Invariant PDEs of Conformal Galilei Algebra as deformations: cryptohermiticity and contractions

We investigate the general class of second-order PDEs, invariant under the $d=1$ $\ell=\frac{1}{2}+{\mathbb N}_0$ centrally extended Conformal Galilei Algebras, pointing out that they are deformations of decoupled systems. For $\ell=\frac{3}{2}$ the unique deformation parameter $\gamma$ belongs to the fundamental domain $\gamma\in ]0,+\infty[$. We show that, for any $\gamma\neq 0$, invariant PDEs with discrete spectrum (either bounded or unbounded) induce cryptohermitian operators possessing the same spectrum as two decoupled oscillators, provided that their frequencies are in the special ratio $r=\frac{\omega_2}{\omega_1}=\pm\frac{1}{3},\pm 3$ (the negative energy solutions correspond to a special case of Pais-Uhlenbeck oscillator), where $\omega_1,\omega_2$ are two different parameters of the invariant PDEs. We also consider the $\gamma=0$ decoupled system for any value $r$ of the ratio. It possesses enhanced symmetry at the critical values $r=\pm \frac{1}{3}, \pm 1,\pm 3$. Two inequivalent $12$-generator symmetry algebras are found at $r =\pm\frac{1}{3},\pm 3$ and $r=\pm 1$, respectively. The $\ell=\frac{3}{2}$ Conformal Galilei Algebra is not a subalgebra of the decoupled symmetry algebra. Its $\gamma\rightarrow 0$ contraction corresponds to a $8$-generator subalgebra of the decoupled $r=\pm\frac{1}{3},\pm 3$ symmetry algebra. The features of the $\ell\geq \frac{5}{2}$ invariant PDEs are briefly discussed.


Introduction
In [1] and [2] second-order PDEs, invariant under the cga ℓ (ℓ = 1 2 + N 0 ) centrally extended Conformal Galilei Algebra [3], were constructed. They were shown to possess a spectrum which is either continuous [1] or discrete (positive and bounded) [2]. In [1] the invariant PDEs were obtained via Verma module representation, while in [2] the so called on-shell condition was used (for the cases at hand the two approaches are proven to be equivalent).
In this paper we address several important issues that were not touched in these two previous works. We name a few: the identification of the general class of invariant PDEs (which turns out to depend on real parameters belonging to a fundamental domain), the existence of a contraction algebra, the reason for the cryptohermiticity (we use here the word adopted in [4]) of the discrete spectrum, the construction of the Hilbert space connected with Pais-Uhlenbeck oscillators with unbounded spectrum, etc.
Specifically, the following list of results is derived in the present paper (we limit here in the Introduction to discuss the first non-trivial case obtained for ℓ = 3 2 , the ℓ > 3 2 cases are commented in Section 8): two special differential realizations of cga ℓ produce, as invariant PDEs, Schrödinger equations with continuous and respectively discrete spectrum. Both realizations depend on a parameter γ = 0. Unitarily inequivalent theories are recovered for γ belonging to the fundamental domain γ ∈]0, +∞[. The γ = 0 PDEs are decoupled equations. The continuum spectrum case corresponds to the free Schrödinger equation in 1 + 1 dimensions, while the discrete spectrum case corresponds to a system of two decoupled cryptohermitian oscillators (namely, despite being non hermitian, possessing the same spectrum as two decoupled oscillators with the given frequencies). The parameter γ can therefore be regarded as a deformation parameter and as a coupling constant.
Without loss of generality we can fix ω 1 = 1 to be the energy mode of the first oscillator in the coupled PDE. Then, the cga 3 2 invariance of the PDE is recovered if the energy mode of the second oscillator possesses the critical values ω 2 = ± 1 3 , ±3 (ω 2 = 3 is the solution given in [2]). The negative values correspond to an unbounded spectrum and, as explained later, are connected with special cases of the Pais-Uhlenbeck oscillators. At fixed ω 1,2 , the spectrum of the operators does not depend on the value of γ.
