Outgoing Wave Boundary Conditions Research Articles

ULDM self-interactions, tidal effects and tunnelling out of satellite galaxies

It is well-known that Dark Matter (DM) inside a satellite galaxy orbiting a host halo experiences a tidal potential. If DM is ultra-light, given its wave-like nature, one expects it to tunnel out of the satellite — if this happens sufficiently quickly, then the satellite will not survive over cosmological timescales, severely constraining this dark matter model. In this paper, we study the effects of the inevitable quartic self-interaction of scalar Ultra-Light Dark Matter (ULDM) on the lifetimes of satellite galaxies by looking for quasi-stationary solutions with outgoing wave boundary conditions. For a satellite with some known core mass and orbital period, we find that, attractive (repulsive) self-interactions decrease (increase) the rate of tunnelling of DM out of it. In particular, for satellite galaxies with core mass ∼𝒪(107–108) M⊙ and orbital period ∼𝒪(1) Gyr, one can impose constraints on the strength of self-interactions as small as λ∼𝒪(10-92). For instance, for ULDM mass m = 10-22 eV, the existence of the Fornax dwarf galaxy necessitates attractive self-interactions with λ≲ -2.12 × 10-91.

Open-boundary cluster model with a parameter-free complex absorbing potential.

In quantum chemical calculations of heterogeneous structures in solids, e.g., when an impurity is located on the surface, the conventional cluster model is insufficient to describe the electronic structure of substrates due to its finite size. The open-boundary cluster model (OCM) overcomes this problem by performing cluster calculations under the outgoing-wave boundary condition. In this method, a complex absorbing potential (CAP) is used to impose the boundary condition, but the CAP used in the previous studies required parameter optimization based on the complex variational principle. This study proposes and applies a parameter-free CAP to OCM calculations. This approach makes it possible to uniquely determine the band-specific CAP based on the surface Green's function theory. Using this CAP, we conducted OCM calculations of the tight-binding model of a one-dimensional semi-infinite chain, and we found that the calculated density of states agreed with the exact one. Surface states of the Newns-Anderson-Grimley model were also computed using the CAP, and the projected density of states on the adsorbed atom was successfully reproduced.

Atomic Siegert states in a rotating electric field

We show that the time-dependent Schr\"odinger equation describing in the dipole approximation the interaction of a one-electron atom with a monochromatic circularly polarized electromagnetic field can be reduced to a stationary Schr\"odinger equation in the form which allows to separate variables in the asymptotic region and explicitly formulate outgoing-wave boundary conditions. The solutions to this equation are called atomic Siegert states (SSs) in a rotating electric field. We develop the theory of such states and propose an efficient method to construct them using powerful techniques of stationary scattering theory. The method yields not only the SS eigenvalue defining the Stark-shifted energy and total ionization rate of the state, but also the SS eigenfunction defining partial ionization amplitudes and the photoelectron momentum distribution. The theory is illustrated by calculations for a model potential.

Tidal response and near-horizon boundary conditions for spinning exotic compact objects

Teukolsky equations for $|s|=2$ provide efficient ways to solve for curvature perturbations around Kerr black holes. Imposing regularity conditions on these perturbations on the future (past) horizon corresponds to imposing an ingoing (outgoing) wave boundary condition. For exotic compact objects (ECOs) with external Kerr spacetime, however, it is not yet clear how to physically impose boundary conditions for curvature perturbations on their boundaries. We address this problem using the membrane paradigm, by considering a family of zero-angular-momentum fiducial observers (FIDOs) that float right above the horizon of a linearly perturbed Kerr black hole. From the reference frame of these observers, the ECO will experience tidal perturbations due to ingoing gravitational waves, respond to these waves, and generate outgoing waves. As it also turns out, if both ingoing and outgoing waves exist near the horizon, the Newman-Penrose (NP) quantity ${\ensuremath{\psi}}_{0}$ will be numerically dominated by the ingoing wave, while the NP quantity ${\ensuremath{\psi}}_{4}$ will be dominated by the outgoing wave---even though both quantities contain full information regarding the wave field. In this way, we obtain the ECO boundary condition in the form of a relation between ${\ensuremath{\psi}}_{0}$ and the complex conjugate of ${\ensuremath{\psi}}_{4}$, in a way that is determined by the ECO's tidal response in the FIDO frame. We explore several ways to modify gravitational-wave dispersion in the FIDO frame and deduce the corresponding ECO boundary condition for Teukolsky functions. Using the Starobinsky-Teukolsky identity, we subsequently obtain the boundary condition for ${\ensuremath{\psi}}_{4}$ alone, as well as for the Sasaki-Nakamura and Detweiler functions. As it also turns out, the reflection of spinning ECOs will generically mix between different $\ensuremath{\ell}$ components of the perturbation fields, and it will be different for perturbations with different parities. It is straightforward to apply our boundary condition to computing gravitational-wave echoes from spinning ECOs, and to solve for the spinning ECOs' quasinormal modes.

