Non-Canonical Phase-Space Noncommutativity and the Kantowski-Sachs singularity for Black Holes

# Non-Canonical Phase-Space Noncommutativity and the Kantowski-Sachs singularity for Black Holes

Catarina Bastos Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico,
Avenida Rovisco Pais 1, 1049-001 Lisboa, PortugalDepartamento de Física e Astronomia,
Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre, 687,4169-007 Porto, PortugalDepartamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,
Avenida Campo Grande, 376, 1749-024 Lisboa, PortugalGrupo de Física Matemática, UL,
Avenida Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal.
Orfeu Bertolami Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico,
Avenida Rovisco Pais 1, 1049-001 Lisboa, PortugalDepartamento de Física e Astronomia,
Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre, 687,4169-007 Porto, PortugalDepartamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,
Avenida Campo Grande, 376, 1749-024 Lisboa, PortugalGrupo de Física Matemática, UL,
Avenida Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal.
Nuno Costa Dias Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico,
Avenida Rovisco Pais 1, 1049-001 Lisboa, PortugalDepartamento de Física e Astronomia,
Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre, 687,4169-007 Porto, PortugalDepartamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,
Avenida Campo Grande, 376, 1749-024 Lisboa, PortugalGrupo de Física Matemática, UL,
Avenida Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal.
and João Nuno Prata Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico,
Avenida Rovisco Pais 1, 1049-001 Lisboa, PortugalDepartamento de Física e Astronomia,
Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre, 687,4169-007 Porto, PortugalDepartamento de Matemática, Universidade Lusófona de Humanidades e Tecnologias,
Avenida Campo Grande, 376, 1749-024 Lisboa, PortugalGrupo de Física Matemática, UL,
Avenida Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal.
###### Abstract

We consider a cosmological model based upon a non-canonical noncommutative extension of the Heisenberg-Weyl algebra to address the thermodynamical stability and the singularity problem of both the Schwarzschild and the Kantowski-Sachs black holes. The interior of the black hole is modelled by a noncommutative extension of the Wheeler-DeWitt equation. We compute the temperature and entropy of a Kantowski-Sachs black hole and compare our results with the Hawking values. Again, the noncommutativity in the momenta sector allows us to have a minimum in the potential, which is relevant in order to apply the Feynman-Hibbs procedure. For Kantowski-Sachs black holes, the same model is shown to generate a non-unitary dynamics, predicting vanishing total probability in the neighborhood of the singularity. This result effectively regularizes the Kantowski-Sachs singularity and generalizes a similar result, previously obtained for the case of Schwarzschild black hole.

## 1 Introduction

Noncommutativity has been recurrently considered as a fundamental feature of space-time at the Planck scale CS (). Indeed, it is advocated that noncommutative geometry (NCG) captures the fuzzy nature of space-time at this scale, and that it should emerge as one of the main effects of quantum gravity Connes (); Madore (). In the context of string and M-theory, a configuration space noncommutativity arises as the low-energy effective theory of a D-brane in the background of a Neveu-Schwarz B field living in a space with spatial noncommutativity CS (). More recently, various aspects of noncommutative quantum field theory DS (); OBCZ () and quantum mechanics Bastos1 (); OB (); ZA (); Bastos9 () have been investigated.