The γ → 0 limit of the cga 3 2 algebra produces a contraction algebra which is a symmetry subalgebra of the decoupled systems. For the decoupled oscillators (without loss of generality the analysis can be limited to the ω 1 = 1, ω 2 ≥ 1 domain), the PDE possesses a 9-generator symmetry algebra at generic values, with enhanced symmetry at the critical values ω 2 = 1 and ω 2 = 3 (two different 12-generator symmetry algebras are obtained at these special points). The γ → 0 contraction algebra is a 8-generator subalgebra of the ω 2 = 3, 12-generator decoupled symmetry.
In this paper we discuss the subtle connection with Pais-Uhlenbeck oscillators. The Pais-Uhlenbeck model is a higher derivative system [5,4,6] which admits, via the Ostrogradskiȋ construction [7], a Hamiltonian formulation. The Ostrogradskiȋ Hamiltonian is canonically equivalent to a set of decoupled harmonic oscillators with alternating (positive and negative) energy modes. In a series of papers [8,9,10,11,12] the Pais-Uhlenbeck oscillators with energy modes given (up to a normalization factor) by the arithmetic progression ω i = 2i − 1 were linked to the Conformal Galilei Algebras cga ℓ (with ℓ = n − 1 2 ). As discussed in Section 8 the connection is rather subtle. To avoid any confusion, we should stress that the symmetry operators considered here are required to be realized as first-order differential operators.
The scheme of the paper is as follows: in Section 2 we present the (γ = 0-dependent) differen-tial realization for the deformation of the free Schrödinger equation at ℓ = 3 2 and the differential realization for the coupled oscillator. The connection of the two differential realizations obtained by similarity transformations and change of the time coordinate is shown in Section 3. In Section 4 the most general solution of the cga 3 2 -invariant oscillator is given. The symmetry of the decoupled oscillator (with enhanced critical points at ω 2 = 1 and ω 2 = 3) is presented in Section 5. The ℓ = 3 2 contraction algebra in the γ → 0 limit is given in Section 6. The Hilbert space for the oscillators and the relation with PT-symmetry is discussed in Section 7. In Section 8 the extension to the ℓ > 3 2 cases and the relation to Pais-Uhlenbeck oscillators are commented. Generalizations of the present construction are discussed in the Conclusions. consists of eight generators (z 0 , z ± , w ±1 , w ±3 , c), obeying the following non-vanishing commutation relations We introduce in this Section two differential realizations of the above algebra. As it will be explained in the following, the first differential realization corresponds to a symmetry of a deformed free Schrödinger equation in 1 + 1 dimensions, while the second differential realization corresponds to a symmetry of another system (possessing the same spectrum as two decoupled oscillators). The first differential realization is a one-parameter (γ = 0) extension of the differential realization obtained in [1] via standard left action on the space of functions defined on the coset (see also [13]). The presence of γ can be understood by the fact that the combined rescalings of the space and time coordinates which preserve the conformal Galilei algebra structure lead to just one free parameter. In this case we obtain first-order differential operators acting on functions of τ, x, y. They are given by The second differential realization is the γ extension of the differential realization presented in [2]. It is given by the first-order differential operators acting on functions of t, x, y, We consider now the three elements Ω 0 , Ω ±1 of the cga 3 2 enveloping algebra of degree 0, ±1 (measured by the degree generator − i 2 z 0 ), respectively: , , where the curly brackets on the r.h.s. denote anticommutators. The three operators Ω ±1,0 close the sl(2) algebra, with Ω 0 the Cartan element: It is important to note that for both differential realizations (2) and (3), Ω 0 , Ω ±1 are presented as second order differential operators. For the differential realization (2) we have The cga 3 2 on-shell invariant condition for Ω ±1,0 (see [14,15,2] for a definition) is guaranteed by the fact that their only non-vanishing commutators with the cga 3 2 generators are expressed as The degree 1 invariant equation is a Schrödinger equation with τ playing the role of a time coordinate. The parameter γ is a coupling constant. This equation can be regarded as a γ-deformation of the free Schrödinger equation in 1 + 1 dimensions. For the differential realization (3), Ω 0 , Ω ±1 are given by The on-shell invariant condition is guaranteed by the fact that their only non-vanishing commutators with the cga 3 2 generators are expressed as The degree 0 invariant equation is a Schrödinger equation with t playing the role of the time coordinate. The non-hermitian operator in the right hand side possesses a discrete and positive spectrum. The parameter γ is a coupling constant. The equation (11) can be regarded as a γ-deformation of a decoupled "cryptohermitian oscillator" discussed in the following.
