The logarithmic phase singularity in the inverted harmonic oscillator

Intriguing physical phenomena manifest themselves in the proximity of singularities. Of particular interest are singularities induced by gravity, where spacetime itself becomes singular. In an outstanding work,¹ it has been demonstrated that a black hole, a particular spacetime singularity, is a robust prediction of general relativity.² Evidence for the existence of such a supermassive object at the center of the Milky Way has been provided by the exploration³ of Sagittarius A^∗. Recently, gravitational wave detection has even enabled the first observation⁴ of a merging binary black hole system.

However, due to their weak nature, unobserved so far are quantum effects that cause a thermal evaporation of black holes. At the very heart of this so-called Hawking radiation⁵ lies a logarithmic phase singularity that manifests itself at the event horizon encoded in the metric of spacetime. Its origins are to be found in the hyperbolic motion in curved spacetime.⁶ 

There are many situations when physical systems display phenomena connected to black hole evaporation.⁷ Arguably, the most prominent example is acceleration radiation,⁸ as suggested by Unruh, which is related to Hawking radiation via the equivalence principle. However, even two-level atoms falling into a black hole would emit acceleration radiation which resembles, but is different from, Hawking radiation for a distant observer. Also, here, a logarithmic phase singularity appears in combination with an event horizon, resulting in a particular form of the excitation probability of the atoms.^9–11 Related features emerge in the context of quantum chaos,¹² the quantum Hall effect,^13,14 and the Riemann Zeta function.¹⁵ 

By now, various systems analogous to black holes have been identified^16,17 that are experimentally accessible and enable a direct study of the phenomena noted above. They range from the presence of a sonic horizon for sound waves,¹⁸ over the field of optics,¹⁹ as expressed in the quantum catastrophe of slow light,^20,21 and Bose–Einstein condensates,^22–24 to setups employing water waves.^25,26

Insight into this plethora of physical systems can be provided by simple models that cover the main features of the underlying effects. In the view of Hawking radiation, such an elementary model^12,13,27,28 is the inverted harmonic oscillator.^29–33 We point out that the analogy of both systems has already been studied in detail in Ref. 13. In contrast to this work, we take a different viewpoint and focus on a quantum mechanical perspective that becomes most evident within a phase space formulation. By providing an in-depth analysis of the inverted harmonic oscillator, we concentrate on the key ingredients of the apparent similarities in both systems. We trace them back to the presence of horizons and a logarithmic phase singularity.

In our article, we present a detailed study of a particle exposed to an inverted parabolic potential. By providing a short review, we relate relevant phenomena of quantum field theory such as Hawking radiation to this simple quantum system. Similar as the event horizon of a black hole separates spacetime into two disjunct regions, the inverted harmonic oscillator also displays horizons, which are now located in phase space. These horizons, appearing already in the classical dynamics of a particle subject to an inverted parabolic potential, crucially influence the corresponding quantum mechanical situation. In fact, this feature becomes most evident in Wigner phase space, which provides an overall and intuitive picture of the problem. By deriving the Wigner functions of the energy eigenstates, we demonstrate that they only live on a half plane of phase space that is isolated by a horizon.

However, it is surprising that the position representation of the energy eigenstates does not display such a horizon. We show that only by choosing a particular representation, the horizons can be recovered. In these rotated quadratures, the quantum waves are governed by a logarithmic phase singularity in combination with an amplitude singularity. We argue that the occurrence of a logarithmic phase singularity is of central relevance in establishing a connection between this system and the evaporation of a black hole. Indeed, the mode function of a massless scalar field also displays a logarithmic phase singularity at the event horizon of a black hole.

Moreover, it is the Fourier transform of this mode function that leads to the appearance of a Bose–Einstein statistics that governs the number of particles emitted by a black hole.^27,34,35 While this is a result of the second quantization of a scalar field, a related dependency arises in the single particle problem of an inverted harmonic oscillator. Indeed, the transmission and reflection coefficients of the inverted harmonic oscillator resemble a Fermi–Dirac distribution.^29,30,36 We demonstrate that this dependency results from the Fourier transform of a logarithmic phase singularity in combination with an amplitude singularity that arises at a horizon in phase space.

With the help of an elementary quantum system, our analysis thus reveals underlying effects of Hawking radiation. We hope that our results might stimulate further research into this direction and allow us to connect at first glance seemingly unrelated fields.

Our article is structured as follows. Section II provides a short overview on pair production in the presence of an electric field and the evaporation of a black hole. It serves the purpose to establish a link of these quantum effects to the system of the inverted harmonic oscillator. In Sec. III, we concentrate on the inverted harmonic oscillator in the classical domain. In particular, we present the emergence of horizons in phase space and the origin of the logarithmic singularity. Next, we turn to the quantum domain in Sec. IV. Here, we address the singularities present in the energy eigenfunctions of the inverted harmonic oscillator. We establish the relation to features familiar from Hawking radiation emitted at the event horizon of a black hole. Especially, we show that a logarithmic phase singularity in combination with an amplitude singularity leads to a distribution that resembles the Fermi–Dirac statistics. In Sec. V, we illustrate our results in phase space. For this purpose, we make use of Wigner functions corresponding to the energy eigenstates of the inverted harmonic oscillator. Due to the presence of horizons, these functions only live on a half plane of phase space. We conclude in Sec. VI by summarizing our results and provide an outlook to future topics of interest.

