Odinger-like Equation Research Articles

Dynamical screening and excitonic bound states in biased bilayer graphene

Excitonic bound states are characterized by a binding energy ${\ensuremath{\epsilon}}_{b}$ and a single-particle band gap ${\mathrm{\ensuremath{\Delta}}}_{b}$. This work provides a theoretical description for both strong (${\ensuremath{\epsilon}}_{b}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}_{b}$) and weak (${\ensuremath{\epsilon}}_{b}\ensuremath{\ll}{\mathrm{\ensuremath{\Delta}}}_{b}$) excitonic bound states, with particular application to biased bilayer graphene. Standard description of excitons is based on a wave function that is determined by a Schr\"odinger-like equation with screened attractive potential. The wave function approach is valid only in the weak-binding regime ${\ensuremath{\epsilon}}_{b}\ensuremath{\ll}{\mathrm{\ensuremath{\Delta}}}_{b}$. The screening depends on frequency, i.e., dynamical screening, and this implies retardation. In the case of strong binding, ${\ensuremath{\epsilon}}_{b}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}_{b}$, a wave function description is not possible due to the retardation. Instead we appeal to the Bethe-Salpeter equation, written in terms of the electron-hole Green's function, to solve the problem. So far only the weak-binding regime has been achieved experimentally. Our analysis demonstrates that the strong-binding regime is also possible and we specify conditions in which it can be achieved for the prototypical example of biased bilayer graphene. The conditions concern the bias, the configuration of gates, and the substrate material. To verify the accuracy of our analysis we compare with available data for the weak-binding regime. We anticipate applying the developed dynamical screening Bethe-Salpeter techniques to various 2D materials with strong binding.

Constraining modifications of black hole perturbation potentials near the light ring with quasinormal modes

In modified theories of gravity, the potentials appearing in the Schr\"odinger-like equations that describe perturbations of nonrotating black holes are also modified. In this paper we ask: can these modifications be constrained with high-precision gravitational-wave measurements of the black hole's quasinormal mode frequencies? We expand the modifications in a small perturbative parameter regulating the deviation from the general-relativistic potential, and in powers of $M/r$. We compute the quasinormal modes of the modified potential up to quadratic order in the perturbative parameter. Then we use Markov-chain Monte Carlo methods to recover the coefficients in the $M/r$ expansion in an ``optimistic'' scenario where we vary them one at a time, and in a ``pessimistic'' scenario where we vary them all simultaneously. In both cases, we find that the bounds on the individual parameters are not robust. Because quasinormal mode frequencies are related to the behavior of the perturbation potential near the light ring, we propose a different strategy. Inspired by Wentzel-Kramers-Brillouin (WKB) theory, we demonstrate that the value of the potential and of its second derivative at the light ring can be robustly constrained. These constraints allow for a more direct comparison between tests based on black hole spectroscopy and observations of black hole ``shadows'' by the Event Horizon Telescope and future instruments.

Sound from extra dimensions: Quasinormal modes of a thick brane

In this work, we investigate the quasinormal modes of a thick brane system. Considering the transverse-traceless tensor perturbation of the brane metric, we obtain the Schr\"odinger-like equation of the Kaluza-Klein modes of the tensor perturbation. Then we use the Wentzel-Kramers-Brillouin approximation and the asymptotic iteration method to solve this Schr\"odinger-like equation. We also study the numeric evolution of an initial wave packet against the thick brane. The results show that there is a set of discrete quasinormal modes in the thick brane model. These quasinormal modes appear as the decaying massive gravitons for a brane observer. They are characteristic modes of the thick brane and can reflect the structure of the thick brane.

Time-dependent generator coordinate method study of fission: Dissipation effects

Starting from a quantum theory of dissipation for nuclear collective motion introduced by Kerman and Koonin [Phys. Scr. 10, 118 (1974)], the time-dependent generator coordinate method (TDGCM) is extended to allow for dissipation effects in the description of induced fission dynamics. The extension is based on a generalization of the GCM generating functions that includes excited states, and the resulting equation of motion in the collective coordinates and excitation energy. With the assumption of a narrow Hamiltonian kernel, an expansion in a power series in collective momenta leads to a Schr\"odinger-like equation that explicitly includes a dissipation term, proportional to the momentum of the statistical wave function. An illustrative calculation is performed for induced fission of $^{228}\mathrm{Th}$. The three-dimensional model space includes the axially symmetric quadrupole and octupole shape variables, and the nuclear temperature. When compared with data for photoinduced fission of $^{228}\mathrm{Th}$, the calculated fission yields demonstrate the important role of the additional term in the Hamiltonian that explicitly takes into account the dissipation of energy of collective motion into intrinsic degrees of freedom.