3 Connection of the two differential realizations The two differential realizations (2) and (3) of the cga 3 2 algebra introduced in Section 2 induce Schrödinger equations from, respectively, degree 1 and degree 0 invariant operators.
The two differential realizations are connected via a similarity transformation coupled with a redefinition of the time coordinate.
Let us denote as g an operator entering (3) or (9) and as g its corresponding operator entering (2) or (6). For convenience we introduce the operator X + by setting, for z + in (3), The connection is explicitly realized by the similarity transformation g → g = e S ge −S , (e S = e S 2 e S 1 ), supplemented by the redefinition of the time coordinate The first similarity transformation (induced by S 1 ) allows to map so that with Due to the commutators where we obtain The remarkable identity implies that H +1 does not depend on the time coordinate (either t or τ ). The second similarity transformation (induced by S 2 ) allows to express in the form which reduces, in the γ → 0 limit, to the standard free Schrödinger equation in 1 + 1 dimensions. One should observe that the similarity transformation preserves the symmetry properties of the equations, mapping first-order invariant operators into first-order invariant operators.
The following commutative diagram is obtained: The left (right) part of the diagram denotes the equations obtained from the differential realizations (2) and (3), respectively. The horizontal arrows indicate the similarity transformation together with the change of the time coordinate, τ and t respectively.
The three invariant PDEs (at degree 0, ±1) are mapped into each other.
In the left part, the deformed Schrödinger invariant PDEs correspond to deg 1 and possess a continuous spectrum.
In the right part the deformed Schrödinger invariant PDEs correspond to deg 0. They possess a real, discrete spectrum which, as shown in Section 7, coincides with the spectrum of two decoupled harmonic oscillators. We will call this system the ℓ = 3 2 oscillator. The vertical arrows denote the mapping to the decoupled systems. This mapping can be reached in two ways: i) the singular similarity transformation (such that γ → e −α γ) in the α → ∞ limit. * Despite the singularity of the limit, the invariant equations of the upper part of the diagram admits as non-singular limit the decoupled equations of the lower part of the diagram. This similarity transformation preserves the symmetry of the equations, mapping first-order invariant operators into first-order invariant operators; ii) the non-singular similarity transformation This non-singular transformation does not preserve the symmetry of the equation because some of the transformed generators are no longer first-order differential operators. The four Schrödinger equations associated with the commutative diagram, starting from the upper right corner and proceeding clockwise, are: (I) the ℓ = 3 2 oscillator (11), (II) the decoupled (i.e. γ = 0) ℓ = 3 2 oscillator, (III) the free Schrödinger equation in 1 + 1 dimensions and, finally, (IV) the deformed free Schrödinger equation (8). The three inequivalent (with constant, linear and quadratic potential, see [15,14,16,17]) Schrödinger equations in 1 + 1 dimensions invariant under the Schrödinger algebra are recovered as restrictions of the ℓ = 3 2 invariant PDEs. Indeed, if we introduce the x, y separation of variables, the equation of the harmonic oscillator and the free Schrödinger equation are recovered by setting ∂ y ≡ 0 from, respectively, equations (I) and (IV). The linear Schrödinger equation is recovered from equation (IV) after setting Ψ(τ, x, y) = ψ(τ, x)φ(y), with the restriction ∂ y φ(y) = kφ(y).