Our analysis is supplemented by two  Appendixes which contain transformations between different basis states as employed for the description of the inverted harmonic oscillator. In  Appendix A, we focus on the position representation of rotated quadrature states. In  Appendix B, we derive the position representation of the energy eigenstates.

In the following, we review two quantum field effects. In particular, we consider the pair production induced by the presence of a static electric field and the emission of Hawking radiation at the event horizon of a black hole. We focus on the singularities encoded in the mode functions for these problems and establish a relation to the simple quantum system of an inverted harmonic oscillator.

In the early days of quantum mechanics, it has already been argued that an electric field changes the vacuum in such a way that pairs of particles and antiparticles can be created.^37,38 In order to obtain a deeper insight into this mechanism, we relate this problem to the system of an inverted harmonic oscillator, which is at the heart of our article.

For the sake of simplicity, we restrict ourselves to one spatial dimension x and follow the approach presented in Refs. 7 and 39 for the case of a scalar field $Ψ$⁠. In the presence of a constant electric field $E$⁠, the Klein–Gordon equation takes the form

where m and q are the mass and charge, respectively, c is the speed of light, and $ℏ$ denotes the reduced Planck constant.

Next, due to the stationary nature of the electric field $E$⁠, we identify solutions of Eq. (1) by separating the dependence of $Ψ=Ψ(t,x)$ with regard to time t and position x. For a mode $ΨΩ(t,x)=exp(−iΩt)fΩ(x)$ of frequency Ω, the mode function $fΩ(x)$ then satisfies the equation

resulting from Eq. (1). Here, we have introduced the parameters $ω≡qE/(mc), x0=−ℏΩ/(qE)$⁠, and $ε=−mc2/(2ℏω)$⁠.

It is intriguing to ascertain that Eq. (2) actually corresponds to the stationary Schrödinger equation of a quantum particle of mass m. This particle is exposed to a one-dimensional inverted harmonic oscillator of steepness ω which is centered at the coordinate x₀. As a consequence, the mode function $fΩ(x)$ coincides with the eigenfunction of energy $ℏωε$ of an inverted harmonic oscillator.

In the succeeding Secs. III and IV, we demonstrate that the behavior of these eigenfunctions is governed by horizons in phase space at which a logarithmic phase singularity occurs. For this reason, we anticipate that a better understanding of an inverted harmonic oscillator might lead to a deeper insight into the pair creation in the presence of a strong electric field.

In the following, we show that not only the pair creation mechanism but also the evaporation of a black hole displays analogies to the inverted harmonic oscillator. For this purpose, we consider a scalar quantum field with massless quanta in the presence of the gravitational field of a static, uncharged, and homogeneous black hole of mass M. For further details, we refer to Refs. 7 and 13.

The modes of the scalar field ψ can be obtained by solving the covariant Klein–Gordon equation

where $g=det(gμν)$ denotes the determinant of the metric tensor $gμν$⁠.

We analyze the field in the Schwarzschild space, that is for the radius $r>rs$⁠, where the Schwarzschild radius $rs=2GM/c2$ determines the event horizon of the black hole in terms of the gravitational constant G and the speed of light c. In this domain, spacetime is governed by the Schwarzschild metric with the line element

Here, $dΩs2=dθ2+sin2θdϕ2$ denotes the square of the differential for the solid angle Ω_s as determined by the polar angle θ and the azimuthal angle $ϕ$ in spherical coordinates.

Next, we introduce the Regge–Wheeler tortoise coordinate

with

such that the line element in Eq. (4) simplifies to

By making use of the determinant $g=−1/(r4 sin2θ)$ of the metric tensor $gμν$⁠, we recast the Klein–Gordon equation (3) as⁷ 

where we have introduced the angular momentum operator $L̂$⁠.

For spherical symmetric waves with angular momentum quantum number l = 0, we arrive at

for $r≈rs$⁠, where the last term in Eq. (8) can be neglected.

The separation of variables $ψ0(t,r)=q(t)R0(r)$ in Eq. (9) then yields the relation

Thus, in the temporal domain, we obtain the oscillator equation

for the function q(t), and in radial direction, we arrive at

for $R0(r)$ with $k=Ω/c$⁠.

The radial equation (12) then determines the mode functions

which display a logarithmic phase singularity at the event horizon located at $r=rs$⁠. Here, we have made use of the relation between the tortoise coordinate $r∗$ and the radial coordinate r presented in Eq. (5). In addition, the occupation of a mode relies on the quantization of the time-dependent Eq. (11), leading to the appearance of the creation and annihilation operators that are familiar from the harmonic oscillator.

