Analytic Perturbation Theory Research Articles

Intuitive understanding of extinction of small particles in absorbing and active host media within the MLWA

In an absorbing or an active host medium characterized by a complex refractive index n2=n2′+in2′′, our previously developed modified dipole long-wave approximation (MLWA) is shown to essentially overlie with the exact Mie theory results for localized surface plasmon resonance of spherical nanoparticles with radius a≲25nm (a≲20nm) in the case of Ag and Au (Al and Mg) nanoparticles. The agreement for Au and Ag (Al and Mg) nanoparticles, slightly better in the case of Au than Ag, continues to be acceptable up to a∼50nm (a∼40nm), and can be used, at least qualitatively, up to a∼70nm (a∼50nm) correspondingly. A first order analytic perturbation theory (PT) in a normalized extinction coefficient, κ¯=n2′′/n2′, around a nonabsorbing host is developed within the dipole MLWA and its properties are investigated. It is shown that, in a suitable parameter range, the PT can reliably isolate and capture the effect of host absorption or host gain on the overall extinction efficiency of various plasmonic nanoparticles.

Analytical Inverse QCD Coupling Constant Approach and Its Result for αs

We propose a model for the QCD running coupling constant based on the analytical inverse QCD coupling constant concept with an additional regularization in the low momentum region. Analyticity in the q2-complex plane, where q is the four-momentum transfer, is imposed by methods of the Analytic Perturbation Theory. The model incorporates a peculiar low-momentum behavior for αs(q2) as a divergence at q2=0 to retrieve color confinement, without spoiling its correct high-momentum behavior. This was achieved by means of a two-parameter regularization function, for which we considered three possible analytic expressions. In fact, within the framework of the Analytic Perturbation Theory, αs(q2) assumes a finite value for q2=0, at all perturbative orders (infrared stability), hence the infrared divergence cannot be implemented. For this reason, we found it more straightforward to work with its reciprocal, namely, εs(q2)=1/αs(q2), imposing its vanishing at the origin of the q2-complex plane via the multiplication of the aforementioned regularizing functions and the spectral density. Once the two free parameters of the regularization functions were settled by fitting to the experimental values of αs(q2) at the momenta where these data were available and reliable, the model could reproduce the QCD running coupling constant at any other momentum transferred.

Continuous dependence of eigenvalues on potential functions for nonlocal Sturm–Liouville equations

In this paper, we discuss the continuous dependence of eigenvalues on the potential function for a nonlocal Sturm–Liouville equation with truncated Marchaud fractional derivative term by two‐parameter method. To this end, we first study the properties of eigenvalues for a two‐parameter nonlocal Sturm–Liouville eigenvalue problem. We obtain the two‐parameter nonlocal Sturm–Liouville eigenvalue problem that has countable number of simple real eigenvalues. Meanwhile, the asymptotic behavior of eigenvalues is studied by aid of the analytic perturbation theory.

Direct and alternating magnon spin currents across a junction interface irradiated by linearly polarized laser

The developments in the field of quantum optics raise expectations that laser-matter coupling is a promising building block for magnonics. Here, we propose a method for the generation of direct and alternating spin currents of magnons across the junction interface irradiated by linearly polarized laser. In a junction of ferromagnetic insulators with a large electronic gap, the spin angular momentum is exchanged during the tunneling process of magnons across the junction interface. The advanced technology in the field of plasmonics and metamaterials realizes that spins irradiated by the laser field interact only with the magnetic component of the laser through the Zeeman coupling. Using an analytic perturbation theory, we provide a general formula for magnon transport induced by the inversion symmetry breaking across the junction interface. Then, we show that those spin currents are enhanced by the ferromagnetic resonance, and the period of the ac spin current is one-half of that of the laser magnetic field. Finally, we estimate the magnitude of the spin current, and find that it will be within experimental reach.

Correlated Insulators, Semimetals, and Superconductivity in Twisted Trilayer Graphene

Motivated by recent experiments indicating strong superconductivity and intricate correlated insulating and flavor-polarized physics in mirror-symmetric twisted trilayer graphene, we study the effects of interactions in this system close to the magic angle, using a combination of analytical and numerical methods. We identify asymptotically exact correlated many-body ground states at all integer filling fractions $\nu$ of the flat bands. To determine their fate when moving away from these fine-tuned points, we apply self-consistent Hartree-Fock numerics and analytic perturbation theory, with good agreement between the two approaches. This allows us to construct a phase diagram for the system as a function of $\nu$ and the displacement field, the crucial experimental tuning parameter of the system, and study the spectra of the different phases. The phase diagram is dominated by a correlated semimetallic intervalley coherent state and an insulating sublattice-polarized phase around charge neutrality, $\nu=0$, with additional spin-polarization being present at quarter ($\nu=-2$) or three quarter ($\nu=+2$) fillings of the quasi-flat bands. We further study the superconducting instabilities emerging from these correlated states, both in the absence and in the additional presence of electron-phonon coupling, also taking into account possible Wess-Zumino-Witten terms. In the experimentally relevant regime, we find triplet pairing to dominate, possibly explaining the observed violation of the Pauli limit. Our results have several consequences for experiments as well as future theoretical work and illustrate the rich physics resulting from the interplay of almost flat bands and dispersive Dirac cones in twisted trilayer graphene.