The Fredholm determinant for Hulthén-distorted non-local separable potential: Application to $$ \alpha {-}\alpha $$ elastic scattering

Exact analytical expression of the Fredholm determinant with outgoing wave boundary condition for motion in Hulthen-distorted non-local separable potential is constructed and expressed in the maximum reduced form. Using boundary conditions (regular and irregular), two approximate energy-dependent interactions corresponding to the parent non-local potential are also constructed. The phase shifts for the $$ \alpha $$ – $$\alpha $$ elastic scattering are computed by using (i) exact expression for the Fredholm determinant and (ii) energy-dependent local interactions by exploiting the phase function method. The merits of our constructed equivalent energy-dependent potentials are judged by comparing the $$ \alpha $$ – $$\alpha $$ elastic scattering phases with our exact calculation and standard data.

RESONANT STATES IN A ONE-DIMENSIONAL QUANTUM SYSTEM

In this work, the concept of resonant states (RSs) in a finite square quantum well is presented. We first derive the analytic secular transcendental equations for even and odd states by applying the outgoing wave boundary conditions into the one-dimensional Schrödinger’s wave equation. The complex solution of these equations is found using the numerical Newton-Raphson method implemented in MATLAB. We can see in particular, that the RSs present a general class of Eigenstates, which includes bound states, anti-bound states, and normal RSs.

Starting behaviors of the TM-mode gyrotrons

This work shows the derivations and calculations of the starting behaviors using the Laplace method and the numerical method. Calculated results based on these two methods agree well when dealing with a uniform structure, while the numerical method is advantageous for the non-uniform and practical structure. The applicability of the zero-field and the outgoing-wave boundary conditions at the collector end is discussed. These two boundary conditions agree well when the beam-wave resonant line and the waveguide dispersion curve intercept at the backward-wave region but differ significantly at the forward-wave region. The beam-wave coupling strength of the TM-mode gyrotron is found to be strongly correlated with the starting current, which can be utilized to avoid the potential competition from the transverse electric (TE) modes. The starting current of the TM11 gyrotron exhibits an additional operating condition at the low-beam voltage that may facilitate the development of low-cost and tabletop gyrotron systems. The beam-voltage and magnetic-field tunings are investigated for an open-cavity structure with the numerical method. Interestingly, the TM11 gyrotron as well as the TE01 gyrotron exhibits a similar starting behavior, which warrants the potential applications of the TM-mode gyrotrons.

Resonant states in double and triple quantum wells

The full set of resonant states in double and triple quantum well/barrier structures is investigated. This includes bound, anti-bound and normal resonant states which are all eigensolutions of Schrödinger’s equation with generalized outgoing wave boundary conditions. The transformation of resonant states and their transitions between different subgroups as well as the role of each subgroup in observables, such as the quantum transmission, is analyzed. The quantum well potentials are modeled by Dirac delta functions; therefore, as part of this study, the well known problem of bound states in delta-like potentials is also revisited.

Resonant-state expansion of light propagation in nonuniform waveguides

A new rigorous approach for precise and efficient calculation of light propagation along non-uniform waveguides is presented. Resonant states of a uniform waveguide, which satisfy outgoing-wave boundary conditions, form a natural basis for expansion of the local electromagnetic field. Using such an expansion at fixed frequency, we convert the wave equation for light propagation in a non-uniform waveguide into an ordinary second-order matrix differential equation for the expansion coefficients depending on the coordinate along the waveguide. We illustrate the method on several examples of non-uniform planar waveguides and evaluate its efficiency compared to the aperiodic Fourier modal method and the finite element method, showing improvements of one to four orders of magnitude. A similar improvement can be expected also for applications in other fields of physics showing wave phenomena, such as acoustics and quantum mechanics.