The general ℓ = 3 2 oscillator
The cga 3 2 conformal Galilei invariance requires the coupling parameter γ = 0. Since a unitary transformation changes its phase, we can assume without loss of generality that γ belongs to the fundamental domain γ ∈]0, +∞[. For γ real, the invariant PDEs in the left part of the commutative diagram (23) are hermitian. This is not the case for the invariant PDEs in the right part of the diagram. Under hermitian conjugation, the ℓ = 3 2 oscillator equation is transformed into its conjugate All operators for any arbitrary constant C and any γ = 0, induce a Schrödinger equation with ℓ = 3 2 Conformal Galilei symmetry, if ω is restricted to the values The ω ↔ −ω change of sign is explained by the hermitian conjugation. Understanding the ω ↔ 1 ω transformation is subtler. One should note at first that in the γ = 0 decoupled case the role of the space coordinates x, y can be exchanged by performing the canonical transformation Next, the coupling term is introduced in terms of the non-singular similarity transformation given by the inverse of equation (25). As it turns out, this procedure guarantees the conformal Galilei invariance of the resulting PDE. An explicit check of the symmetries of this class of PDEs proves that, in order to have the on-shell invariant equations [ z ± , Ω 0 ] = f ± · Ω 0 , with f ± arbitrary functions of the coordinates and symmetry generators of the form z ± = e ±iλt (∂ t + X ± ), (X ± time-independent operators and λ = 0), the following necessary and sufficient condition has to be satisfied: the two equations must be simultaneously solved. The only non-vanishing solutions for λ are encountered at ω = ±3 and ω = ± 1 3 . Therefore, the ω = ± 1 3 , ±3 critical values are special points of enhanced symmetry.
The critical values ω = 1 and ω = 3 are points of enhanced symmetry for the decoupled system.

The enhanced symmetry for the decoupled ω = 1 system
At the critical value ω = 1 three extra generators are found at degree 0 and −1: They have to be added to the previous set of (generic) symmetry generators where we denoted, for ω = 1, w ±ω entering (31) as "w ±1b ".
The extra non-vanishing commutation relations involving the q i 's generators are The symmetry algebra closes as a non semi-simple, 12-generator, Lie algebra.

The enhanced symmetry for the decoupled ω = 3 system
At the ω = 3 critical value the three extra generators r −j , j = 1, 2, 3, of degree −j, are encountered. We have, explicitly, At ω = 3 the symmetry algebra is a 12-generator algebra which differs from the 12-generator symmetry algebra of the ω = 1 decoupled system. The extra non-vanishing commutation relations involving the r −j generators are given by 6 The contraction algebra A contraction algebra is recovered from (3) by taking the γ → 0 limit and by suitably rescaling the generators. The contraction requires the rescaling g → g = γ s g ( g is any generator entering (3)), with the power s given by The contracted 8-generator algebra expressed by z ± , z 0 , c, w k (k = ±1, ±3) is a subalgebra of the full 12-generator symmetry algebra of the ω = 3 decoupled system with generators given by (31) and (37). The identification goes as follows with the similarity transformation given by S = − 3 2 it. The contraction algebra corresponds to the two-dimensional Euclidean algebra acting on two sets of creation/annihilation operators. We have the e(2)⊃ +h(2) algebra, with non-vanishing commutators given by

Non-hermitian deformed oscillators with real eigenvalues
The non-hermitian operator derived from the degree 0 invariant equation (11) is It may be written (see (3)) in the form A discrete spectrum can be calculated algebraically from a lowest weight representation. The ground state ψ 0,0 is defined by The solution of the above equations gives the explicit expression of the ground state (up to normalization) The excited states are given by The corresponding eigenvalue E n,m is computed by the commutation relations One finds that the operator H 0 (γ) has real discrete eigenvalues which are identical to the decoupled harmonic oscillator eigenvalues: The explicit form of the eigenfunctions are readily obtained from (3). We list some of them.
In general, the excited state wavefunctions have the form of ψ n,m = P n,m (x, y)e −x 2 /2 e −(n+3m)it , where P n,m (x, y) is a degree n+m polynomial in x, y. The phase factor e −(n+3m)it is not essential to the eigenvalue problem. Following [4], we can call "cryptohermitian" any non-hermitian operator (such as (42)) with a real spectrum. The operator (42), being invariant under the transformations x → −x, y → y, i → −i, is PT-symmetric. Therefore, it belongs to the class of non-hermitian PT-symmetric operators (see [18] and [19] and [20] for a review) with real spectrum. It was pointed out in [21], [22] and [23] that PT-symmetric operators are often pseudo-hermitian. It turns out that this is not the case for the operator (42), since the similarity transformation (25) is not realized by a hermitian operator. For this reason the stronger notion of quasi-hermiticity, as well as the constructions based upon that (see [24], [25] and [26]), is not applicable in our context.
In Section 8 we discuss the connection of the operator (42) with a special class of Pais-Uhlenbeck oscillators, recovered at given algebraic frequencies. It is worth mentioning that PT-symmetry in the context of Pais-Uhlenbeck oscillators (at arbitrary, unequal, frequencies) was investigated in [27], [28] and [29].
