^{p/2}$, $\ell=p\,\ell_{3}$, $\rho$ becomes a $p$th root of unity, and the intertwiner shifts $(g_{S},g_{L},\ell)\to(g_{S}{+}1,g_{L}{+}1,\ell{-}p)$. The wave-function formulæ (4.10) and (5.10) generalize without any change after the first line.

6 $AD_{3}$ model

A much richer and less trivial situation appears one dimension higher, i.e. at rank 3. To reduce index cluttering, we redenote the coordinates as

[TABLE]

The set of positive roots can be chosen as

[TABLE]

We consider the $A_{3}$ (or $D_{3}$) Calogero model spherically reduced to what we have named the tetrahexahedric model. Here, a particle moves on the 2-sphere with the potential

[TABLE]

Since $W=S_{4}$ has just one orbit on the root system, again we have a single coupling, $g_{\alpha}=g$. It can be generated by

[TABLE]

The three basic $S_{4}$-invariant homogeneous polynomials read

[TABLE]

and therefore $\{\ell\}=(\ell_{3},\ell_{4})$ and $\ell=3\ell_{3}+4\ell_{4}$, and the energy levels take the form

[TABLE]

for $S_{4}$-invariant states, with a degeneracy given by

[TABLE]

The Vandermonde reads

[TABLE]

and the Dunkl operators in the potential-free frame are given by

[TABLE]

with the Coxeter reflections

[TABLE]

These ingredients enter the $S_{4}$-invariant energy eigenfunctions v^{(g)}_{\{\ell\}}=r^{-q}{\Delta\vphantom{\big{|}}}^{g}h^{(g)}_{\{\ell\}}(x) in

[TABLE]

for which we cannot offer a more explicit expression. The lowest-energy wave functions are given in the table of Appendix A.3. Their degeneracies and corresponding quantum numbers $(\ell_{3},\ell_{4})$ are listed below, where the notation $(\ell_{3},\ell_{4})^{*}$ identifies the $q{<}0$ states.

[TABLE]

The Dunkl-deformed angular momenta,

[TABLE]

with {\cal D}_{i}=\widetilde{{\cal D}}_{i}-g\,\partial_{i}\ln{\Delta\vphantom{\big{|}}} (amounting to dropping the ‘1’s in (6.9)), get permuted under the action of $S_{4}$, with an odd number of sign flips thrown in. The ring of Weyl invariant polynomials in $\{{\cal L}_{x},{\cal L}_{y},{\cal L}_{z}\}$ (case D) is generated by

[TABLE]

where

[TABLE]

giving rise to three algebraically independent conserved quantities, $C_{k}=\textrm{res}({\cal C}_{k})$ for $k=2,4,6$, see also [12]. Their algebra seems to be freely generated, modulo the center spanned by $C_{2}$.

The basic Weyl antiinvariants built from $\{{\cal L}_{x},{\cal L}_{y},{\cal L}_{z}\}$ (case A) are

[TABLE]

and all higher ones are words in these and the ${\cal C}_{k}$. Their restriction to $S_{4}$-symmetric functions produces two independent intertwiners, $M_{3}$ and $M_{6}$, which obey the same relations (4.14). Their potential-free version444 Note that $\widetilde{M}_{s}$ are not the restrictions of the corresponding polynomials $\widetilde{{\cal M}}_{s}$ in the potential-free frame.

[TABLE]

can be employed to step up the energy eigenfunctions in the coupling,

[TABLE]

In this way, eventually all states with positive integer coupling can be reached. This may not be true for the (more numerous) negative integer coupling states, some of which can be found by applying the adjoint intertwiner. In contrast to the previous section, $\widetilde{M}_{s}$ now depend on the value of $g$, which prevents a nice closed formula like (4.10) for the polynomials $h_{\{\ell\}}^{(g)}$.

What are the possibilities for a linear realization of ${\cal PT}$ transformations? The Coxeter group $W=S_{4}$ contains one rank-zero involution (the identity), 6 rank-one involutions (the Coexeter reflections), and 3 rank-two involutions ($\pi$ rotations on one of the three basic planes). The unique rank-three involution (the negative identity) is the outer automorphism of $A_{3}$, hence it is not in $S_{4}$ but generates its double cover. Vanishing rank or co-rank of $P_{-}$ does not admit a compatible complex deformation. The three-dimensional coset $\textrm{SO}(3,\mathbb{C})/\textrm{SO}(3,\mathbb{R})$ is parametrized as

[TABLE]

where

[TABLE]

Clearly, any nonvanishing $G$ is of rank two. Degeneracy in the singular locus $\alpha\cdot x(\epsilon)=0$ occurs only when $e$ is parallel to some root $\alpha$.