Noncommutativity has also been considered in the context of Black Holes (BH) to address the issues of thermodynamical stability and the removal/regularization of singularities. In this framework, we have argued in a series of papers that an additional momentum-momentum noncommutativity leads to interesting and relevant properties Bastos2 (); Bastos5 (); Bastos3 ()111The impact of configuration space noncommutativity on BH has been throughly reviewed in Ref. Nicolini (). In these works the BH is modelled with the Kantowski-Sachs (KS) metric Kantowski () and, through the ADM procedure, the corresponding Wheeler-de Witt (WDW) equation is obtained and solved. In Refs. Bastos2 (); Bastos5 () we have considered a canonical phase-space noncommutativity for the two scale factors and the conjugate momenta of the KS geometry. There are two noteworthy consequences of this choice of the algebra. To start with, the solution of the WDW equation factorizes into an oscillatory part of a “time” variable and a “spatial” part which obeys an ordinary Schrödinger-like differential equation. The associated potential of this equation exhibits a local minimum. This allows for an expansion around the minimum and the saddle point evaluation of the partition function in accordance with the Feynman-Hibbs procedure Feynman (). From the rules of statistical mechanics, we can then compute the temperature and entropy of the BH. The results are tantamount to the Hawking-Bekenstein quantities plus stringy and noncommutative corrections. However, the essential point is that the local minimum of the potential only appears if momenta noncommutativity is assumed. The second consequence of the noncommutative algebra is that the “spatial” part of the wave function displays a pronounced damping behavior, and shown, that in fact, vanishes at infinity Bastos2 (). However, the obtained rate of decay is not sufficient to ensure square-integrability of the wave-function. But this is only marginally so. This prompted the search for alternative noncommutative algebras which might lead to a full-fledged square-integrable wave-function and the corresponding regularization of the BH singularity. In Ref. Bastos3 () we have proposed a non-canonical phase-space noncommutative algebra, which delivers a normalizable “spatial” wave function. This algebra is a parsimonious extension of the Heisenberg-Weyl (HW) algebra in the sense that: (i) it includes both position-position and momentum-momentum noncommutativity; (ii) it only requires one extra noncommutative parameter; (iii) it is globally isomorphic with the Heisenberg-Weyl algebra. In Ref. Bastos3 () we have shown that this algebra effectively regularizes the singularity of the Schwarzshild BH at the quantum level.

In the present work we further investigate the implications of this algebra. We compute the thermodynamical quantities of the BH, which are shown to have extra non-canonical noncommutative corrections. Moreover, we prove that this algebra also regularizes the singularity of the KS BH.

This paper is organized as follows. In section 2, we review the main aspects of the non-canonical noncommutative algebra of Ref. Bastos3 () and derive the Schrödinger-type equation satisfied by the “spatial” part of the wave function. In section 3, we evaluate the partition function of the BH by the Feynman-Hibbs method and compute the temperature and entropy of the BH. In section 4, we study the probability of the BH near the KS singularity. Finally, in section 5, we discuss some of the mathematical subtleties of the regularization result and present our conclusions. In this work we use units .

## 2 Phase-space non-canonical noncommutative quantum cosmology

The Schwarzschild BH is described by the metric,

 ds2=−(1−2Mr)dt2+(1−2Mr)−1dr2+r2dΩ2 , (1)

where is the radial coordinate and . For the time and radial coordinates are interchanged () so that space-time is described by the metric:

 ds2=−(2Mt−1)−1dt2+(2Mt−1)dr2+t2dΩ2 . (2)

This is an anisotropic metric, thus for , the interior of a Schwarzschild BH can be described as an anisotropic cosmological space-time. Indeed, the metric (2) can be mapped into the KS metric Dominguez (), which, in the Misner parameterization, can be written as

 ds2=−N2dt2+e2√3βdr2+e−2√3βe−2√3ΩdΩ2 , (3)

where and are scale factors, and is the lapse function. The following identification for Bastos5 (); Bastos3 ():

 N2=(2Mt−1)−1,e2√3β=(2Mt−1),e−2√3βe−2√3Ω=t2 . (4)

is a surjective mapping that transforms the metric Eq. (3) into the metric Eq. (2). Moreover, we also have from (4) that

 e−2√3Ω=t(2M−t) (5)

and so the Schwarzschild singularity () corresponds to

 t⟶0+⟹Ω,β⟶+∞. (6)

On the other hand, the KS singularity is attained at (cf. Eq.(3)) .