Superradiant instability of extremal black holes in STU supergravity

We investigate the superradiant (in)stability of the extremal multicharge static black holes in the STU supergravity model, which reduce to the Reissner-Nordstr\"om (RN) black hole when all the charges are equal. We first show that the frequency of quasibound states is necessarily complex and obtain the corresponding superradiant condition. We then study the effective potential of the Schr\"odinger-like equation associated with the radial function of the charged scalar field. We find that trapping-well configurations can emerge with either a single peak or double peaks. We numerically obtain the corresponding unstable quasibound states, organized under the overtone number, as well as how the charged black holes deviate from the RN black hole. We find that the STU black holes are superradiantly unstable as long as not all the charges are equal, indicating that the superradiant stability of the extremal RN black hole is a fine-tuning result in the framework of the STU supergravity model.

Gravitational ringing of rotating black holes in higher-derivative gravity

We study gravitational perturbations of slowly rotating black holes in a general effective-field-theory extension of general relativity that includes up to eight-derivative terms. We show that two Schr\"odinger-like equations with spin-dependent effective potentials govern the odd- and even-parity master variables. These equations are coupled for parity-violating corrections, and this coupling affects the quasinormal modes even at linear order in the higher-derivative corrections, due to their isospectrality in general relativity. We provide results for the shifts in the fundamental quasinormal mode frequencies at linear order in the spin, which we expect to be valuable for high-precision phenomenology through future gravitational wave observations.

Asymptotics of linear differential systems and application to quasinormal modes of nonrotating black holes

The traditional approach to perturbations of nonrotating black holes in General Relativity uses the reformulation of the equations of motion into a radial second-order Schr\"odinger-like equation, whose asymptotic solutions are elementary. Imposing specific boundary conditions at spatial infinity and near the horizon defines, in particular, the quasi-normal modes of black holes. For more complicated equations of motion, as encountered for instance in modified gravity models with different background solutions and/or additional degrees of freedom, we present a new approach that analyses directly the first-order differential system in its original form and extracts the asymptotic behaviour of perturbations, without resorting to a second-order reformulation. As a pedagogical illustration, we apply this treatment to the perturbations of Schwarzschild black holes and then show that the standard quasi-normal modes can be obtained numerically by solving this first-order system with a spectral method. This new approach paves the way for a generic treatment of the asymptotic behaviour of black hole perturbations and the identification of quasi-normal modes in theories of modified gravity.

Black hole perturbations in modified gravity

We study the linear perturbations about nonrotating black holes in the context of degenerate higher-order scalar-tensor (DHOST) theories, using a systematic approach that extracts the asymptotic behaviour of perturbations (at spatial infinity and near the horizon) directly from the first-order radial differential system governing these perturbations. For axial (odd-parity) modes, this provides an alternative to the traditional approach based on a second-order Schr\"odinger-like equation with an effective potential, which we also discuss for completeness. For polar (even-parity) modes, which contain an additional degree of freedom in DHOST theories, and are thus more complex, we use a direct treatment of the four-dimensional first-order differential system (without resorting to a second order reformulation). We illustrate our study with two specific types of black hole solutions: 'stealth' Schwarzschild black holes, with a non trivial scalar hair, as well as a class of non-stealth black holes whose metric is distinct from Schwarzschild. The knowledge of the asymptotic behaviours of the perturbations enables us to compute numerically quasi-normal modes, as we show explicitly for the non-stealth solutions. Finally, the asymptotic form of the modes also signals some pathologies in the stealth and non-stealth solutions considered here.