For rank$(P_{-})=1$, without loss of generality we choose ${\cal P}$ to permute $x$ and $y$, i.e.

[TABLE]

with free real parameters $u$ and $v$. Compatibility of (6.18) with the rank-one involution (6.20) requires merely $u=v$. The simplest option is $(u,v,w)=(0,0,1)$, which copies the $n{=}2$ case into the $xy$ plane,

[TABLE]

Since no root is orthogonal to this plane, our option is generic, and each singular locus has a nontrivial imaginary part. This is not the case for another option, $(u,v,w)=\pm(1,1,0)/\sqrt{2}$, since this unit vector is parallel to a root.

For rank$(P_{-})=2$, we may take ${\cal P}$ to rotate by $\pi$ in the $yz$ plane, so effectively

[TABLE]

The deformation (6.18) is consistent with (6.22) precisely if $u=0$. Specializing once more to $(u,v,w)=(0,0,1)$, without loss of generality, we again arrive at the boost (6.21). Also in this case, there are some degenerate options, namely $(u,v,w)=\pm(0,1,1)/\sqrt{2}$ and $(u,v,w)=\pm(0,1,-1)/\sqrt{2}$.

The singular set of the deformed potential

[TABLE]

consists of 6 antipodal pairs $(x_{\alpha},-x_{\alpha})$ of points,

[TABLE]

where the first two pairs coincide. With increasing deformation parameter $\epsilon$, the other four pairs move from the location of the roots (outside the $xy$ plane) to the north and south poles. Clearly, the singular Vandermonde factor {\Delta\vphantom{\big{|}}}_{\epsilon}^{g} keeps the energy eigenstates unphysical for negative values of $g$. Hence, only the free state spaces at $g{=}0$ and $g{=}1$ should be combined, so that its degeneracy becomes

[TABLE]

When the potential is turned on, the linear ${\cal PT}$ deformation hence does not alter the degeneracy of the energy spectrum but smoothly modifies the states.

With a nonlinear ${\cal PT}$ deformation of the type (3.20) we may, however, completely remove the wave-function and potential singularities. For the case at hand, it reads

[TABLE]

[TABLE]

For $\epsilon_{1}=0$ we come back to the linear complex boost in the $xy$ plane. The ${\cal P}$ involution is chosen as the outer automorphism

[TABLE]

and it is easy to see that the deformed Hamiltonian $H(\epsilon)$ is ${\cal PT}$ symmetric. We should note, however, that the above deformation modifies the kinetic term,

[TABLE]

Because with this deformation the Vandermonde is nowhere vanishing,

[TABLE]

all state spaces at $g<0$ become physical, and so we should combine the tower for any $g{>}\frac{1}{2}$ with the one at $1{-}g$. In contrast to the $A_{2}$ model, both branches for $1{-}g<0$ contribute, and for positive integral $g$ one finds (demanding $S_{4}$ invariance) that

[TABLE]

which demonstrates the doubling (for large enough energy) compared to (6.7).

In this situation we encounter additional ‘odd’ conserved charges (suppressing $\epsilon$)

[TABLE]

They square to polynomials in the even charges $C_{2}$, $C_{4}$ and $C_{6}$, e.g.

[TABLE]

7 $BC_{3}$ model

To understand the non-simply-laced situation at rank-three, we study the model based on the $BC_{3}$ Coxeter system. It is obtained by extending the $AD_{3}$ root system to

[TABLE]

yielding the potential

[TABLE]

The Coxeter group $W=S_{4}\ltimes\mathbb{Z}_{2}$ enlarges the previous $S_{4}$ by reflections on the basic coordinate planes, and it may be generated by

[TABLE]

The basic invariant polynomials are555 The choice of $\sigma_{3}$ is not unique. Other options are  $x^{6}{+}y^{6}{+}z^{6}$  or  $\sigma_{2}\sigma_{4}-(x^{6}{+}y^{6}{+}z^{6})$.

