oschl-Teller Potential Research Articles

Black hole greybody factors from Korteweg–de Vries integrals: Computation

It has recently been shown that the dynamics of perturbed nonrotating black holes (BHs) admits an infinite number of symmetries that are generated by the flow of the Korteweg-de Vries (KdV) equation. These symmetries lead to an infinite number of conserved quantities that can be obtained as integrals of differential polynomials in the potential appearing in the gauge-invariant master equations describing the BH perturbations, the KdV integrals. These conserved quantities are the same for all the possible potentials, which means that they are invariant under Darboux transformations, and they fully determine the BHs transmission amplitudes, or greybody factors, via a moment problem. In this paper we introduce a new semianalytical method to obtain the greybody factors associated with BH scattering processes by solving the moment problem using only the KdV integrals. The method is based on the use of Pad\'e approximants and we check it first by comparing with results from the case of a P\"oschl-Teller potential, for which we have analytical expressions for the greybody factors. Then, we apply it to the case of a Schwarzschild BH and compare with results from computations based on the Wentzel--Kramers--Brillouin (WKB) approximation. It turns out that the new method provides accurate results for the BH greybody factors for all frequencies. The method is also computationally very efficient.

Black hole greybody factors from Korteweg–de Vries integrals: Theory

The dynamics of perturbed non-rotating black holes (BHs) can be described in terms of master equations of the wave type with a potential. In the frequency domain, the master equations become time-independent Schr\"odinger equations with no discrete spectrum. It has been recently shown that these wave equations possess an infinite number of symmetries that correspond to the flow of the infinite hierarchy of Korteweg-de Vries (KdV) equations. As a consequence, the infinite set of associated conserved quantities, the KdV integrals, are the same for all the different master equations that we can consider. In this paper we show that the BH scattering reflection and transmission coefficients characterizing the continuous spectrum can be fully determined via a moment problem, in such a way that the KdV integrals provide the momenta of a distribution function depending only on the reflection coefficient. We also discuss the existence and uniqueness of solutions, strategies to solve the moment problem, and finally show the case of the P\"oschl-Teller potential where all the steps can be carried out analytically.

Quasinormal modes from bound states: The numerical approach

It is known that the spectrum of quasinormal modes of potential barriers is related to the spectrum of bound states of the corresponding potential wells. This property has been widely used to compute black hole quasinormal modes, but it is limited to a few ``approximate'' potentials with certain transformation properties for which the spectrum of bound states must be known analytically. In this work we circumvent this limitation by proposing an approach that allows one to make use of potentials with similar transformation properties, but where the spectrum of bound states can also be computed numerically. Because the numerical calculation of bound states is usually more stable than the direct computation of the corresponding quasinormal modes, the new approach is also interesting from a technical point of view. We apply the method to different potentials, including the P\"oschl-Teller potential for which all steps can be understood analytically, as well as potentials for which we are not aware of analytic results but provide independent numerical results for comparison. As a canonical test, all potentials are chosen to match the Regge-Wheeler potential of axial perturbations of the Schwarzschild black hole. We find that the new approximate potentials are more suitable to approximate the exact quasi-normal modes than the P\"oschl-Teller potential, particularly for the first overtone. We hope this work opens new perspectives to the computation of quasinormal modes and finds further improvements and generalizations in the future.

Pseudospectrum and Black Hole Quasinormal Mode Instability

We study the stability of quasinormal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals the following: (i) the stability of the slowest-decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an "infrared" effect; (ii) the instability of all overtones under small-scale ("ultraviolet") perturbations of sufficiently high frequency, which migrate towards universal QNM branches along pseudospectra boundaries, shedding light on Nollert's pioneer work and Nollert and Price's analysis [H. P. Nollert and R. H. Price, Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40, 980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint operator. In this setting, spectral (in)stability is naturally addressed through the pseudospectrum notion that we construct numerically via Chebyshev spectral methods and foster in gravitational physics. After illustrating the approach with the P\"oschl-Teller potential, we address the Schwarzschild black hole case, where QNM (in)stabilities are physically relevant in the context of black hole spectroscopy in gravitational-wave physics and, conceivably, as probes into fundamental high-frequency spacetime fluctuations at the Planck scale.

