Quartic Oscillator Research Articles

Exploring high multiplicity amplitudes: The quantum mechanics analogue of the spontaneously broken case

Calculations of high multiplicity Higgs amplitudes exhibit a rapid growth that may signal an end of perturbative behavior or even the need for new physics phenomena. As a step towards this problem we consider the quantum mechanical equivalent of $1 \to n$ scattering amplitudes in a spontaneously broken $\phi^4$-theory by extending our previous results on the quartic oscillator with a single minimum to transitions $\langle n \lvert \hat{x} \rvert 0 \rangle$ in the symmetric double-well potential with quartic coupling $\lambda$. Using recursive techniques to high order in perturbation theory, we argue that these transitions are of exponential form $\langle n \lvert \hat{x} \rvert 0 \rangle \sim \exp \left( F (\lambda n) / \lambda \right)$ in the limit of large $n$ and $\lambda n$ fixed. We apply the methods of "exact perturbation theory" put forward by Serone et al. to obtain the exponent $F$ and investigate its structure in the regime where tree-level perturbation theory violates unitarity constraints. We find that the resummed exponent is in agreement with unitarity and rigorous bounds derived by Bachas.

Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential

In this paper, we study the weak asymptotic in the $$\mathbb {C}$$-plane of some wave functions resulting from the WKB-techniques applied to a Schrodinger equation with quartic oscillator and having some boundary condition. As a first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions. Especially, we investigate the properties of the Cauchy transform of the root counting measure of re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form $${\mathcal {C}}^{2}(z) +r(z){\mathcal {C}}(z)+s(z)=0$$, where r, s are also polynomials. As a second step, we discuss the existence of solutions (in the form of Cauchy transform of a signed measure) of this algebraic equation. It remains to describe the critical graph of a related quadratic differential $$-p(z)dz^{2}$$ where p(z) is a quartic polynomial. In particular, we discuss the existence (and their number) of finite critical trajectories of this quadratic differential.

TBA equations and resurgent Quantum Mechanics

We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in mathcal{N} = 2 gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators.

Quasi-exactly solvable polynomial extensions of the quantum harmonic oscillator

The (analytic) sextic oscillator is often considered as the prototype of quasi-exactly solvable (QES) Schrödinger equations, i.e., those Schrödinger equations for which at some ad hoc couplings a finite number of eigenstates can be found explicitly by algebraic means, while the remaining ones remain unknown. Recently, a (non-analytic) QES symmetrized quartic oscillator was introduced and shown to complete the list of (analytic) QES anharmonic oscillators, which does not contain any quartic one. Here we prove that such a quartic oscillator is amenable to an sl(2, ℝ) description. Furthermore, we generalize it to a symmetrized sextic oscillator. The latter is obtained by parting the real line into two subintervals ℝ− and ℝ+ on which the corresponding Schrödinger equations are solved by using the functional Bethe ansatz method, and the resulting wavefunctions and their first derivatives are matched at x = 0. Two categories of QES potentials are obtained: the first one containing the well-known analytic sextic potentials as a subset, and the second one of novel potentials with no counterpart in such a class.

Quantum corrections of work statistics in closed quantum systems

We investigate quantum corrections to the classical work characteristic function (CF) as a semiclassical approximation to the full quantum work CF. In addition to explicitly establishing the quantum-classical correspondence of the Feynman-Kac formula, we find that these quantum corrections must be in even powers of ℏ. Exact formulas of the lowest corrections (ℏ^{2}) are proposed, and their physical origins are clarified. We calculate the work CFs for a forced harmonic oscillator and a forced quartic oscillator respectively to illustrate our results.

Ergodic Isoenergetic Molecular Dynamics for Microcanonical-Ensemble Averages

Considerable research has led to ergodic isothermal dynamics which can replicate Gibbs' canonical distribution for simple ( small ) dynamical problems. Adding one or two thermostat forces to the Hamiltonian motion equations can give an ergodic isothermal dynamics to a harmonic oscillator, to a quartic oscillator, and even to the Mexican-Hat ( double-well ) potential problem. We consider here a time-reversible dynamical approach to Gibbs' microcanonical ( isoenergetic ) distribution for simple systems. To enable isoenergetic ergodicity we add occasional random rotations to the velocities. This idea conserves energy exactly and can be made to cover the entire energy shell with an ergodic dynamics. We entirely avoid the Poincare-section holes and island chains typical of Hamiltonian chaos. We illustrate this idea for the simplest possible two-dimensional example, a single particle moving in a periodic square-lattice array of scatterers, the cell model.

Perturbation theory for arbitrary coupling strength?

We present a new formulation of perturbation theory for quantum systems, designated here as: “mean field perturbation theory” (MFPT), which is free from power-series-expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of arbitrary strength and to compute system-properties having non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT for the case of the quartic- and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since “renormalon”-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case.

Resummation of the Brillouin‐Wigner Perturbation Series

We present a new method of resumming the Brillouin‐Wigner perturbation series with Epstein‐Nesbet partitioning. Both the nondegenerate and quasi‐degenerate cases are considered. We illustrate the accuracy of the proposed resummed Brillouin‐Wigner perturbation theory for selected atoms and small molecules, as well as a simple model system like the quartic anharmonic oscillator. For the electronic structure problems, it is found that the present theory provides essentially the same results as the corresponding configuration‐interaction calculations at the level of the fourth‐order calculations.