[TABLE]

which leads to $\ell=6\ell_{3}+4\ell_{4}$ and $W$-invariant energy levels

[TABLE]

and a degeneracy deg$(E_{\ell})=0$ when $\ell$ is odd and

[TABLE]

when $\ell$ is even. Putting $g_{S}=0$, we are back to the $AD_{3}$ case, but its states with odd $\ell_{3}$ and thus odd $\ell$ are absent here. The Vandermonde splits,

[TABLE]

The Dunkl operators $\widetilde{{\cal D}}_{i}$ can be obtained from (6.9) by specifying $g\to g_{L}$ and adding a term $\frac{g_{S}}{x^{i}}(1{-}s_{i})$ with the additional Coxeter reflections

[TABLE]

complementing (6.10). For the $W$-invariant energy eigenfunctions v^{(g)}_{\{\ell\}}=r^{-q}{\Delta\vphantom{\big{|}}}^{g}h^{(g)}_{\{\ell\}}(x) we must construct the degree-$\ell$ homogeneous polynomials

[TABLE]

Comparing with the $AD_{3}$ case, apart from the extended Dunkl operators this formula is very similar to (6.11), but all odd-$\ell$ states have disappeared. The following tables show the states and degeneracy at small values of the energy for a few values of $g_{S}$ and $g_{L}$, where again a $*$ denotes the $q{<}0$ states. We see that the latter appear even when only one of the couplings is negative. Some of the wave functions can be calculated explicity from the table in Appendix A.4.

[TABLE]

[TABLE]

The Dunkl-deformed angular momenta

[TABLE]

do not differ much from those of the $AD_{3}$ model. The Coxeter reflections permute them and can flip the sign of any number of them. Therefore, the Weyl invariant polynomials in $\{{\cal L}_{x},{\cal L}_{y},{\cal L}_{z}\}$ are the same as in the $AD_{3}$ case, generated by $\{C_{0},C_{2},C_{4},C_{6}\}$, and the conserved charges agree with the previous ones, except that the constituting Dunkl operators have been extended by the short-root terms. What about Weyl antiinvariants, corresponding to cases A, B or C in Section 2? Unfortunately, because

[TABLE]

there do not exist ${\cal L}_{i}$ polynomials which are antiinvariant under the short-root reflections. Besides, an intertwiner shifting $g_{S}$ by unity would connect states with an even value of $q$ to states with an odd one, which is incompatible with (7.5). Therefore, besides case D (the invariants) we can only realize case B, which copies the $AD_{3}$ intertwining situation. As a result, the two basic $AD_{3}$ intertwiners $M_{3}$ and $M_{6}$, based on (6.15) with the ${\cal L}_{i}$ pertaining to the $BC_{3}$ system, will obey the relations

[TABLE]

but do not shift the $g_{S}$ value. Therefore, iterating the $\widetilde{M}_{s}$ action, we can produce the polynomials $h^{(g_{L},g_{S})}_{\{\ell\}}$ from $h^{(0,g_{S})}_{\{\ell^{\prime}\}}$.

The discussion of ${\cal PT}$ deformations may be completely borrowed from the previous section. The additional $\textrm{rank}(P_{-}){=}1$ option of ${\cal P}=s_{x}$ does not produce anything new. Under the nonlinear deformation (6.26), again the Vandermonde loses its zeros, and the negative-$g$ state spaces become physical. So for positive integral values of $g_{L}$ and $g_{S}$, we must combine two state towers at

[TABLE]

where one pair has states only at even $q$ and the other pair only at odd $q$. For $q\geq 6(g_{L}{-}1)+3(g_{S}{-}1)$, the irregularities due to missing low-energy states disappear, and the degeneracy grows approximately like $\frac{\ell}{6}$ both for even and odd $q$ values. For $g_{L}\in\mathbb{Z}$ there appear ‘odd’ conserved charges $Q^{(g_{L},g_{S})}_{\{s\}}$ mapping $(1{-}g_{L},g_{S})\mapsto(g_{L},g_{S})$. They are formally identical to those of the $AD_{3}$ model. Analogous odd operators connecting the states at $1{-}g_{S}$ and $g_{S}$ do not exist since the two pairs of towers have disjoint spectra.