Inverse problem for Hawking radiation

In this work we study the inverse problem related to the emission of Hawking radiation. We first show how the knowledge of greybody factors of different angular contributions $l$ can be used to constrain the width of the corresponding black hole perturbation potentials. Afterwards we provide a framework to recover the greybody factors from the actual energy emission spectrum, which has to be treated as sum over all multipole numbers. The underlying method for the reconstruction of the potential widths is based on the inversion of the Gamow formula, a parabolic expansion and the P\"oschl-Teller potential. We define a `normalized' energy emission spectrum that turns out to be very beneficial for the numerical fitting process, as well as for an improved qualitative understanding of how much information of the black hole potentials are actually imprinted in the spectrum. The connection to recent studies on the inverse problem using the quasi-normal spectra of ultra compact stars and exotic compact objects is discussed as well. In the appendix we show that the spectrum can be approximated surprisingly well and simply with a parabolic expansion of the peak of the classical black hole scattering potentials.

Phase-space quantum profile of Pöschl-Teller two-level systems

The quantum phase-space dynamics driven by hyperbolic P\"oschl-Teller (PT) potentials is investigated in the context of the Weyl-Wigner quantum mechanics. The obtained Wigner functions for quantum superpositions of ground and first-excited states exhibit some non-classical and non-linear patterns which are theoretically tested and quantified according to a non-gaussian continuous variable framework. It comprises the computation of quantifiers of non-classicality for an anharmonic two-level system where non-Liouvillian features are identified through the phase-space portrait of quantum fluctuations. In particular, the associated non-gaussian profiles are quantified by measures of {\em kurtosis} and {\em negative entropy}. As expected from the PT {\em quasi}-harmonic profile, our results suggest that quantum wells can work as an experimental platform that approaches the gaussian behavior in the investigation of the interplay between classical and quantum scenarios. Furthermore, it is also verified that the Wigner representation admits the construction of a two-particle bipartite quantum system of continuous variables, $A$ and $B$, identified by $\sum_{i,j=0,1}^{i\neq j}\vert i_{_A}\rangle\otimes\vert j_{_B}\rangle$, which are shown to be separable under gaussian and non-gaussian continuous variable criteria.

Nonadiabatic evolution of primordial perturbations and non-Gaussinity in hybrid approach of loop quantum cosmology

While loop quantum cosmology predicts a robust quantum bounce of the background evolution of a Friedmann-Robertson-Walker spacetime prior to the standard slow-roll inflation, whereby the big bang singularity is resolved, there are several different quantization procedures to cosmological perturbations, for instance, the deformed algebra, dressed metric, and hybrid quantizations. This paper is devoted to studying the quantum bounce effects of primordial perturbations in the hybrid approach. The main discrepancy of this approach is the effective positive mass at the quantum bounce for the evolution of the background that is dominated by the kinetic energy of the inflaton field at the bounce, while this mass is always nonpositive in the dressed metric approach. It is this positivity of the effective mass that violates the adiabatic evolution of primordial perturbations at the initial moments of the quantum bounce. With the assumption that the evolution of the background is dominated by the kinetic energy of the inflaton at the bounce, we find that the effective potentials for both scalar and tensor perturbations can be well approximately described by a P\"oschl-Teller potential, which allows us to find analytical solutions of perturbations, and from these analytical expressions we are able to study the nonadiabatic evolution of primordial perturbations in detail. In particular, we derive their quantum bounce effects and investigate their observational constraints. In addition, the impacts of quantum bounce effects on the non-Gaussianity and their implication on the explanations of the observed power asymmetry in the cosmic microwave background are also explored.

Time evolution of linear and generalized Heisenberg algebra nonlinear Pöschl-Teller coherent states

We analyze the time evolution of two kinds of coherent states for a particle in a P\"oschl-Teller potential. We find a pair of canonically conjugate operators and compare the behavior of their time evolution for both coherent states. The nonlinear ones are more localized. The trajectory in the phase space of the mean values of these two operators is a kind of generalization of the Rose algebraic curves. The new pair of canonically conjugate variables leads to a fourth-order Schr\"odinger equation which has the same energy spectrum as the P\"oschl-Teller system.