Hadron spectroscopy using the light-front holographic Schrödinger equation and the ’t Hooft equation

Light-front holographic QCD provides a successful first approximation to hadron spectroscopy in the chiral limit of $(3+1)$-dim light-front QCD, where a holographic Schr\"odinger-like equation, with an emerging confining scale, $\kappa$, governs confinement in the transverse direction. In its supersymmetric formulation, light-front holography predicts that each baryon has two superpartners: a meson and a tetraquark, with their degenerate masses being generated by the same scale, $\kappa$. In nature, this mass degeneracy is lifted by chiral symmetry breaking and longitudinal confinement. In this paper, we show that the latter can be successfully captured by the 't Hooft equation of $(1+1)$-dim, large $N_c$, QCD. Together, the holographic Schr\"odinger equation and the 't Hooft equation, provide a good global description of the data across the full hadron spectrum with a universal $\kappa$.

Correspondence of eikonal quasinormal modes and unstable fundamental photon orbits for a Kerr-Newman black hole

In this work, we study the relation of the eikonal quasinormal modes (EQNMs) and the unstable fundamental photon orbits (UFPOs) in the Kerr-Newman spacetime. We find that in the eikonal limit the gravitational and electromagnetic perturbations of the Kerr-Newman black hole are naturally decoupled, and a single one-dimensional Schr\"odinger-like equation encoding the QNM spectrum can be derived. We then show that the decoupled Teukolsky master equation and the Klein-Gordon equation for the massless scalar field in the Kerr-Newman spacetime are of the same form in the eikonal limit. As a direct consequence, taking into account of the boundary conditions for EQNMs we show an exact correspondence between EQNMs and UFPOs, that is, EQNM/UFPO correspondence. More precisely, similar to the Kerr case, the real part of EQNM's frequency is a linear combination of the precessional and (polar) orbital frequencies, while the imaginary part of the frequency is proportional to the Lyapunov exponent of the UFPO.

Conditional wave theory of environmental interactions with a quantum particle

We present an alternative formulation of quantum decoherence theory using conditional wave theory (CWT), which was originally developed in molecular physics (also known as exact factorisation methods). We formulate a CWT of a classic model of collisional decoherence of a free particle with environmental particles treated in a long-wavelength limit. In general, the CWT equation of motion for the particle is non-linear, where the non-linearity enters via the CWT gauge fields. For Gaussian wave packets the analytic solutions of the CWT equations are in exact agreement with those from the density matrix formalism. We show that CWT gauge terms that determine the dynamics of the particle's marginal wave function are related to a Taylor series expansion of the particle's reduced density matrix. Approximate solutions to these equations lead to a linear-time approximation that reproduces the ensemble width in the limits of both short and long times, in addition to reproducing the long-term behaviour of the coherence length. With this approximation, the non-linear equation of motion for the particle's marginal wave function can be written in the form of the logarithmic Schr\"odinger equation. The CWT formalism may lead to computationally efficient calculations of quantum decoherence, since it involves working with wave-function level terms instead of evolving a density matrix via a master equation.

Sharp variation-norm estimates for oscillatory integrals related to Carleson’s theorem

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schr\"odinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain-Guth iteration and the Bourgain-Demeter $\ell^2$ decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions $n \ge 4$.

Schrödinger formulation of the nondipole light-matter interaction consistent with relativity

An alternative and powerful Schr\odinger-like equation for describing beyond dipole laser-matter interactions and leading relativistic corrections is derived. It is shown that this particular formulation is numerically very efficient with respect to computational effort and convergence rate of the solutions. Furthermore, its nonrelativistic form turns out to be more compatible with relativity than what seems to be the case with the more common formulations of the nonrelativistic light-matter interaction. Moreover, the extension of this interaction form into the relativistic region preserves, to a large extent, the numerical efficiency. In this work, the formulation is applied to study beyond dipole corrections and relativistic corrections in multiphoton ionization of a hydrogen atom in the x-ray regime.