The logarithmic singularity in Eq. (13) plays a crucial role in the case of a two-level atom falling into a black hole and emitting acceleration radiation.¹⁰ There it has been shown that it is the Fourier transform of the logarithmic phase that determines the excitation probability such that it follows a Bose–Einstein distribution.

In the case of Hawking radiation,⁵ the pair creation at the event horizon of a black hole is described in terms of a scattering problem for ingoing and outgoing waves. However, it is the Fourier transform of the logarithmic phase in the mode functions that determines the number of particles emitted in a mode of frequency Ω as

times the number of particles that would have been absorbed. Here, κ denotes the surface gravity of the black hole. Clearly, Eq. (14) corresponds to a Bose–Einstein statistics.

To conclude, the relationship of the pair creation mechanism and the evaporation of a black hole to the system of the inverted harmonic oscillator might reveal a deeper insight into both phenomena. In the following, we demonstrate this relationship with the help of a detailed analysis of the inverted harmonic oscillator.

As counterpart of the harmonic oscillator, the inverted harmonic oscillator^13,29–33 has found great application in modeling systems that display a point of instability. For instance, it has been applied in the context of nuclear fission^40,41 or to explore dissipative quantum systems with a potential barrier.⁴² Yet, many interesting features have not been analyzed to the same extent as those emerging in the harmonic oscillator.

In the following, we focus on the dynamics of a classical particle in the presence of an inverted harmonic oscillator potential. We identify horizons in phase space in combination with a logarithmic singularity. Our findings provide the basis for the analysis of the corresponding quantum effects.

A classical particle of mass m exposed to the parabolic potential $V(x)≡−(1/2)mω2x2$ is described by the Hamiltonian

characterizing an inverted harmonic oscillator of steepness ω, as depicted in Fig. 1(a). Here, p denotes the momentum and x is the position.

Accordingly, the dynamics of the particle is governed by the Hamilton equations of motion

whose solution gives raise to the classical phase space trajectories

with x₀ and p₀ being the initial position and momentum at time t = 0, respectively.

Since the Hamiltonian H(x, p), Eq. (15), is time-independent, each phase space trajectory is associated with a particular energy $E=H(x0,p0)$ as determined by the initial condition for the respective motion. Due to the quadratic dependence on x and p, the Hamiltonian H(x, p) is invariant³¹ under the transformations (i) $p→−p$ and (ii) $x→−x$⁠. Hence, there are two distinct motions corresponding to the same value of the energy E, which however differ by the incoming direction of the particles as shown in Fig. 1. For negative energies E < 0 (red and orange), the particle is reflected at the potential barrier, while for positive energies E > 0 (blue and green), the particle is transmitted.

In Fig. 1(b), we present the corresponding dynamics in phase space by the hyperbolas ${xcl(t),pcl(t)}$ for different initial conditions x₀ and p₀ according to Eqs. (18) and (19). For each trajectory associated with an incoming particle from the left of energy E < 0 (red) and E > 0 (blue), there is an equivalent trajectory for an incoming particle from the right with energy E < 0 (orange) and E > 0 (green), respectively. For negative energies E < 0, the motion of the particle is restricted to the domain x < 0 (incoming from the left) or x > 0 (incoming from the right). For positive energies E > 0, its motion is instead restricted to the domain p > 0 (incoming from the left) and p < 0 (incoming from the right).

In order to characterize these different domains in phase space, we introduce the coordinates

and

where we have made use of the Planck constant $ℏ$ to arrive at dimensionless quantities. Consequently, with Eqs. (20) and (21), the Hamiltonian H(x, p), Eq. (15), can be expressed in the form

which is invariant under the exchange of the coordinates ξ and η. Here, we have maintained the specific order in which these coordinates enter.

Next, we focus on the separatrices ξ = 0 and η = 0, depicted by the black diagonal lines in Fig. 1(b), which divide phase space into four disjunct regions. Analogous to an event horizon of a black hole that leads to a causal separation of different regions in spacetime, the lines ξ = 0 and η = 0 restrict the accessible regions in phase space. For this reason, we refer to them as horizons in phase space.

We are now in the position to reveal the intimate relation between the inverted harmonic oscillator and the logarithmic dependence that is characteristic for this system.

For this purpose, we recall the Hamilton equations of motion given by Eqs. (16) and (17). In the coordinates ξ and η, they read

By inserting the Hamiltonian H, Eq. (22), of the inverted harmonic oscillator, Eqs. (23) and (24) reduce to the logarithmic derivatives

As solution of Eqs. (25) and (26), the classical phase space trajectory in the coordinates ξ and η takes the simple form

where the initial values ξ₀ and η₀ are assumed at time t = 0. We note that Eqs. (27) and (28) agree with the expressions for the classical phase space trajectories presented in Eqs. (18) and (19) with regard to position x and momentum p, as could be shown by using Eqs. (20) and (21).