In this paper we shall study the thermodynamics of the interior of the black hole by considering the following non-canonical extension of the HW algebra Bastos3 ():

 [^Ω,^β] = iθ(1+ϵθ^Ω+ϵθ21+√1−ξ^Pβ) (7) [^PΩ,^Pβ] = i(η+ϵ(1+√1−ξ)2^Ω+ϵθ(1+√1−ξ)^Pβ) [^Ω,^PΩ] = [^β,^Pβ]=i(1+ϵθ(1+√1−ξ)^Ω+ϵθ2^Pβ),

where , and are positive constants and . Notice that this condition is experimentally satisfied in at least two distinct situations OB (); Bastos10 (). The remaining commutation relations vanish. For it implies that the noncommutative commutation and uncertainty relations are themselves position and momentum dependent. Notice that corresponds to the canonical phase-space noncommutativity Bastos1 (); Bastos2 (); Bastos5 (). We point out that the well known Darboux’s Theorem ensures that we can always find a system of coordinates, where locally the algebra can be written as the HW algebra. For this algebra, this statement also holds globally, so that the algebra is isomorphic to the HW algebra. Actually, algebra Eqs. (7) is the simplest non-canonical extension that is globally isomorphic to the HW algebra. It is an open issue whether this algebraic structure can be physically motivated or derived from a fundamental theory. On the other hand, this non-canonical extension has a direct impact on the singularity problem as shown in Ref. Bastos3 ().

The isomorphism between the algebra Eqs. (7) and the HW algebra is referred to as a Darboux (D) transformation. D transformations are not unique. Indeed, the composition of a D transformation with a unitary transformation yields another D transformation. However, the physical predictions are invariant under different choices of the D map. Suppose that obey the HW algebra:

 [^Ωc,^PΩc]=[^βc,^Pβc]=i . (8)

The remaining commutation relations vanish. Then a suitable transformation is:

 ^Ω = λ^Ωc−θ2λ^Pβc+E^Ω2c,^β=λ^βc+θ2λ^PΩc ^PΩ = (9)

Here, , are real parameters such that , and we choose the positive solution (given the invariance of the physics under different choices of the D map Bastos1 ()), and set

 E=−θ1+√1−ξF,F=−λμϵ√1−ξ(1+√1−ξ) . (10)

The inverse D map is easily computed:

 ^Ωc = 1√1−ξ(μ^Ω+θ2λ^Pβ) ^PΩc = 1√1−ξ(λ^PΩ−η2μ^β) ^βc = 1√1−ξ(μ^β−θ2λ^PΩ) (11) ^Pβc = 1√1−ξ⎡⎣λ^Pβ+η2μ^Ω−Fμ√1−ξ(^Ω+θ1+√1−ξ^Pβ)2⎤⎦.

It can be shown using Eqs. (2) and (2) that the algebra Eq. (7) implies the HW algebra, Eq. (8), and vice-versa. Of course, as already pointed out, we could consider other quadratic transformations relating the two algebras, which would, nevertheless, lead to the same physical predictions Bastos1 ().

We consider now the Hamiltonian associated to the WDW equation for the KS metric and a particular factor ordering Bastos2 ()

 ^H=−^P2Ω+^P2β−48e−2√3^Ω. (12)

Upon substitution of Eqs. (2), we obtain:

 ^H = −μ2^P2Ωc−η24μ2^β2c−η^βc^PΩc+μ2^P2βc+η24μ2^Ω2c+F2^Ω4c−η^Ωc^Pβc (13) + 2μF^Ω2c^Pβc−ηFμ^Ω3c−48exp(−2√3λ^Ωc+√3θλ^Pβc−2√3E^Ω2c) .