In order to pave the way to discuss the connection between Conformal Galilei Algebras and Pais-Uhlenbeck oscillators at special frequencies, we need to introduce the operator K(γ), obtained from (42) via a non-canonical transformation which preserves the canonical commutation relations, but does not preserve the hermitian conjugation property. Up to a vacuum energy constant, we can set through the positions The two independent creation/annihilation operators (a, a † and b, b † ) have non vanishing commutators: [a, The similarity transformation (25) maps K(γ) into the decoupled hermitian operator K(0) through Since, as mentioned before, R is not hermitian, K(γ), for γ = 0, is not a pseudo-hermitian operator.
The operator K(γ) acts on the Hilbert space L 2 (R 2 ), the space of square-integrable functions defined on the coordinates x, z.
One can read from the commutators which excited modes are created. For any γ = 0, the solutions of the (54) equation are obtained for λ = ±3, ± 1 3 . The corresponding modes are In this basis the non-vanishing commutators are The non-hermitian operator K(γ) commutes with the "non-hermitian analog of the Number operator", N (γ). In terms of the A k modes, the operators are given by The Fock vacuum |vac > satisfies a|vac >= b|vac >= 0 , A −1 |vac >= A −3 |vac >= 0.
The Hilbert space L 2 (R 2 ) can be spanned by both sets of (unnormalized) states, harmonic oscillators with energy modes 1, −3 continues to be L 2 (R 2 ), obtained by applying the creation operators a † , b † to the Fock state |0, 0 > (a|0, 0 >= b|0, 0 >= 0). Due to the unboundedness of the spectrum, |0, 0 > can no longer be interpreted as the vacuum state.
The system with unbounded spectrum is related to the Pais-Uhlenbeck oscillators. We recall [5,4] that the Pais-Uhlenbeck model is a higher derivative system. It admits, via the Ostrogradskiȋ construction [7] (see [30] for a review), a Hamiltonian formulation. The resulting Ostrogradskiȋ Hamiltonian is canonically equivalent to a set of decoupled harmonic oscillators with alternating (positive and negative) energy modes. The n-oscillator Pais-Uhlenbeck system is canonically expressed as where ω i ∈ R and the constraint ω i < ω i+1 is satisfied. The harmonic oscillator with energy modes 1, −3 is a special case of the 2-oscillator Pais-Uhlenbeck model. In a series of papers [8,9,10,11,12] the Pais-Uhlenbeck oscillators with energy modes given (up to a normalization factor) by the arithmetic progression ω i = 2i − 1 were linked to the Conformal Galilei Algebras cga ℓ (with ℓ = n − 1 2 ). The present analysis proves that this association is rather subtle. We would like to stress that in this paper we consider (as it is standard in PDE's theory) the symmetry generators to be at most first-order differential operators. In this respect the Pais-Uhlenbeck PDE given by the decoupled harmonic oscillators does not possess any enhanced symmetry (not even at the special 1, −3 energy modes). The PDE, invariant under the Conformal Galilei Algebra, is obtained for the coupled operator only for γ = 0. The decoupled operator (γ = 0) possesses the 12-generator symmetry algebra (introduced in Section 6) which does not contain the Conformal Galilei Algebra as a subalgebra.
Even so, these results are not in contradiction with the findings in previous works on Pais-Uhlenbeck oscillators. For example, in [9] it was shown that on the Hamiltonian level the ω 1 = 1, ω 2 = −3 Pais-Uhlenbeck oscillators possess the (centrally extended) 3 2 -conformal Galilei symmetry in terms of Poisson brackets among conserved charges. It can be shown, on the other hand, that at the quantum level, some of the Noether charges entering formula (39) in [9] are second-order differential operators.
For general half-integer ℓ, the invariant PDEs which possess the Conformal Galilei algebra cga ℓ (for a definition, see [3]) as a symmetry algebra, depend on ℓ + 3 2 coordinates. The invariant PDEs are deformations of decoupled equations, depending on ℓ − 1 2 deformation parameters γ j = 0 (j = 1, . . . , ℓ − 1 2 ). The decoupled systems are recovered in the limit, for any j,γ j → 0. The invariant PDEs with continuous spectrum are The invariant PDEs with discrete spectrum are The energy modes ω i are normalized so that |ω i | = 2i − 1. The solution ω i = ǫ i |ω i | with all positive signs (∀i, ǫ i = +1) corresponds to the bounded discrete spectrum discussed in [2]. By taking the hermitian conjugation, the solution with flipped signs, ǫ i = −1 for all i, also leads to a cga ℓ -invariant PDE.