Green-function approach to the theory of tunneling ionization

We solve the problem of tunneling ionization of a multielectron atom in a static electric field by using the Green's function for the Stark-Coulomb problem. This allows us to incorporate the outgoing-wave boundary conditions at infinity. The interaction of the active electron with the atomic residue is described either by a model potential or by the $l$-dependent pseudopotential which prevents virtual transitions to orbitals occupied by inner electrons. The method works well in the broad range of electric fields including the region above the classical ionization threshold (the barrier-suppression region). Calculations of ionization of Ar demonstrate a noticeable difference between the model potential approach and the pseudopotential approach, but both sets of results agree with experimental data.

Properties of Stark Resonant States in Exactly Solvable Systems

Properties of Stark resonant states are studied in two exactly solvable systems. These resonances are shown to form a biorthogonal system with respect to a pairing defined by a contour integral that selects states with outgoing wave boundary conditions. Analytic expressions are derived for the pseudonorm, dipole moment, and coupling matrix elements which relate systems with different strengths of the external field. All results are based on explicit calculations made possible by a newly designed integration method for combinations of Airy functions representing resonant eigenstates. Generalizations for one-dimensional systems with short-range potentials are presented, and relations are identified which are likely to hold in systems with three spatial dimensions.

Exterior complex scaling method in TDDFT: HHG of Ar atoms in intense laser fields

Exterior complex scaling (ECS) method is applied in the framework of the time-dependent density functional theory to study high-order harmonic generation (HHG) of multielectron atoms in intense laser fields. ECS allows to impose correct (outgoing-wave) boundary conditions on the wave functions at large distances. In the present work, ECS is combined with the time-dependent generalized pseudospectral method for accurate and efficient solution of the time-dependent Kohn-Sham equations. Results for HHG of Ar atoms are presented.

Exterior complex scaling method in TDDFT: HHG of Ar atoms in intense laser fields

The exterior complex scaling (ECS) method is applied in the framework of time-dependent density-functional theory (TDDFT) to study high-order harmonic generation (HHG) of multielectron atoms in intense laser fields. With the help of ECS, correct outgoing-wave boundary conditions can be imposed on the wave functions at large distances. In our implementation, ECS is combined with the time-dependent generalized pseudospectral method for accurate and efficient solution of the time-dependent Kohn-Sham equations. We make use of LB94 exchange-correlation potential which appears quite accurate in calculations of unperturbed electronic structure of Ar. Calculations of HHG are performed for the laser fields with the wavelength of 800 nm and several peak intensities. The HHG spectrum exhibits an intensity-independent minimum corresponding to the photon energy of about 51 eV which is closely related to the Cooper minimum in the photoionization cross section of Ar. We found that HHG spectra calculated with the frozen-core potential (not including dynamic response of the electron density) differ significantly from those obtained by TDDFT.

Exterior complex scaling method in time-dependent density-functional theory: Multiphoton ionization and high-order-harmonic generation of Ar atoms

The exterior complex scaling (ECS) method is applied in the framework of time-dependent density-functional theory (TDDFT) to study multiphoton ionization (MPI) and high-order-harmonic generation (HHG) of multielectron atoms in intense laser fields. ECS allows one to impose correct (outgoing-wave) boundary conditions on the wave functions at large distances. In our implementation, ECS is combined with the time-dependent generalized pseudospectral method for accurate and efficient solution of the time-dependent Kohn-Sham equations. We make use of LB94 exchange-correlation potential which proved accurate in calculations of unperturbed electronic structure of Ar. Calculations of MPI and HHG are performed for the laser pulses with the wavelength of 800 nm and several peak intensities. The HHG spectrum exhibits an intensity-independent minimum corresponding to the photon energy of about 51 eV which is closely related to the Cooper minimum observed in the photoionization cross section of Ar. We found that MPI probabilities and HHG spectra calculated with the frozen-core potential (that is, not including dynamic response of the electron density) differ significantly from those obtained by TDDFT.