8 $A_{1}^{\oplus 3}$ model

The previous section reduced the $AD_{3}$ system to the $A_{1}^{\oplus 3}$ system of short roots,

[TABLE]

When the radial excitations are included, this model is reducible and decomposes into three copies of the rank-one system with inverse-square potential and coinciding couplings $g_{s}=g$. However, the spherical reduction couples the three subsystems to a potential

[TABLE]

The Coxeter group $W=\mathbb{Z}_{2}^{3}$ consists merely of the 3 reflections about the elementary coordinate planes,

[TABLE]

and the basic invariant polynomials can be taken as666 The choice of $\sigma_{3}$ and $\sigma_{4}$ is ambiguous; other possibilities are $\sigma_{3}=x^{2}{-}y^{2}$ and $\sigma_{4}=x^{2}{+}y^{2}{-}2z^{2}$ or $\sigma_{3}=x^{2}$ and $\sigma_{4}=x^{2}{+}y^{2}$.

[TABLE]

and thus

[TABLE]

for the $W$-invariant states, with a degeneracy

[TABLE]

This is consistent with the fact that only the spherical-harmonic combinations

[TABLE]

are Weyl invariant. The Vandermonde is simply {\Delta\vphantom{\big{|}}}=x\,y\,z, and the potential-free wave functions arise from

[TABLE]

with

[TABLE]

With the above choice of symmetric polynomials we could find the following formulæ for the states with $\ell_{4}=0$,

[TABLE]

and due to the symmetry $\ell_{3}\leftrightarrow\ell_{4}$ plus $x\leftrightarrow y$ we can obtain the $\ell_{3}=0$ states,

[TABLE]

Below we present the low-lying degeneracies and quantum numbers at $g\geq-2$. Their explicit form can be found in Appendix A.5, where without loss of generality we restrict to $\ell_{3}\geq\ell_{4}$.

[TABLE]

The Dunklized angular momenta have the simple form

[TABLE]

and any word in ${\cal L}_{i}^{2}$ and ${\cal L}_{x}{\cal L}_{y}{\cal L}_{z}$ (and permutations) will restrict to a conserved quantity. As was argued in the previous section, there exist neither Weyl antiinvariant polynomials in ${\cal L}_{i}$ nor intertwiners shifting $g$ by unity.777 The intertwiners proposed in [2] are not $W$ invariant. As a consequence, an ‘odd’ conserved charge for integral $g$ cannot be constructed in this way.

A linear ${\cal PT}$ deformation of the type (6.21) (but with a non-coordinate plane) still leaves three pairs of singular points in the potential $U_{\epsilon}$, while the nonlinear deformation (6.26) yields the fully regularized potential

[TABLE]

This revives the negative-$g$ state spaces and lets us combine the towers at $1{-}g$ and $g$. The result is a linearly (with $q$) growing $W$-invariant spectrum both for even and odd values of $q$,

[TABLE]

The $A_{1}^{\oplus 3}$ model is the simplest of an infinite reducible series, based on $A_{1}\oplus I_{2}(p)$. We leave it to the reader to work out the details for $p>2$.

9 $H_{3}$ model

Finally, to fully cover the rank-3 landscape, let us turn to the non-crystallographic case of $H_{3}$. Abbreviating the golden ratio and its algebraic conjugate,

[TABLE]

the set of 15 positive roots,888 All four sign combinations appear. These roots do not lie in a half-space, but this is irrelevant here.

[TABLE]

Accordingly, the potential takes the form

[TABLE]

with 15 double poles. The Coxeter group is the icosahedral group $I$ of 120 elements, and it may be generated by the elements

[TABLE]

We can choose three basic invariant polynomials of degrees 2, 6 and 10, for instance

[TABLE]

Hence, $\ell=6\ell_{3}+10\ell_{4}$, and the $I$-invariant energy levels are given by

[TABLE]

and a degeneracy deg$(E_{\ell})=0$ when $\ell$ is odd and

[TABLE]

when $\ell$ is even. The Vandermonde factor {\Delta\vphantom{\big{|}}}={\Delta\vphantom{\big{|}}}_{1}{\Delta\vphantom{\big{|}}}_{2} is split in terms of  {\Delta\vphantom{\big{|}}}_{1}=x\,y\,z  and