In Fig. 2, we depict the phase space trajectories for particles with different initial conditions ξ₀ and η₀. In agreement with Eqs. (27) and (28), we observe that the motion of a classical particle in the inverted harmonic oscillator can be obtained by dilations of the phase space coordinates ξ and η. While the coordinate ξ₀ is scaled by $exp (−ωt)$⁠, the coordinate η₀ experiences a scaling with regard to the reciprocal factor $exp (ωt)$ yielding the phase space coordinates of the particle at a later time t.

As a consequence, for any time t, each classical trajectory is restricted to a particular quadrant of phase space that is determined by the sign of the initial values ξ₀ and η₀. According to Eq. (22), these phase space trajectories are associated with the energy $E=−ℏωξ0η0$⁠. Consequently, a particle located initially on the horizon $ξ0=0$ or $η0=0$ travels along the respective horizon with a vanishing energy E = 0.

First, we focus on particles that travel forward in time and share the same initial value $ξ0>0$ at time t = 0, as displayed in Fig. 2(a). According to Eq. (25), it then takes the time

to reach the coordinate ξ₁. Consequently, at this time, all particles with initial coordinate ξ₀ reach the same ξ-coordinate ξ₁, as evident from Fig. 1. Moreover, due to the logarithmic dependence in Eq. (29), these particles never reach the horizon ξ = 0 in a finite amount of time.³¹ 

Next, we turn to particles traveling backwards in time with an identical value $η0>0$ at time t = 0, as depicted in Fig. 2(b). Here, it takes the time

for all particles to reach the η-coordinate η₁ according to Eq. (28). Thus, the horizon η = 0 cannot be reached within a finite duration.

This behavior is a consequence of the fact that time reversal leads to a reflection of the momentum $p→−p$ which, according to Eqs. (20) and (21), corresponds to the exchange $ξ↔η$ of the phase space coordinates ξ and η. Indeed, the reflection $p→−p$ establishes the connection between the phase space trajectories displayed in Figs. 2(a) and 2(b).

In conclusion, we emphasize three important properties of the inverted harmonic oscillator: (i) the dynamics of a classical particle in the inverted harmonic oscillator can be obtained by dilations of the phase space coordinates ξ and η. Here, the scale factor is governed by the time t. For one of these phase space coordinates, the scale factor is the reciprocal of the other one. (ii) The horizons ξ = 0 and η = 0 separate different domains in phase space which distinguish overall aspects of a particular motion. In particular, these are the direction of the incoming particle, as expressed by the initial momentum p < 0 and p > 0, and the behavior at the potential barrier, as determined by the energy E < 0 and E > 0. Indeed, the horizons of an inverted harmonic oscillator, separating disjunct regions of phase space, remind of the event horizon of a black hole that cuts apart different domains of spacetime. (iii) The emergence of the logarithmic singularity in time, as presented in Eqs. (29) and (30), is a characteristic feature that occurs at the horizons in phase space of an inverted harmonic oscillator. Analogously, at the event horizon of a black hole, a logarithmic singularity also occurs in the mode functions of emitted Hawking radiation as shown in Sec. II.

In Sec. III, we have demonstrated that already a classical inverted harmonic oscillator points to several features that are familiar from Hawking radiation. The relationship between Hawking radiation and the inverted harmonic oscillator becomes even more evident when we turn to the quantum domain. Due the quadratic dependence of this particular potential, the classical and quantum dynamical equations of motion in the inverted harmonic oscillator are closely related to each other. However, the energy eigenstates as particular quantum states are the key to a deeper understanding of the logarithmic singularity.

In the following, we first introduce the relevant operators and quantum states that are required for our analysis. Next, we focus on the form of the energy eigenstates of the inverted harmonic oscillator, displaying a horizon in combination with a logarithmic phase singularity when represented in a suitable basis. As a consequence of the Fourier transform of a logarithmic phase, we then demonstrate the emergence of a functional dependence that resembles the Fermi–Dirac statistics.

As shown in Sec. II, a horizon, a logarithmic phase, and a Fourier transform of the mode function also play a crucial role in the context of Hawking radiation. Here, they lead to the appearance of a Bose–Einstein statistics. For this reason, these three ingredients establish a close relationship between the evaporation of a black hole and an inverted harmonic oscillator. However, in case of the inverted harmonic oscillator, the corresponding energy eigenstates do not display a horizon nor a logarithmic phase singularity in position representation. Indeed, only remnants of these features remain visible and appear, for instance, in the asymptotic expansion of the corresponding wave function. In the present section, we reveal these secrets of the energy eigenstates of the inverted harmonic oscillator.

We start our analysis by considering the Hamiltonian

for a quantum particle of mass m subject to an inverted harmonic oscillator of steepness ω as resulting from Eq. (15). Here, the position $x̂$ and the momentum operator $p̂$ are subject to the familiar commutation relation

In order to reflect the phase space structure presented in Fig. 1(b), we introduce the operators^29,31

and

in analogy to Eqs. (20) and (21).

Although $ξ̂$ and $η̂$ are Hermitian operators, their form reminds of the annihilation and creation operator of the harmonic oscillator as obtained up to an overall phase for an imaginary value of ω. Moreover, these operators satisfy the commutation relation

according to Eqs. (32)–(34). Thus, besides a rescaling, the operators $ξ̂$ and $η̂$ reflect a canonical transformation of the position $x̂$ and the momentum operator $p̂$⁠.