Theory of linear spin wave emission from a Bloch domain wall

We report an analytical theory of linear emission of exchange spin waves from a Bloch domain wall, excited by a uniform microwave magnetic field. The problem is reduced to a one-dimensional Schr\"odinger-like equation with a P\"oschl-Teller potential and a driving term of the same profile. The emission of plane spin waves is observed at excitation frequencies above a threshold value, as a result of a linear process. The height-to-width aspect ratio of the P\"oschl-Teller profile for a domain wall is found to correspond to a local maximum of the emission efficiency. Furthermore, for a tailored P\"oschl-Teller potential with a variable aspect ratio, particular values of the latter can lead to enhanced or even completely suppressed emission.

Bosonic interactions with a domain wall

We consider here the interaction of scalar bosons with a topological domain wall. Not only is there a continuum of scattering states, but there is also an interesting "quasi-discretuum" of positive energy bosonic bound states, describing bosons entrapped within the wall's core. The full spectrum of the scattering and bound state energies and eigenstates is obtainable from a Schr\"odinger-type of equation with a P\"oschl-Teller potential. We also consider the presence of a boson gas within the wall and high energy boson emission.

Two-dimensional transmission through modified Pöschl-Teller potential in bilayer graphene

We study two-dimensional transport of quasiparticles in bilayer graphene through a modified P\oschl-Teller potential, a potential with hyperbolic profile. Both scattering by well and tunneling through a barrier are investigated. At normal incidence the oscillating and evanescent wave states are completely decoupled, so that only one mode participates in transmission. At oblique incidence a combined contribution of the two modes leads to reduction or enhancement of transmission probability. In both cases the chirality can forbid transition to oscillating modes existing within the well or barrier region, which results in a cloak effect. In the barrier tunneling it appears at normal incidence. In the scattering by the well, however, it arises at an oblique incidence as a consequence of coupling between the positive- and negative-energy states, which we interpret as a secondary cloak effect. When a gap is present in the barrier tunneling, because of the coupling, the transmission probability at and near normal incident angles is greatly enhanced. We also examine the effect of trigonal warping in the barrier tunneling and find valley-dependent transmissions and broadening of resonant-transmission regions. As a result, the resonance curve of conductance has a thicker width and a lower peak-to-bottom contrast than the case without trigonal warping. We demonstrate these phenomena with numerical simulations and explain the results using the chirality-dependent transition probability and semiclassical argument. The numerical results are also compared with those of the corresponding rectangular potential. We observe that the two models display large quantitative differences in their two-dimensional transport.

The Higgs oscillator on the hyperbolic plane and light-front holography

The Light Front Holographic (LFH) wave equation, which is the conformal scalar equation on the plane, is revisited from the perspective of the supersymmetric quantum mechanics, and attention is drawn to the fact that it naturally emerges in the small hyperbolic angle approximation to the "curved" Higgs oscillator on the hyperbolic plane, i.e. on the upper part of the two-dimensional hyperboloid of two sheets, a space of constant negative curvature. Such occurs because the particle dynamics under consideration reduces to the one dimensional Schr\"odinger equation with the second hyperbolic P\"oschl-Teller potential, whose flat-space (small-angle) limit reduces to the conformally invariant inverse square distance plus harmonic oscillator interaction, on which LFH is based. In consequence, energies and wave functions of the LFH spectrum can be approached by the solutions of the Higgs oscillator on the hyperbolic plane in employing its curvature and the potential strength as fitting parameters. Also the proton electric charge form factor is well reproduced within this scheme by means of a Fourier-Helgason hyperbolic wave transform of the charge density. In conclusion, in the small angle approximation, the Higgs oscillator on the hyperbolic plane is demonstrated to satisfactory parallel essential outcomes of the Light Front Holographic QCD. The findings are suggestive of associating the hyperboloid curvature of the with a second scale in LFH, which then could be employed in the definition of a chemical potential.