Turbulence and scale relativity

We develop a new formalism for the study of turbulence using the scale relativity framework (applied in v-space, following de Montera’s proposal). We first review some of the various ingredients which are at the heart of the scale relativity approach (scale dependence and fractality, chaotic paths, irreversibility) and recall that they indeed characterize fully developed turbulent flows. Then, we show that, in this framework, the time derivative of the Navier-Stokes equation can be transformed into a macroscopic Schrödinger-like equation. The local velocity Probability Distribution Function (PDF), Pv(v), is given by the squared modulus of a solution of this equation. This implies the presence of null minima Pv(vi) ≈ 0 in this PDF. We also predict a new acceleration component, Aq(v)=±Dv ∂v⁡lnPv, which is divergent in these minima. Then, we check these theoretical predictions by data analysis of available turbulence experiments: (1) Empty zones are in effect detected in observed Lagrangian velocity PDFs. (2) A direct proof of the existence of the new acceleration component is obtained by identifying it in the data of a laboratory turbulence experiment. (3) It precisely accounts for the intermittent bursts of the acceleration observed in experiments, separated by calm zones which correspond to Aq ≈ 0 and are shown to remain perfectly Gaussian. (4) Moreover, the shape of the acceleration PDF can be analytically predicted from Aq, and this theoretical PDF precisely fits the experimental data, including the large tails. (5) Finally, numerical simulations of this new process allow us to recover the observed autocorrelation functions of acceleration magnitude and the exponents of structure functions.

A SU(5)xZ2 kink solution and its local stability

A non-abelian kink inducing asymptotically the breaking pattern $SU(5)\times Z_2\rightarrow SU(4)\times U(1)/Z_4$ is obtained. We consider a fourth order Higgs potential in a $1+1$ theory where the scalar field is in the adjoint representation of $SU(5)$. The perturbative stability of the kink also is evaluated. A Schr\"odinger-like equation for the excitations along each $SU(5)$ generator is determined and in none of the cases negative eigenvalues compromising the stability of solution are found. In particular, several bounded scalar states are determined among them the translational zero mode of the flat space $SU(5)\times Z_2$ kink.

Informational entropic Regge trajectories of meson families in AdS/QCD

Bulk mesons propagating in chiral and gluon condensates, in a gravity background, are scrutinized in holographic soft wall AdS/QCD models, involving deformed dilatonic backgrounds. The configurational entropy of the $a_1$ axial vector, the $\rho$ vector, and the $f_0$ scalar meson families is then computed. Two types of informational entropic Regge trajectories are then obtained, where the logarithm of the mesons configurational entropy is expressed in terms of both the experimental meson mass spectra and their excitation number as well. Therefore the mass spectra of the next generation of elements in each meson family, besides being predicted as eigenvalues of Schr\"odinger-like equations, are estimated with better accuracy and discussed.

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.

Universal description of channel plasmons in two-dimensional materials

Channeling surface plasmon-polaritons to control their propagation direction is of the utmost importance for future optoelectronic devices. Here, we develop an effective-index method to describe and characterize the properties of 2D material's channel plasmon-polaritons (CPPs) guided along a V-shaped channel. Focusing on the case of graphene, we derive a universal Schr\"odinger-like equation from which one can determine the dispersion relation of graphene CPPs and corresponding field distributions at any given frequency, since they depend on the geometry of the structure alone. The results are then compared against more rigorous theories, having obtained a very good agreement. Our calculations show that CPPs in graphene and other 2D materials are attractive candidates to achieve deep subwavelength waveguiding of light, holding potential as active components for the next generation of tunable photonic devices.

Space-time-symmetric extension of nonrelativistic quantum mechanics

In quantum theory we refer to the probability of finding a particle between positions $x$ and $x+dx$ at the instant $t$, although we have no capacity of predicting exactly when the detection occurs. In this work, first we present an extended non-relativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between $x$ and $x+dx$ during [$t$,$t+dt$]. Then, we find a Schr\"odinger-like equation for a "mirror" wave function $\phi(t,x)$ associated with the probability of measuring the system between $t$ and $t+dt$, given that detection occurs at $x$. In this framework, it is shown that energy measurements of a stationary state display a non-zero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to {\it ad hoc} assumptions, is a natural, built-in part of the formalism presented here.

Sharp Strichartz estimates in spherical coordinates

We prove almost Strichartz estimates found after adding regularity in the spherical coordinates for Schr\"odinger-like equations. The estimates are sharp up to endpoints. The proof relies on estimates involving spherical averages. Sharpness is discussed making use of a modified Knapp-type example.