Based on the Hermitian operators $ξ̂$⁠, Eq. (33), and $η̂$⁠, Eq. (34), we define the states $|ξ⟩$ and $|η⟩$ via the eigenvalue equations

and

with the real-valued eigenvalues ξ and η, respectively.

Due to the commutation relation Eq. (35), the states $|ξ⟩$ and $|η⟩$ are related by the Fourier transform

These particular basis states define the representation that is used in our subsequent analysis.⁴³ 

With the help of Eqs. (33)–(35), the Hamiltonian $Ĥ$⁠, Eq. (31), can then be cast into the form

in agreement with Eq. (22). We point out the invariance of the Hamiltonian $Ĥ$ under the exchange of the operators $ξ̂$ and $η̂$⁠. Moreover, we note that $Ĥ$ corresponds to the Hermitian form of the Berry–Keating Hamiltonian.¹⁵ 

Consequently, the energy eigenstates $|ε⟩$ of the inverted harmonic oscillator are defined as solutions of the eigenvalue equation

for the Hermitian Hamiltonian $Ĥ$⁠, Eq. (39), with real-valued dimensionless energy ε. We note that, more generally, eigenstates of the unbounded Hamiltonian $Ĥ$ with complex eigenvalues can also be constructed.^44,45

We are now in the position to determine the energy eigenstates $|ε⟩$ of the inverted harmonic oscillator by considering their projection onto the states $|ξ⟩$⁠, Eq. (36), and $|η⟩$⁠, Eq. (37). With the help of Eqs. (35) and (39), the eigenvalue equation (40) in ξ-representation takes the form

Analogously, the eigenvalue equation (40) in η-representation reads

We emphasize that the factor $−1/2$ in Eqs. (41) and (42) results from the non-commutativity of the operators $ξ̂$ and $η̂$⁠, as presented in Eq. (35). In addition, due to the invariance of the Hamiltonian $Ĥ$⁠, Eq. (39), under the exchange $ξ̂↔η̂$⁠, both the ξ- and η-representation lead to the almost identical form of the differential equations (41) and (42). However, they differ by the sign in front of the eigenvalue ε.

Furthermore, we point out the non-analytic behavior of $⟨ξ|ε⟩$ at ξ = 0 due to the right-hand side of Eq. (41), and similarly of $⟨η|ε⟩$ at η = 0 in Eq. (42). Indeed, this is due to a branch point that occurs at the horizons in phase space. As a consequence, there exist two linear independent solutions to each of these equations.

In agreement with Ref. 29, every energy eigenstate $|ε⟩$ can then be expressed as a linear combination

of two orthogonal energy eigenstates $|Ψε±⟩$ with coefficients $α±(ε)$⁠. We define these particular states by their ξ-representation $Ψε±(ξ)≡⟨ξ|Ψε±⟩$⁠, where

is a solution of Eq. (41). Here, we have introduced the Heaviside step function

The particular set of eigenstates $|Ψε±⟩$ of the Hamiltonian $Ĥ$⁠, Eq. (39), with energy ε is characterized by the fact that the ξ-representation $Ψε±(ξ)$⁠, Eq. (44), vanishes for (+) negative and (−) positive values of ξ, as demonstrated in Fig. 3. There we depict in red the real and imaginary part of the two orthogonal energy eigenstates (a) $Ψε+(ξ)$ and (b) $Ψε−(ξ)$ of the inverted harmonic oscillator as a function of the coordinate ξ.

We clearly observe the horizon at ξ = 0 which separates the non-vanishing part of the wave function from the zero solution. When approaching the horizon ξ = 0 from the direction $ξ>0$ (+) and $ξ<0$ (−), both the amplitude and the phase of $Ψε±(ξ)$ diverge and result in an increasing radius, presented by the orange envelope, and a faster winding of the spiral. In addition, we point out that the dimensionless energy ε only enters as magnitude of the logarithmic phase $−ε ln |ξ|$ of the energy eigenfunctions $Ψε±(ξ)$⁠, while the amplitude is independent of ε.

Alternatively, every energy eigenstate $|ε⟩$ can be expressed as a superposition

of the two orthogonal energy eigenstates $|Φε±⟩$ with coefficients $β±(ε)$⁠. We define these states by their η-representation $Φε±(η)≡⟨η|Φε±⟩$⁠, where

is a solution of Eq. (42) with the Heaviside step function Θ as defined by Eq. (45). Consequently, the eigenstates $|Φε±⟩$ are characterized by the fact that their η-representation vanishes for negative or positive values of η, as facilitated by the non-analytic behavior at η = 0 in Eq. (42).

Each set of energy eigenstates forms a complete basis with the completeness relations

and

where $1$ denotes the identity.