Improved calculation of the γ*γ → π process at low Q2 using LCSR’s and renormalization-group summation

We study two versions of lightcone sum rules to calculate the γ*γ → π0 transition form factor (TFF) within QCD. While the standard version is based on fixed-order perturbation theory by means of a power-series expansion in the strong coupling, the new method incorporates radiative corrections by renormalization-group summation and generates an expansion within a generalized fractional analytic perturbation theory involving only analytic couplings. Using this scheme, we determine the relative nonperturbative parameters and the first two Gegenbauer coefficients of the pion distribution amplitude (DA) to obtain TFF predictions in good agreement with the preliminary BESIII data, while the best-fit pion DA satisfies the most recent lattice constraints on the second moment of the pion DA at the three-loop level.

Laplace–Beltrami spectrum of ellipsoids that are close to spheres and analytic perturbation theory

Abstract We study the spectrum of the Laplace–Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $L^2$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first 16 eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to our results from the analytic perturbation theory, and approximations for the eigenvalues of ellipsoids that degenerate into infinite cylinders or two-dimensional disks. We propose a conjecture on the exact number of nodal domains of near-sphere ellipsoids.

Instability theory of kink and anti-kink profiles for the sine–Gordon equation on Josephson tricrystal boundaries

The aim of this work is to establish an instability study for stationary kink and antikink/kink profiles solutions for the sine–Gordon equation on a metric graph with a structure represented by a Y-junction so-called a Josephson tricrystal junction. By considering boundary conditions at the graph-vertex of δ′-interaction type, it is shown that kink profiles which are continuous at the vertex, as well as anti-kink/kink profiles possibly discontinuous at the vertex, are linearly (and nonlinearly) unstable. The extension theory of symmetric operators, Sturm–Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. The local well-posedness of the sine–Gordon model in H1(Y)×L2(Y) is also established. The theory developed in this investigation has prospects for the study of the (in)-stability of stationary wave solutions of other configurations for kink-solitons profiles.

Linear instability criterion for the Korteweg–de Vries equation on metric star graphs

The aim of this work is to establish a novel linear instability criterion for the Korteweg–de Vries (KdV) model on metric graphs. In the case of balanced graphs with a structure represented by a finite collection of semi-infinite edges and with boundary condition of δ-type interaction at the graph-vertex, we show that the continuous tail and bump profiles are linearly unstable. In this case, the use of the analytic perturbation theory of operators as well as the extension theory of symmetric operators is fundamental in our stability analysis. The arguments showed in this investigation have prospects in the study of the instability of stationary waves solutions for nonlinear evolution equations on metric graph.

Instability of Static Solutions of the sine-Gordon Equation on a $${\mathcal {Y}}$$-Junction Graph with $$\delta $$-Interaction

The aim of this work is to establish a linear instability result of stationary, kink and kink/anti-kink soliton profile solutions for the sine-Gordon equation on a metric graph with a structure represented by a $\mathcal Y$-junction. The model considers boundary conditions at the graph-vertex of $\delta$-interaction type, or in other words, continuity of the wave functions at the vertex plus a law of Kirchhoff-type for the flux. It is shown that kink and kink/anti-kink soliton type stationary profiles are linearly (and nonlinearly) unstable. For that purpose, a linear instability criterion that provides the sufficient conditions on the linearized operator around the wave to have a pair of real positive/negative eigenvalues, is established. As a result, the linear stability analysis depends upon of the spectral study of this linear operator and of its Morse index. The extension theory of symmetric operators, Sturm-Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. A comprehensive study of the local well-posedness of the sine-Gordon model in $\mathcal E(\mathcal Y) \times L^2(\mathcal{Y})$ where $\mathcal E(\mathcal Y) \subset H^1(\mathcal{Y})$ is an appropriate energy space, is also established. The theory developed in this investigation has prospects for the study of the instability of stationary wave solutions of other nonlinear evolution equations on metric graphs.

Nonlinear dispersive equations: classical and new frameworks

The purpose of this manuscript associated to the Golden Jubilee of the IME-USP is to present selected material from the author’s scientific contribution dealing with nonlinear phenomena of dispersive type. That study has been a source for modern research in the dynamic of traveling wave solutions of different type and it has been disseminated through publication of scientific articles and/or books. Here, we provide the reader with information about current research in stability theory for nonlinear dispersive equations and possible developments. Also, I hope it inspires future developments in this important and fascinating subject. In this manuscript we consider the following topics: stability theory of solitary waves and the applicability of the concentration–compactness principle, the existence and orbital (in)stability of periodic traveling wave solutions for nonlinear dispersive models, nonlinear Schrodinger and Korteweg–de Vries models on star-shaped metric graphs. The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. The methods presented in this manuscript have prospect for the study of the dynamic of solutions for nonlinear evolution equations around of different traveling waves profiles which may appear in non-standard environments such as star-shaped metric graphs.