Localization of bulk matter fields, the hierarchy problem and corrections to Coulomb’s law on a pure de Sitter thick braneworld

In this paper we investigate the localization and mass spectra of various matter fields with spin 0, 1 and $1/2$ on a geometric thick brane generated by pure four-dimensional (4D) and five-dimensional positive cosmological constants without bulk scalar fields. This model possesses a 4D cosmological constant that can be made as small as one desires without fine-tuning it with the bulk cosmological constant. The Randall-Sundrum model was obtained as an analytic continuation of the flat brane limit of this braneworld configuration when the Hubble parameter disappears. We also show that within the framework of this inflating braneworld model, it is possible to formulate a mechanism for obtaining TeV mass scales from Planck ones by adding a positive thin brane, where the Standard Model fields are trapped, located at a distance ${y}_{2}$ from the origin, where the Planck thick brane resides. The brane separation must be of the same order as the inverse thickness parameter of the model in order for the mechanism to generate the desired hierarchy. This result is obtained by imposing the recovery of both the correct 4D gravitational couplings and the actually observed accelerated expansion of the Universe in our de Sitter braneworld. Regarding the localization of matter in the purely geometric thick braneworld, for spin-0 massless and massive scalar fields as well as for spin-1 vector fields, the potentials of the Kaluza-Klein (KK) modes in the corresponding Schr\"odinger equations are modified P\"oschl-Teller potentials, which lead to the localization of the scalar and vector zero modes on the brane as well as to mass gaps in the mass spectra. We also compute the corrections to Coulomb's law coming from massive KK vector modes. For spin-$1/2$ fermions, we introduce the bulk mass term $MF(z)\overline{\ensuremath{\Psi}}\ensuremath{\Psi}$ in the action, and four different cases are investigated. Localization of the massless left-chiral fermion zero mode is feasible for just two cases of $F(z)$. In the first one, we obtain Schr\"odinger equations with modified P\"oschl-Teller potentials with the corresponding mass gaps for both left- and right-chiral fermions; the number of massive KK bound states is finite (determined by the ratio $M/b$) and is the same for left- and right-chiral modes. In the second case, we get a family of solutions with an infinite number of bound states where the mass spectra for left- and right-chiral KK fermion modes are discrete. A special case resembles the one-dimensional quantum harmonic oscillator problem. For both of these cases, the mass spectrum of right-chiral fermions is shifted with respect to the mass spectrum of the left-chiral ones.

Domain wall brane in squared curvature gravity

We suggest a thick braneworld model in the squared curvature gravity theory. Despite the appearance of higher order derivatives, the localization of gravity and various bulk matter fields is shown to be possible. The existence of the normalizable gravitational zero mode indicates that our four-dimensional gravity is reproduced. In order to localize the chiral fermions on the brane, two types of coupling between the fermions and the brane forming scalar is introduced. The first coupling leads us to a Schr\"odinger equation with a volcano potential, and the other a P\"oschl-Teller potential. In both cases, the zero mode exists only for the left-hand fermions. Several massive KK states of the fermions can be trapped on the brane, either as resonant states or as bound states.

Dirac equation with spin symmetry for the modified Pöschl–Teller potential in D dimensions

We present solutions of the Dirac equation with spin symmetry for vector and scalar modified P\"oschl-Teller potential within framework of an approximation of the centrifugal term. The relativistic energy spectrum is obtained using the Nikiforov-Uvarov method and the two-component spinor wavefunctions are obtain are in terms of the Jacobi polynomials. It is found that there exist only positive-energy states for bound states under spin symmetry, and the energy levels increase with the dimension and the potential range parameter $\alpha$.

Thermalization of Nonequilibrium Electrons in Quantum Wires

We study the problem of energy relaxation in a one-dimensional electron system. The leading thermalization mechanism is due to three-particle collisions. We show that for the case of spinless electrons in a single channel quantum wire the corresponding collision integral can be transformed into an exactly solvable problem. The latter is known as the Schrödinger equation for a quantum particle moving in a Pöschl-Teller potential. The spectrum for the resulting eigenvalue problem allows for bound-state solutions, which can be identified with the zero modes of the collision integral, and a continuum of propagating modes, which are separated by a gap from the bound states. The inverse gap gives the time scale at which counterpropagating electrons thermalize.