Moreover, as evident from Eqs. (44) and (47), the energy eigenstates are normalized according to the conditions

and

with the indices $s,s′=±$ labeling the two degenerate solutions. Here, $δs,s′$ denotes the Kronecker delta and $δ(ε−ε′)$ is the Dirac delta function. In this regard, the functions $Ψε±(ξ)$ and $Φε±(η)$ are not square-integrable.

Despite the seemingly similar form of the solutions $Ψε±(ξ)$⁠, Eq. (44), and $Φε±(η)$⁠, Eq. (47), we emphasize that each of them corresponds to a different eigenstate $|Ψε±⟩$ and $|Φε±⟩$ of the Hamiltonian $Ĥ$⁠, Eq. (39), with dimensionless energy ε.

In summary, the inverted harmonic oscillator displays the following characteristic features: (i) it has a continuous spectrum and a two-fold degeneracy with regard to the energy ε, as expressed by the orthogonal solutions $Ψε±(ξ)$⁠, Eq. (44), or $Φε±(η)$⁠, Eq. (47), respectively. (ii) As a consequence of the Heaviside step function defined by Eq. (45), the solutions $Ψε±(ξ)$ and $Φε±(η)$ only possess non-vanishing contributions in a half-plane of phase space with regard to the ξ- and η-domain, respectively. This feature is reminiscent of the event horizon of a black hole in spacetime. However, here two horizons emerge in phase space. (iii) For $ε≠0$⁠, the energy eigenstates display both a phase singularity and an amplitude singularity at the horizons ξ = 0 and η = 0, respectively. While the logarithmic phase singularity is reminiscent of the logarithmic singularity in the mode function of a quantized bosonic field at the event horizon of a black hole,⁵ an additional amplitude singularity induced by the reciprocal square-root dependency arises in the system of the inverted harmonic oscillator.

In the context of a black hole, it is the Fourier transform of a mode function with a logarithmic phase singularity that determines the number of particles emitted by Hawking radiation in this mode.⁵ For a massless scalar field of integer spin, this number follows a Bose–Einstein statistics that is governed by the ratio of the frequency of the wave and the surface gravity of the black hole as presented in Sec. II. We now demonstrate that a related feature is present in the simple model of an inverted harmonic oscillator. Also, here, a Fourier transform of a logarithmic phase singularity is of crucial relevance.

In order to illustrate this fact, we consider the η-representation

of the energy eigenstate $|Ψε+⟩$⁠, where we have inserted the identity with regard to the basis states $|ξ⟩$ for $ξ∈ℝ$⁠, Eq. (36), on the right-hand side.

By making use of Eq. (38), we express Eq. (52) as the Fourier transform

of the energy eigenfunction $Ψε+(ξ)$⁠, Eq. (44), displaying a logarithmic phase singularity at $ξ→0+$ in combination with an amplitude singularity induced by the reciprocal square root.

Due to the particular form of the energy eigenfunction $Ψε+(ξ)$⁠, Eq. (53) corresponds not only to the Fourier transform of $|ξ|−1/2−iεΘ(ξ)$ but also to the Mellin transform of the plane wave $exp (−iξη)$⁠. As such, Eq. (53) indeed constitutes an example of the quantum Mellin transform.⁴⁶ In addition, we point out that it also represents a fractional Weyl integral.⁴⁷ This interesting link between the inverted harmonic oscillator with its horizons in phase space and the field of fractional calculus requires further exploration.

Next, we evaluate the integral in Eq. (53) and arrive at the expression

where Γ denotes the gamma function. Accordingly, the state

can be expressed as a superposition of the energy eigenstates $|Φε±⟩$⁠, whose η-representation is given by Eq. (47), where we have introduced the coefficients

Analogously, we obtain the decomposition

for the energy eigenstate $|Ψε−⟩$⁠, whose ξ-representation is given by Eq. (44).

The coefficients $S±(ε)$⁠, Eq. (56), constitute the elements of the unitary scattering matrix

which establishes the relation

for the vectors $α(ε)≡[α+(ε),α−(ε)]T$ and $β(ε)≡[β+(ε),β−(ε)]T$ in Eqs. (43) and (46), as evident from Eqs. (55) and (57). We point out that the S-matrix $S(ε)$⁠, Eq. (58), has already been related to scattering off a black hole.^27,34

Next, we turn to the probability density

in η-representation by making use of the orthogonality of the states $|Φε±⟩$⁠, Eq. (51).

By applying the relation

for the gamma function, we identify the weights

being displayed in Fig. 4.

Finally, we make use of Eqs. (47) and (62) in order to recast Eq. (60) in the form

determining the absolute value squared for the overlap of the state $|η⟩$ and the energy eigenstate $|Ψε+⟩$⁠.

It is a fascinating coincidence that the weights $|S±(ε)|2$⁠, Eq. (62), actually resemble the Fermi–Dirac distribution⁴⁸ as governed by the dimensionless energy $±ε$⁠. In contrast to thermodynamics where the characteristic energy of the system is given by the product $kBT$ of the Boltzmann constant $kB$ and the temperature T, it is the product $hω$ of the Planck constant h and the steepness ω divided by $4π2$⁠, which defines the energy scale of the inverted harmonic oscillator as evident from Eqs. (40) and (63).