Stability of equilibrium solutions of a double power reaction‐diffusion equation with a Dirac interaction

AbstractIn this paper, information about the instability of equilibrium solutions of a nonlinear family of localized reaction‐diffusion equations in dimension one is provided. More precisely, explicit formulas to the equilibrium solutions are computed and, via analytic perturbation theory, the exact number of positive eigenvalues of the linear operator associated to the stability problem is analyzed. In addition, sufficient conditions for blow up of the solutions of the equation are also discussed.

Stability properties of standing waves for NLS equations with the [formula omitted]-interaction

We study the orbital stability of standing waves with discontinuous bump-like profile for the nonlinear Schrödinger model with the repulsiveδ′-interaction on the line. We consider the model with power non-linearity. In particular, it is shown that such standing waves are unstable in the energy space under some restrictions for parameters. The use of extension theory of symmetric operators by Krein–von Neumann is fundamental for estimating the Morse index of self-adjoint operators associated with our stability study. Moreover, for this purpose we use Sturm oscillation results and analytic perturbation theory. The Perron–Frobenius property for the repulsive δ′-interaction is established.The arguments presented in this investigation have prospects for the study of the stability of stationary waves solutions of other nonlinear evolution equations with point interactions.

Bjorken sum rule with analytic QCD coupling

We present details of study of the polarized Bjorken sum rule carried out recently in [1]. Four QCD coupling versions are considered: perturbative QCD (pQCD) in the scheme, Analytic Perturbation Theory, and 2δ and 3δ analytic QCD versions. In contrast to pQCD, these QCD variants do not have Landau singularities at low positive Q2. In general, when 2δ and 3δ QCD coupling is used the fitted curves give the best results.

Non-perturbative quantum chromodynamics and short strings in electron-positron pair annihilation to hadrons

We develop a new mathematical method to study the non-perturbative quantum chromodynamics effects in low-energy processes using ordinary and analytic perturbation theory (PT and APT). We apply this tool to extract the lowest dimensional condensates from the high-precision fits of the data on e+e–-annihilation to pion channels:  e+e− → π+π−, e+e− → 2π+2π−, e+e− → π+π−2π0, e+e− → 3π+3π− and e+e− → 2π+2π−2π0. The special attention is devoted to the existence of the contribution of dimension 2 operator. Our consideration is based on the dependence of Adler function corresponding to e+e−-annihilation into pions on both Q and Borel transform parameter M2. It is shown that within the framework of the proposed method the C2 coefficient of the dimension 2 operator is negative and is closer to zero for analytic perturbation theory, which may therefore be partially responsible for the effect of short strings.

Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.

Coulomb scattering in the massless Nelson model II. Regularity of ground states

For the massless Nelson model, we provide detailed information about the dependence of the normalized ground states [Formula: see text] of the fiber single-electron Hamiltonians [Formula: see text] on the total momentum [Formula: see text] and the infrared cut-off [Formula: see text]. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form [Formula: see text] for some constant [Formula: see text] and a function of the maximal admissible coupling constant [Formula: see text] such that [Formula: see text]. These results hold both in the infrared-regular and infrared-singular cases. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of the Haag–Ruelle scattering theory. They should also be relevant to the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion.

Form factor π0γ*γ in lightcone sum rules combined with renormalization-group summation vs experimental data.

We consider the lightcone sum-rule (LCSR) description of the pionphoton transition form factor in combination with the renormalization group of QCD. The emerging scheme represents a certain version of Fractional Analytic Perturbation Theory and significantly extends the applicability domain of perturbation theory towards lower momenta Q2 ≲ 1 GeV2. We show that the predictions calculated herewith agree very well with the released preliminary data of the BESIII experiment, which have very small errors just in this region, while the agreement with other data at higher Q2 is compatible with the LCSR predictions obtained recently by one of us using fixed-order perturbation theory.

Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

Abstract We study properties of the nodal sets of high-frequency eigenfunctions and quasimodes for radial perturbations of the harmonic oscillator. In particular, we consider nodal sets on spheres of large radius (in the classically forbidden region) for quasimodes with energies lying in intervals around a fixed energy $E$. For well-chosen intervals we show that these nodal sets exhibit quantitatively different behavior compared to those of the unperturbed harmonic oscillator. These energy intervals are defined via a careful analysis of the eigenvalue spacings for the perturbed operator, based on analytic perturbation theory and linearization formulas for Laguerre polynomials.

Sachdev-Ye-Kitaev Model with Quadratic Perturbations: The Route to a Non-Fermi Liquid

We study stability of the Sachdev-Ye-Kitaev (SYK_{4}) model with a large but finite number of fermions N with respect to a perturbation, quadratic in fermionic operators. We develop analytic perturbation theory in the amplitude of the SYK_{2} perturbation and demonstrate stability of the SYK_{4} infrared asymptotic behavior characterized by a Green function G(τ)∝1/τ^{3/2}, with respect to weak perturbation. This result is supported by exact numerical diagonalization. Our results open the way to build a theory of non-Fermi-liquid states of strongly interacting fermions.