In the previous expression, the HW operators have the usual representation: , , and . Let us now define the operator , which corresponds to a constant of motion of the classical problem as discussed in Ref. Bastos2 () (which considers the same space-time setup, but assumes a canonical phase-space noncommutative algebra):

 ^A=μ^Pβc+η2μ^Ωc. (14)

A simple calculation reveals that for the noncommutative algebra Eq. (7), also commutes with the Hamiltonian , given by Eq. (12), i.e. . This quantity corresponds to the momentum shifted by , which can be seen as analogous to the canonical conjugate momentum in the presence of a gauge field, where corresponds to the electric charge and to the gauge field component.

Thus, one seeks for solutions of the WDW equation,

 ^Hψ=0 , (15)

which are simultaneous eigenstates of . The most general solution of the eigenvalue equation with real is Bastos5 (); Bastos3 ()

 ψa(Ωc,βc)=R(Ωc)exp[iβcμ(a−η2μΩc)], (16)

where is an arbitrary function of , which is assumed to be real.

¿From Eqs. (13), (15), and (16), one gets after some algebraic manipulation

 μ2R′′−48exp(−2√3μΩc−2√3EΩ2c+√3θaμλ)R−2ημ(Ωc+FΩ3c)R+ +F2Ω4cR+a2R+(η2μ2+2aF)Ω2cR=0 . (17)

The dependence on has completely disappeared and one is effectively left with an ordinary differential equation for . Through the substitution, , and one obtains a second order linear differential equation, which is a Schrödinger-like equation

 −ϕ′′a(z)+V(z)ϕa(z)=0 , (18)

where the potential function, V(z), reads:

 V(z)=−(Fμ2z2+ηz−a)2+48exp(−2√3z−2√3μ2Ez2+√3θaμλ). (19)

Equation (18) depends explicitly on the noncommutative parameters , , and the eigenvalue .

## 3 Thermodynamics of phase-space non-canonical noncommutative black hole

In this section we compute de thermodynamical properties of the noncommutative BH using the previous phase-space non-canonical noncommutative quantum cosmological model. The procedure used here is the one developed in Ref. Bastos5 () for the canonical noncommutativity. We consider the NCWDW equation to study the quantum behaviour of the interior of the Scharwzschild BH. To evaluate the noncommutative temperature and entropy of the BH, we compute the partition function through the Feynman-Hibbs procedure, applied to the minisuperspace potential function, Eq. (19) depicted in Fig. 1.

We present four different cases. The first case (a) describes the potential function for the commutative case , a strictly decreasing exponential function without a local minimum. The second case (b), the non-canonical noncommutativity in the configuration variables is considered, which reveals a potential function also without a local extremum. Finally, in the last two cases the noncommutativity is non-canonical and imposed (c) on the momentum sector , and (d) on the full phase-space. The last two cases admit a local minimum. Thus, as pointed out in Ref. Bastos5 () the potential has a local minimum only when the noncommutativity in the momentum sector is introduced. In the three last cases “the non-canonical” parameter was set to . This value is typical as are the values of the other noncommutative parameters. For some other values of these three parameters we encounter a qualitatively similar behavior for the potential function. When we set in the potential, we recover the results of Ref. Bastos5 ().

¿From Fig. 2 one concludes that the potential function has a minimum, which is located at small values. Around the minimum the qualitative behavior of the two graphics (Fig. 2a) and b)) is quite similar. This allow us to safely neglect the term in the derivation of the noncommutative temperature and entropy of the BH. We stress that the Feynman-Hibbs method can be employed only when the noncommutative parameter associated with the momenta is non-vanishing. It is only under these circumstances that we have a local minimum of the minisuperspace potential. We can then expand the potential up to second order around the minimum. The potential function is, then,

 V(z)=−(ηz−a)2+48exp[−2√3(z−θa2μλ+μ2Ez2)] . (20)

In order to use the Feynman-Hibbs procedure to evaluate the partition function, one expands the exponential term in the potential Eq. (20) to second order in powers of the variable. The minimum is then obtained by solving the following equation

 dVdz|z0=η(ηz0−a)+48√3(1+2μ2Ez0)exp[−2√3(z0−θa2μλ+μ2Ez20)]=0 . (21)