An explicit computation of the on-shell condition for ℓ = 5 2 (similar to the one presented in Section 4), proves that the cga 5 2 invariance is guaranteed by both choices of signs, ǫ 2 = ±1 and ǫ 3 = ±1. As explained above, the alternating choice (ǫ 2 = −1, ǫ 3 = +1) is related to a special case of three Pais Uhlenbeck oscillators (with 1, −3, 5 energy modes). An open problem is finding a general proof, valid for all half-integer ℓ, that every choice of ǫ i = ±1 signs lead to the cga l symmetry algebra of the PDE equation (65).

Conclusions
We summarize, in the Conclusions, the list of new results obtained in this paper.
We constructed the most general class of second-order PDEs, invariant under the d = 1 centrally extended Conformal Galilei Algebras cga ℓ with half-integer ℓ, proving that they are Schrödinger equations which are deformations of decoupled equations. For ℓ = 3 2 the unique deformation parameter is γ = 0 (the decoupled systems being recovered in the γ → 0 limit).
At ℓ = 3 2 the invariant PDEs with discrete spectrum, besides γ, depend on two frequencies ω 1 , ω 2 (entering the equation in non-symmetric form). The invariance under cga ℓ= 3 2 is only recovered at special critical values of the ratio r = ω 2 ω 1 given by r = ± 1 3 , ±3. The r = 3 value reproduces the bounded spectrum presented in [2], while the negative values r < 0 produce an unbounded spectrum which coincides with the spectrum of two Pais-Uhlenbeck oscillators at the given ratio r.
We further investigated the symmetry algebra of the γ = 0 decoupled systems for a generic value of r = ω 2 ω 1 , obtaining the following results. Enhanced symmetries are encountered at the critical values r = ± 1 3 , ±1, ±3. Two inequivalent 12-generator symmetry algebras are recovered at r = ± 1 3 , ±3 and r = ±1, respectively. The cga ℓ= 3 2 Conformal Galilei Algebra is not a subalgebra of the 12-generator, γ = 0 and r = ± 1 3 , ±3, decoupled symmetry. From cga ℓ= 3 2 , in the γ → 0 limit, a contraction algebra is recovered. The contraction algebra, see Section 6, is an 8-generator subalgebra of the full r = ± 1 3 , ±3 symmetry algebra of the γ = 0 decoupled system. As a corollary of this analysis we showed that the contraction algebra, rather than the Conformal Galilei algebra itself, is an enhanced symmetry of the r = −3 decoupled Pais-Uhlenbeck oscillators.
Concerning the ℓ = 3 2 PDE with discrete spectrum, we showed that, besides the r = 3 real bounded spectrum, a real unbounded spectrum is obtained at r = − 1 3 , −3. For all admissible values r the induced operators on the r.h.s. are not hermitian. They are, nevertheless, PTsymmetric [18]. They are not, on the other hand, pseudo-hermitian [21], since the non-singular similarity transformations (see equations (25) and (53)), mapping the coupled into the decoupled system, are not realized by a hermitian operator. The non-hermitian operator (51) acts on the Hilbert space L 2 (R 2 ). Its real discrete spectrum coincides with the spectrum of two decoupled harmonic oscillators. The Bogoljubov transformations relating the coupled and the decoupled systems are explicitly given.
The extension of this construction to the ℓ = 1 2 + N 0 centrally extended Conformal Galilei Algebras with d > 1 (see [3]) is immediate. The invariant PDEs with discrete spectrum corre-spond to non-hermitian operators whose spectrum is real and given by d copies of the energy modes created in the d = 1 case.
In the class of systems here investigated, the oscillators turn out to have all different frequencies. Equal frequency oscillators are obtained from the different class of second-order PDEs invariant under the d = 2 centrally extended Conformal Galilei Algebras with integer ℓ, see [31].