Coupled-channel study of fine structure in the α decay of 233,235U

The fine structure observed in the α decay of two deformed uranium isotopes 233,235U has been investigated by using the coupled-channel Schrödinger equation with outgoing wave boundary conditions. The coupled-channel effect resulting from nuclear deformation is included, and the internal effect of daughter states is considered in dealing with the interaction matrix and the α-cluster formation. The experimental branching ratios for various daughter states are well reproduced by the multichannel microscopic calculation. Moreover, the α-preformation factor is also discussed by comparison of the calculated half-life with the experimental value.

Theoretical description of fine structure in theαdecay of heavy odd-odd nuclei

The newly developed multichannel cluster model (MCCM), based on the coupled-channel Schrodinger equation with outgoing wave boundary conditions, is extended to study the alpha-decay fine structure in heavy odd-odd nuclei. Calculations are performed for the alpha transitions to favored rotational bands where the unpaired nucleons remain unchanged. The simple WKB barrier penetration formula is also used to evaluate the branching ratios for various daughter states. It is found that the WKB formula seems to overestimate the branching ratios for the second and third members of the favored rotational band, while the MCCM gives a precise description of them without any adjustable parameters. Moreover, the experimental total alpha-decay half-lives are well reproduced within the MCCM. DOI: 10.1103/PhysRevC.87.027602

Systematic calculation of fine structure in theαdecay of heavy odd-mass nuclei

The newly developed multichannel cluster model (MCCM) based on the coupled-channel Schrodinger equation with outgoing wave boundary conditions is extended to study the alpha-decay fine structure in heavy odd-A nuclei. More decay channels are considered for convergence in solving the coupled equations. Calculations are performed for 32 deformed nuclei with atomic numbers Z = 93-102. The branching ratios for various daughter states are well reproduced without any additional free parameters, and the calculated alpha-decay half-lives show good agreement with the experimental data. This indicates that the MCCM has universal applicability and equal validity regardless of even-even or odd-A alpha emitters.

Systematic calculation of fine structure in the α decay of deformed nuclei

We perform an extensive investigation on the a decay of well-deformed even-even nuclei by using the coupled-channel Schrödinger equation with outgoing wave boundary conditions. The internal effect of daughter states is taken into account in dealing with the interaction matrix and the α-cluster formation. In contrast to the traditional α-decay theories, the five-channel microscopic calculations of this work well reproduce the available experimental data concerning α-decay half-lives and fine structures. Some predictions on the fine structure are made for superheavy nuclei. Moreover, the sensitivity of the results to the model quantities is discussed in detail.

Molecular Siegert states in an electric field

The Siegert states of atoms and molecules in a static electric field are the solutions of the stationary Schr\"odinger equation satisfying the regularity and outgoing-wave boundary conditions. Recently, an efficient method for calculating Siegert states in the single-active-electron approximation based on the adiabatic expansion in parabolic coordinates was proposed [P. A Batishchev et al., Phys. Rev. A 82, 023416 (2010); O. I. Tolstikhin et al., Phys. Rev. A 84, 053423 (2011)]. So far, this method has been implemented only for axially symmetric potentials, which corresponds to atoms and linear molecules aligned along the field. In the present work, we extend its implementation to a general class of soft-core molecular potentials. This makes it possible to calculate the Siegert eigenvalue $E=\mathcal{E}\ensuremath{-}i\ensuremath{\Gamma}/2$ defining the energy $\mathcal{E}$ and ionization rate $\ensuremath{\Gamma}$ of the corresponding state as functions of the electric field for arbitrarily oriented polyatomic molecules. The method is illustrated by calculations for the $1s\ensuremath{\sigma}$ and $2p\ensuremath{\pi}$ states of H${}_{2}^{+}$. Comparison of the results with the predictions of perturbation theory for $\mathcal{E}$ and weak-field asymptotic theory for $\ensuremath{\Gamma}$ is discussed.

Generalized effective mode volume for leaky optical cavities

We show explicitly how the commonly adopted prescription for calculating effective mode volumes is wrong and leads to uncontrolled errors. Instead, we introduce a generalized mode volume that can be easily evaluated based on the mode calculation methods typically applied in the literature, and which allows one to compute the Purcell effect and other interesting optical phenomena in a rigorous and unambiguous way.