The minimum is thus defined by the relation

 exp[−2√3(z0−θa2μλ+μ2Ez20)]=−η(ηz0−a)48√3(1+2μ2Ez0) . (22)

Moreover, it should satisfy the second derivative criterion

 6((1+2μ2Ez0)2−√33μ2E)exp[−2√3(z0−θa2μλ+μ2Ez20)]−l2>0 , (23)

where . Using these relations, the second order expansion of the potential function in powers of satisfies:

 V(z)=48B(z−z0)2−(ηz0−a)2+48k , (24)

where and . Notice that the minimum is semiclassically stable as can be shown, following the arguments of Ref. Bastos5 ().

So, we can rewrite the ordinary differential equation resulting from the NCWDW equation as

 −12d2ϕdz2+24[B−l2](z−z0)2ϕ=[12(ηz0−a)2−24k]ϕ . (25)

Comparing Eq. (25) with the Schrödinger equation of the harmonic oscillator, we can identify the noncommutative potential and the energy of the system,

 VNC(y)=24(B−l2)y2 , (26)

in terms of the variable . A quantum correction is introduced to the partition function through the potential, as given by the Feynman-Hibbs procedure Feynman ():

 βBH24V′′NC(y)=2βBH(B−l2) , (27)

where is the inverse of the BH temperature. The noncommutative potential with the corresponding quantum corrections then reads,

 VNC(y)=24(B−l2)(y2+βBH12) . (28)

Finally, the noncommutative partition function is given by

 ZNC=√148(B−l2)1βBH×exp[−2β2BH(B−l2)] . (29)

The thermodynamic quantities of the Schwarzschild BH can now be computed. Starting from the noncommutative internal energy of the BH, ,

 UNC=1βBH+4(B−l2)βBH , (30)

the equality , allows for obtaining an expression for the BH temperature:

 βBH=M8(B−l2)×[1±√1−16M2(B−l2)] . (31)

Inverting this quantity, one obtains the BH temperature. Dropping the term proportional to , as presumably , and considering the positive root, we obtain:

 TBH=4M(B−l2) . (32)

We can clearly see that this quantity is positive (cf. Eq. (23)). If , we recover the results of Ref. Bastos5 (). That is, this non-canonical noncommutativity introduces new corrections to the noncommutative temperature. Furthermore, the mass dependence of the noncommutative temperature remains the same as the Hawking temperature for a BH, . As in Ref. Bastos5 (), we can recover the Hawking temperature for a value. Thus, equating Eq. (32) with the Hawking temperature and using the stationary condition Eq. (22), we obtain for , and :

 z0=2.26369η0=0.487. (33)

And thus, since cannot be exactly equal to zero, we can regard as a reference value which yields the Hawking temperature, and as increases we get a gradual noncommutative deformation of the Hawking temperature. Notice that this value differs from the one obtained using a canonical phase-space noncommutativity Bastos5 (). This occurs as the non-canonical model has more parameters, , and , it is obvious that they will affect the value of .

The entropy is calculated using the expression, . Thus, the entropy for the phase-space noncommutative BH is given by:

 SBH=12+M216B+M216B√1−16BM2−12ln[32M2B(1+√1−16M2B−8BM2)] . (34)

Neglecting as before terms proportional to as , we finally obtain

 SBH≃M28(6k[(1+2μ2Ez0)2−√33μ2E]−l2)−12ln⎛⎜⎝3M26k[(1+2μ2Ez0)2−√33μ2E]−l2⎞⎟⎠ . (35)

For the reference value , we recover the Hawking entropy, plus some “stringy” corrections:

 SBH=4πM2+ln√2π3−ln(8πM) . (36)

In summary, as in the case of the temperature, we obtain that the noncommutative entropy of the BH is the Hawking entropy plus additional contributions and some noncommutative corrections.