We now summarize the main properties of the weights $|S±(ε)|2$ as defined by Eq. (62). First, we identify the relation

which is valid for any value of ε, as evident from Fig. 4, and expected for the two coefficients in the decomposition of the state $|Ψε+⟩$⁠, Eq. (55), due to normalization.

Second, in the classical case of the inverted harmonic oscillator, the particles move along the horizons in phase space for a vanishing energy E = 0 as shown in Fig. 1. In the quantum mechanical case, it is instead the vanishing energy $ε=0$ of the energy eigenstate $|Ψε+⟩$ that leads to equal weights $|S+(0)|2=|S−(0)|2=1/2$⁠.

Third, as depicted in Fig. 4, we obtain the weights $|S±(ε)|2→1$ for the energy $ε→∓∞$⁠, respectively. Accordingly, for $ε→∞$⁠, the main contribution to the state $|Ψε+⟩$ originates from the state $|Φε−⟩$⁠. At first, this seems to be a counter-intuitive result. Indeed, for any value of ε, there are different horizons in phase space associated with the states $|Ψε+⟩$ and $|Φε−⟩$⁠, as shown by Eqs. (44) and (47), respectively. We will resolve this issue in Sec. V.

Finally, we point out that the probabilities $|S±(ε)|2$ in Eq. (62) define the transmission $T(ε)≡|S−(ε)|2$ and reflection coefficient $R(ε)≡|S+(ε)|2$ of the inverted harmonic oscillator,^30,31,36,41 in agreement with the scattering matrix given by Eq. (58). Consequently, the state $|Φε−⟩$ corresponds to the transmitted part of $|Ψε+⟩$⁠, while $|Φε+⟩$ describes the reflected one. The higher the energy eigenvalue ε of the eigenstate state $|Ψε+⟩$⁠, the higher the probability $T(ε)$ of the quantum particle to be transmitted. At the same time, the probability $R(ε)$ of reflection decreases.

In summary, analog to the number of particles for Hawking radiation emitted by black holes, the transmission and reflection coefficient of the inverted harmonic oscillator resemble a particular quantum statistics. This feature is a consequence of the Fourier transform of a logarithmic phase singularity that is contained in the function $Ψε±(ξ)$⁠, Eq. (44). However, the additional amplitude singularity, induced by the reciprocal square root, leads to the emergence of the Fermi–Dirac instead of the Bose–Einstein statistics.^5,13

In Secs. IV B and IV C, we have uncovered the horizons and the logarithmic phase singularity intrinsic to the inverted harmonic oscillator by analyzing the states $|Ψε±⟩$ and $|Φε±⟩$ in ξ- and η-representation, respectively. We have presented the appearance of a functional dependence that resembles the Fermi–Dirac statistics which connects these two representations of the energy eigenstates. However, as demonstrated in the following, these interesting features familiar from the context of Hawking radiation remain hidden in position representation.

For this purpose, we consider the position representation $ψε±(x)≡⟨x|Ψε±⟩$ and $ϕε±(x)≡⟨x|Φε±⟩$ of the energy eigenstates $|Ψε±⟩$⁠, Eq. (44), and $|Φε±⟩$⁠, Eq. (47). Here, $|x⟩$ denotes the eigenstate of the position operator $x̂$ with eigenvalue x.

As derived in  Appendix B, these wave functions take the form

and

Here, $D−1/2±iε$ denotes the parabolic cylinder function⁴⁹ and the normalization constant

is determined by the gamma function Γ. The functions $ψε±(x)$ and $ϕε±(x)$ correspond to well-known solutions^31,33 of the Schrödinger equation for the inverted harmonic oscillator, as expressed in position representation in terms of the parabolic cylinder functions.

First, as evident from Eq. (65), the wave functions $ψε+(x)$ and $ψε−(x)$ are related to each other by a reflection $x→−x$ in position. The same holds for $ϕε+(x)$ and $ϕε−(x)$ according to Eq. (66). Second, we note that the function $ϕε±(x)$ is the complex conjugate of $ψε±(x)$⁠. Apparently, the Hamiltonian $Ĥ$⁠, Eq. (31), remains invariant under the corresponding transformations of the operators $x̂$ and $p̂$⁠.

In Fig. 5, we depict the energy eigenfunctions (a) $ψε+(x)$ and (b) $ψε−(x)$ of the inverted harmonic oscillator in combination with the corresponding probability density $|ψε±(x)|2$⁠. We clearly observe the relation $ψε−(x)=ψε+(−x)$ according to Eq. (65). As apparent from the probability density $|ψε±(x)|2$⁠, the wave function $ψε±(x)$ is associated with an incoming particle approaching the parabolic barrier from (a) right and (b) left. The oscillations in the domains (a) x > 0 and (b) x < 0 are caused by the interference of the incoming wave with the one reflected at the potential barrier. On the other hand, for (a) x < 0 and (b) x > 0 one identifies a decaying amplitude. Indeed, for negative energies $ε<0$⁠, a non-vanishing probability in this domain is caused by tunneling^30,32 through, and for positive energies $ε>0$ by transmission beyond the parabolic barrier.