[TABLE]

The analytical computation of the energy eigenfunctions v^{(g)}_{\{\ell\}}=r^{-q}{\Delta\vphantom{\big{|}}}^{g}h^{(g)}_{\{\ell\}}(x) in

[TABLE]

becomes quite more complicated in contrast with the previous cases. Because of the sum over the 15 positive roots, the Dunkl operators yield quite tedious expressions considering also that the invariant polynomials are given in terms of powers of differential operators of order 6 and 10. We present here the simplest wave functions of order 6, with $(\ell_{3},\ell_{4})=(1,0)$,

[TABLE]

and of order 10, with $(\ell_{3},\ell_{4})=(0,1)$,

[TABLE]

where we defined  ${[rst]}:=x^{r}y^{s}z^{t}+x^{t}y^{r}z^{s}+x^{s}y^{t}z^{r}$  and abbreviated

[TABLE]

It is possible to check that they are symmetric under the simultaneous transposition of variables plus $\tau\rightarrow\bar{\tau}$, concretely

[TABLE]

The low-lying degeneracies and quantum numbers at $g\geq-2$ are presented in the following table and also illustrated in Fig. 11.

[TABLE]

10 Outlook

We have investigated the ${\cal PT}$ deformation of the angular Calogero model firstly in general and secondly in detail for rank-two and rank-three systems. Among the different ways to introduce an antilinear symmetry like ${\cal PT}$, nonlinear complex deformations of the coordinates seem to be more effective for removing the singularities of the potential than linear ones. As a result of such a ‘${\cal PT}$ regularization’, the energy spectrum gets enlarged due to the $g\mapsto 1{-}g$ invariance of the (potential-frame) Hamiltonian: The previously non-normalizable eigenstates at $g{<}0$ become physical and have to be included. In non-simply-laced cases this holds separately for the short- and long-root couplings. For integer (or half-integer) values of $g$, the energy levels at $1{-}g$ concide with those at $g$, increasing the degeneracy of the latter. In this situation, a suitable product of intertwiners produces conserved charges, which act in a regular way thanks to the ${\cal PT}$ regularization. When $g$ is an integer, these charges represent ‘square roots’ of conserved charges defined for any $g$-value, which extends their algebra to a nonlinear $\mathbb{Z}_{2}$-graded one. In the light of our results it is interesting to investigate how the energy spectra get modified for ${\cal PT}$-deformed trigonometric, hyperbolic or elliptic Calogero models. We plan to address these problems in the future.

Acknowledgments

This work was partially supported by the Alexander von Humboldt Foundation under grant CHL 1153844 STP and by the Deutsche Forschungsgemeinschaft under grant LE 838/12. This article is based upon work from COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology). F.C. is grateful for the warm hospitality at Leibniz Universität Hannover, where the main part of this work was done.

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Appendix A Appendix

A.1 $A_{2}$ model

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[TABLE]

Table 1:  Low-lying wave functions v_{\ell}^{(g)}=r^{-\ell-3g}{\Delta\vphantom{\big{|}}}^{g}h_{\ell}^{(g)} of the Pöschl-Teller model with $E_{\ell}={\textstyle\frac{1}{2}}q^{2}$ and $q=\ell{+}3g$.

A.2 $G_{2}$ model

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Table 2:  Deformed harmonic polynomials for low-lying wave functions of the $G_{2}$ model at general couplings $g_{L}$ and $g_{S}$.

A.3 $AD_{3}$ model

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Table 3: Low-lying polynomials for the $AD_{3}$ model. The notation is  $\{rst\}\ :=\ x^{r}y^{s}z^{t}+x^{r}y^{t}z^{s}+x^{s}y^{t}z^{r}+x^{s}y^{r}z^{t}+x^{t}y^{r}z^{s}+x^{t}y^{s}z^{r}$.

A.4 $BC_{3}$ model

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Table 4: Low-lying polynomials for the $BC_{3}$ model. The constants $\beta_{1}$ and $\alpha_{m}$ for $m=1,\ldots,5$ are given below.

[TABLE]

A.5 $A_{1}^{\oplus 3}$ model

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Table 5: Low-lying polynomials for the $A_{1}^{\oplus 3}$ model at coinciding couplings for $\ell_{3}\geq\ell_{4}$.