Filtering of matter-wave vibrational states via spatial adiabatic passage

We discuss the filtering of the vibrational states of a cold atom in an optical trap, by chaining this trap with two empty ones and controlling adiabatically the tunneling. Matter wave filtering is performed by selectively transferring the population of the highest populated vibrational state to the most distant trap while the population of the rest of the states remains in the initial trap. Analytical conditions for two-state filtering are derived and then applied to an arbitrary number of populated bound states. Realistic numerical simulations close to state-of-the-art experimental arrangements are performed by modeling the triple well with time dependent P\"oschl-Teller potentials. In addition to filtering of vibrational states, we discuss applications for quantum tomography of the initial population distribution and engineering of atomic Fock states that, eventually, could be used for tunneling assisted evaporative cooling.

Self-force calculations with matched expansions and quasinormal mode sums

Accurate modeling of gravitational wave emission by extreme-mass ratio inspirals is essential for their detection by the LISA mission. A leading perturbative approach involves the calculation of the self-force acting upon the smaller orbital body. In this work, we present the first application of the Poisson-Wiseman-Anderson method of ``matched expansions'' to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the Green function, which are, respectively, valid in the ``quasilocal'' and ``distant past'' regimes, and which may be matched together within the normal neighborhood. We perform our calculation in a static region of the spherically symmetric Nariai spacetime ($d{S}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{2}$), in which scalar-field perturbations are governed by a radial equation with a P\"oschl-Teller potential (frequently used as an approximation to the Schwarzschild radial potential) whose solutions are known in closed form. The key new ingredients in our study are (i) very high order quasilocal expansions and (ii) expansion of the distant past Green function in quasinormal modes. In combination, these tools enable a detailed study of the properties of the scalar-field Green function. We demonstrate that the Green function is singular whenever $x$ and ${x}^{\ensuremath{'}}$ are connected by a null geodesic, and apply asymptotic methods to determine the structure of the Green function near the null wave front. We show that the singular part of the Green function undergoes a transition each time the null wave front passes through a caustic point, following a repeating fourfold sequence $\ensuremath{\delta}(\ensuremath{\sigma})$, $1/\ensuremath{\pi}\ensuremath{\sigma}$, $\ensuremath{-}\ensuremath{\delta}(\ensuremath{\sigma})$, $\ensuremath{-}1/\ensuremath{\pi}\ensuremath{\sigma}$, etc., where $\ensuremath{\sigma}$ is Synge's world function. The matched-expansion method provides insight into the nonlocal properties of the self-force. We show that the self-force generated by the segment of the worldline lying outside the normal neighborhood is not negligible. We apply the matched-expansion method to compute the scalar self-force acting on a static particle on the Nariai spacetime, and validate against an alternative method, obtaining agreement to six decimal places. We conclude with a discussion of the implications for wave propagation and self-force calculations. On black hole spacetimes, any expansion of the Green function in quasinormal modes must be augmented by a branch-cut integral. Nevertheless, we expect the Green function in Schwarzschild spacetime to inherit certain key features, such as a fourfold singular structure manifesting itself through the asymptotic behavior of quasinormal modes. In this way, the Nariai spacetime provides a fertile testing ground for developing insight into the nonlocal part of the self-force on black hole spacetimes.

Mass gap for gravity localized on Weyl thick branes

We consider thick brane configurations in a pure geometric Weyl integrable 5D space-time, a non-Riemannian generalization of Kaluza-Klein (KK) theory involving a geometric scalar field. Thus, the 5D theory describes gravity coupled to a self-interacting scalar field which gives rise to the structure of the thick branes. We continue the study of the properties of a previously found family of solutions which is smooth at the position of the brane but involves naked singularities in the fifth dimension. Analyzing their graviton spectrum, we find that a particularly interesting situation arises for a special case in which the 4D graviton is separated from the KK gravitons by a mass gap. The corresponding effective Schr\odinger equation has a modified P\oschl-Teller potential and can be solved exactly. Apart from the massless 4D graviton, it contains one massive KK bound state, and the continuum spectrum of delocalized KK modes. We also discuss the mass hierarchy problem, and explicitly compute the corrections to Newton's law in the thin brane limit.

Continuum of extended states in the spectrum of a one-dimensional random potential

We describe a one-dimensional disordered system, based on the P\"oschl-Teller potential, that exhibits a continuum of extended states which is independent of the random or correlated character of the sequence and of the length of the system. The delocalization of the electronic states occurs in the whole positive spectrum where the system shows a perfect transmission.