Furthermore, we notice from Fig. 5 that the wave function $ψε±(x)$ neither displays an amplitude singularity nor a phase singularity as one might have expected from the ξ-representation of the state $|Ψε±⟩$ given by Eq. (44). In addition, the horizon is hidden at which these singularities would occur.

In order to demonstrate that traces of these features are still present in position representation, we consider the asymptotic behavior of the wave function $ψε+(x)$⁠, Eq. (65). By applying the asymptotic expansion⁴⁹ for the parabolic cylinder function $D−1/2±iε$⁠, we obtain for $x≫1$ the expression

and for $x≪−1$⁠, we arrive at

While the asymptotic expansion in Eq. (68) for $x≫1$ is composed of two terms, the one in Eq. (69) for $x≪−1$ only consists of a single term. In Sec. V, we present an illustrative explanation for this manifestation of Stokes phenomenon.

Accordingly, Eqs. (68) and (69) both contain at least one of the terms $|x|−1/2±iε$ which are reminiscent of an amplitude and a logarithmic phase singularity at x = 0. However, these asymptotic expansions are not applicable in the domain $|x|<1$⁠. For this reason, these features remain hidden as evident from Fig. 5. Moreover, due to the slow decrease in the amplitude proportional to $1/|x|$ for $|x|≪1$⁠, the energy eigenstates of the inverted harmonic oscillator remain non-normalizable in position space.

In addition, it is interesting to note that the asymptotic expansions in Eqs. (68) and (69) contain a phase that is governed by a quadratic dependence on the position x. This is a characteristic feature of a wave that displays a focus after a certain time of free propagation.

So far, we have explored the inverted harmonic oscillator in terms of the wave and matrix mechanics formulations of quantum mechanics. In particular, we have studied the emergence of horizons, the logarithmic phase singularity, and the intimidate relation to the Fermi–Dirac statistics. Moreover, we have shown that these features are only apparent in particular representations.

The physics of all these phenomena stands out most clearly when viewed in Wigner phase space,^50,51 which provides an intuitive illustration of quantum mechanics. Indeed, a quasi-probability distribution as the Wigner function does not rely on a particular representation. Instead, it depends on two phase space variables, such as position x and momentum p, or equivalently, the coordinates ξ and η. The Wigner function also builds a bridge between classical statistical mechanics and quantum mechanics in phase space. This intimidate relationship allows us to relate the results of Sec. III, obtained in classical phase space, with our subsequent study in the quantum domain.

In the following, we extend the analysis of the inverted harmonic oscillator in phase space as pursued in Refs. 29 and 30. The focus of these works lays on tunneling phenomena with regard to the parabolic barrier. In contrast, our interest rests on the horizons in phase space in combination with the emergence of a singularity, which relates to Hawking radiation emerging at the event horizon of a black hole. For this purpose, we analyze different characteristic features that are intrinsic to the phase space representation of the energy eigenstates of the inverted harmonic oscillator.

First, we derive the Wigner functions of the energy eigenstates of the inverted harmonic oscillator based on their ξ- and η-representation. Next, we turn to an illustration of the properties of these particular states and relate their shape to the classical phase space trajectories. Subsequently, we review their dependence on the dimensionless energy with regard to the reflection and transmission at a parabolic barrier. We then focus on the horizons in phase space and reveal their relation to the occurrence of singularities in the inverted harmonic oscillator. Moreover, with the help of particular phase space projections, we reveal the emergence of a distribution that resembles the Fermi–Dirac statistics. Finally, we analyze why these features remain hidden in the position representation of quantum mechanics.

In the following, we determine the Wigner functions for the energy eigenstates $|Ψε±⟩$ and $|Φε±⟩$ of the inverted harmonic oscillator with dimensionless energy ε, as introduced in Sec. IV.

First, we focus on the Wigner function^50,51

corresponding to the energy eigenstate $|Ψε±⟩$⁠. Here, we have made use of the fact that, up to a rescaling, the coordinates ξ and η, defined by Eqs. (20) and (21), result from a canonical transformation of the phase space variables x and p under which the Wigner function transforms as a scalar.²⁹ 

By inserting the wave function $Ψε±(ξ)=⟨ξ|Ψε±⟩$⁠, Eq. (44), in ξ-representation, we recast Eq. (70) as

where we have applied the relation

for the Heaviside step function defined by Eq. (45). With the help of the substitution $y′=±y/ξ$ in Eq. (71) we then arrive at the Wigner function

Here, we have introduced the weight function

yielding the contribution to the Wigner function $WΨ,ε±(ξ,η)$⁠, Eq. (73), with regard to the classical energy $E=−ℏωξη$ as given by Eq. (22). With the help of Eq. (61) the weight function can also be expressed as

in terms of the Kummer function of the first kind⁴⁹