## 4 The Singularity Problem

Let us now turn to the singularity problem and further elaborate on the issues discussed in Ref. Bastos3 (). The wave function Eq. (16) solution of the NCWDW equation is oscillatory in and the remaining -sector is square-integrable and, hence, in the following, it is natural to consider hyper-surfaces of constant . This suggests that a measure should be assumed for the definition of the inner product in and the evaluation of probabilities. Starting from this premise, we were able to prove in Ref. Bastos3 () that the probability vanishes in the vicinity of the Schwarzschild singularity (corresponding to , cf. Eq.(6)). Here, we wish to investigate under which circumstances the entire KS singularity () could be regularized by our non-canonical noncommutative model.

It is worthwhile reviewing our arguments. A generic solution of the NCWDW equation can be written as

 ψ(Ωc,βc)=∫C(a)ψa(Ωc,βc)da , (37)

where are (for the time being) arbitrary complex constants and is a simultaneous solution of and ,

 ψa(Ωc,βc)=ϕa(Ωcμ)exp[iβcμ(a−η2μΩc)] , (38)

Here is a solution of Eq. (18). This equation is asymptotically of the form (cf. Fig. 2)

 −ϕ′′a(z)−Kz4ϕa(z)=0,z→+∞ , (39)

( is a positive constant) and so its solutions satisfy, cf. [pag. 75 Voronov ()] ()

 ϕa(z)≃Czexp(±iK1/23z3),z→+∞ , a∈IR , (40)

and are thus square integrable. Hence we may impose the normalization:

 ∫|ψa(Ωc,βc)|2dΩc=∫∣∣ϕa(Ωcμ)∣∣2dΩc=1 . (41)

With the assumed measure, we then have:

Notice that, in general, the functions are not orthogonal as they are solutions of a hyperbolic-type equation (see comment in the discussion below). We may nevertheless apply the Cauchy-Schwartz inequality to obtain:

 ||ψ||L2(IR2,dξ)≤∫|C(a)|||ψa||L2(IR2,dξ)da=∫|C(a)|da . (43)

We conclude that for the wave function Eq. (37) belongs to , i.e. it is square-integrable on constant hypersurfaces. Hence the probability of system reaching the Schwarzschild singularity is:

 P(r=0,t=0)=lim˜Ωc,βc→+∞∫+∞˜Ωc∫+∞−∞|ψ(Ωc,β)|2dξ==lim˜Ωc,βc→+∞∫+∞˜Ωc|ψ(Ωc,βc)|2dΩc=0 (44)

which effectively regularizes the singularity as point out in Ref. Bastos3 ().

We next consider the more general KS singularity, corresponding to , (c.f. Eq.(3)). We thus wish to compute probabilities of the type:

 P(Ωc∈I,βc→+∞)=limβc→+∞∫I∫IR|ψ(Ωc,β)|2dξ=limβc→+∞∫I|ψ(Ωc,βc)|2dΩc , (45)

where is some compact subset of . A simple calculation shows that:

And thus:

 P(Ωc∈I,βc→+∞)=limβc→+∞∫I∣∣∣∫IRC(a)ϕa(Ωcμ)exp(iβcμa)da∣∣∣2dΩc . (47)

Notice that and thus , that is:

 (48)

We have already assumed that . Given the fact that the potential Eq. (19) varies smoothly with respect to , it is reasonable to expect that for a suitable choice of constants , we may admit the following regularity conditions:

1)
2)

We then have:

 ∫I∣∣∣∫IRC(a)ϕa(Ωcμ)exp(iβcμa)da∣∣∣2dΩc≤|I|[∫IR|C(a)|M(a)da]2<∞ , (49)

where we have used the regularity condition 2), and where , since is compact. We thus have a uniform convergence on every compact interval and all . This means that we can safely interchange the integral and the limit to obtain:

 P(Ωc∈I,βc→+∞)=∫Ilimβc→+∞∣∣∣∫IRC(a)ϕa(Ωcμ)exp(iβcμa)da∣∣∣2dΩc . (50)

¿From the regularity condition 2) the functions regarded as functions of (for each fixed ) belong to . Indeed, , for all . On the other hand, the Riemann-Lebesgue Lemma Stein () states that every function has a continuous Fourier transform, which vanishes at infinity, i.e.:

for any . We conclude that:

 P(Ωc∈I,βc→+∞)=0 (52)

for any compact set . This effectively regularizes the KS singularity as well.

## 5 Discussion and Conclusions

In this work we obtain the noncommutative temperature and entropy of the Schwarzschild BH using a non-canonical noncommutative extension of a KS cosmological model. This formulation properly generalizes the full commutative, as well as the configurational and the momentum (canonical) noncommutative, models. We showed that the use of the Feynman-Hibbs method is only meaningful if one admits a non-vanishing noncommutativity in the momentum sector (i.e. ) and we find that the non-canonical noncommutativity introduces further corrections to the noncommutative temperature and entropy obtained in Ref. Bastos5 (). Just as in Ref. Bastos5 (), the Hawking quantities for the BH are recovered for a non-vanishing value of , which in the present setup corresponds to .

In the second part of the paper we have showed that the KS singularity (just as the Schwarzschild singularity, studied in a previous paper) is regularized by the non-canonical, noncommutative regime. This is the main feature of our model and a direct consequence of the fact that the eigenstates are square integrable, in spite of being associated to a continuous set of eigenvalues . This property is shared by all Hamiltonians of the form with a potential asymptotically like (for some ) as Voronov (). The reason for the apparent paradox of having square integrable eigenstates and a continuous spectrum is that the Hamiltonians defined on their maximal domain

 (53)

are not self-adjoint operators. In fact, they are the adjoint operators of the symmetric Hamiltonians

 H(0):S(IR)→L2(IR),ϕ→H(0)ϕ=(p2+V)ϕ (54)

(where is the set of Schwartz functions on ) and act on domains larger than the domains of the self-adjoint extensions of . Since is not self-adjoint, it is not surprising it generates a non-unitary time evolution.

It is worthwhile to compare these results with the full commutative case () and the momentum noncommutative case (, ). In the full commutative case (as also for configurational noncommutativity, i.e. but ) the potential Eq.(19) behaves asymptotically as . Consequently, the Hamiltonian Eq.(54) has deficiency indices , i.e. it is essentially self-adjoint, and its (unique) self-adjoint realization is defined on the maximal domain Eq.(53). Hence, its spectrum is continuous, its eigenstates are not normalizable and the time evolution is unitary.

Most of these properties are also shared by the momentum (canonical) noncommutative case, where the potential behaves asymptotically like . The operator , defined on the domain Eq.(53), is still self-adjoint. Its eigenstates display a pronounced damping behavior which, nevertheless, is not sufficient to make them normalizable. The spectrum is continuous and the evolution unitary Bastos5 ().

Once we give up the self-adjointness of the Hamiltonian, we may become apprehensive that the corresponding eigenvalues may not be real. We stress that a self-adjoint operator is a sufficient condition for the reality of the eigenvalues. However, it is by no means necessary. In fact, a wide class of non self-adjoint Hamiltonian operators has recently been studied which exhibit real eigenvalues Bender (). These Hamiltonians are -symmetric, that is they display an invariance under parity and time reversal transformations. An unbroken -symmetry can be shown to be equally sufficient for a real spectrum. Our Hamiltonian in Eqs. (18) and (19) looks asymptotically like (with ). This Hamiltonian has been analyzed in Bender () and shown to be -symmetric.

### Acknowledgments

The work of CB is supported by Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/62861/2009. The work of OB is partially supported by the FCT grant PTDC/FIS/111362/2009. NCD and JNP were partially supported by the grants PTDC/MAT/69635/2006 and PTDC/MAT/099880/2008 of FCT.

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