Roughness as classicality indicator of a quantum state

We define a new quantifier of classicality for a quantum state, the Roughness, which is given by the L2(R2) distance between Wigner and Husimi functions. We show that the Roughness is bounded and therefore it is a useful tool for comparison between different quantum states for single bosonic systems. The state classification via the Roughness is not binary, but rather it is continuous in the interval [0,1], being the state more classic as the Roughness approaches to zero, and more quantum when it is closer to the unity. The Roughness is maximum for Fock states when its number of photons is arbitrarily large, and also for squeezed states at the maximum compression limit. On the other hand, the Roughness approaches its minimum value for thermal states at infinite temperature and, more generally, for infinite entropy states. The Roughness of a coherent state is slightly below one half, so we may say that it is more a classical state than a quantum one. Another important result is that the Roughness performs well for discriminating both pure and mixed states. Since the Roughness measures the inherent quantumness of a state, we propose another function, the Dynamic Distance Measure (DDM), which is suitable for measure how much quantum is a dynamics. Using DDM, we studied the quartic oscillator, and we observed that there is a certain complementarity between dynamics and state, i.e. when dynamics becomes more quantum, the Roughness of the state decreases, while the Roughness grows as the dynamics becomes less quantum.

Holomorphic anomaly and quantum mechanics

We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory.

Spectral properties of the massless relativistic quartic oscillator

An explicit solution of the spectral problem of the non-local Schrödinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral properties such as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle spectral effect is observed which manifests in an exponentially tight approximation of the spectrum by the zeroes of the dominating term in the Fourier representation of the eigenfunctions and its derivative.

Analytic-continuation approach to the resummation of divergent series in Rayleigh-Schrödinger perturbation theory

The method of analytic continuation is applied to estimate eigenvalues of linear operators from finite order results of perturbation theory even in cases when the latter is divergent. Given a finite number of terms ${E}^{(k)},k=1,2,\ensuremath{\cdots}M$ resulting from a Rayleigh-Schr\odinger perturbation calculation, scaling these numbers by ${\ensuremath{\mu}}^{k}$ ($\ensuremath{\mu}$ being the perturbation parameter) we form the sum $E(\ensuremath{\mu})={\ensuremath{\sum}}_{k}{\ensuremath{\mu}}^{k}{E}^{(k)}$ for small $\ensuremath{\mu}$ values for which the finite series is convergent to a certain numerical accuracy. Extrapolating the function $E(\ensuremath{\mu})$ to $\ensuremath{\mu}=1$ yields an estimation of the exact solution of the problem. For divergent series, this procedure may serve as resummation tool provided the perturbation problem has a nonzero radius of convergence. As illustrations, we treat the anharmonic (quartic) oscillator and an example from the many-electron correlation problem.

Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators

The symmetrized quartic polynomial oscillator is shown to admit an sl(2,$\R$) algebraization. Some simple quasi-exactly solvable (QES) solutions are exhibited. A new symmetrized sextic polynomial oscillator is introduced and proved to be QES by explicitly deriving some exact, closed-form solutions by resorting to the functional Bethe ansatz method. Such polynomial oscillators include two categories of QES potentials: the first one containing the well-known analytic sextic potentials as a subset, and the second one of novel potentials with no counterpart in such a class.

The Schrödinger equation on a Lagrange mesh

In this paper the Lagrange-mesh method is used to solve the two- and three-dimensional Schrödinger equation in spherical coordinates. Matrix elements that involve few free parameters and depend on the grid points for the Hamiltonian, and can be used with any quantum mechanical system, are employed. The accuracy of the matrix elements is tested with the quartic anharmonic oscillator and a double ring-shaped harmonic oscillator potentials. Eigenvalues for the systems are reproduced to within machine accuracy when the free parameters of the matrix elements are chosen appropriately.

Formulation of state projected centroid molecular dynamics: Microcanonical ensemble and connection to the Wigner distribution.

A derivation of quantum statistical mechanics based on the concept of a Feynman path centroid is presented for the case of generalized density operators using the projected density operator formalism of Blinov and Roy [J. Chem. Phys. 115, 7822-7831 (2001)]. The resulting centroid densities, centroid symbols, and centroid correlation functions are formulated and analyzed in the context of the canonical equilibrium picture of Jang and Voth [J. Chem. Phys. 111, 2357-2370 (1999)]. The case where the density operator projects onto a particular energy eigenstate of the system is discussed, and it is shown that one can extract microcanonical dynamical information from double Kubo transformed correlation functions. It is also shown that the proposed projection operator approach can be used to formally connect the centroid and Wigner phase-space distributions in the zero reciprocal temperature β limit. A Centroid Molecular Dynamics (CMD) approximation to the state-projected exact quantum dynamics is proposed and proven to be exact in the harmonic limit. The state projected CMD method is also tested numerically for a quartic oscillator and a double-well potential and found to be more accurate than canonical CMD. In the case of a ground state projection, this method can resolve tunnelling splittings of the double well problem in the higher barrier regime where canonical CMD fails. Finally, the state-projected CMD framework is cast in a path integral form.

Nonlinear description of Yang-Mills cosmology: cosmic inflation and the accompanying Hannay’s angle**Supported by Basic Science Research Program through National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1A09919503)

Hannay’s angle is a classical analogue of the “geometrical phase factor” found by Berry in his research on the quantum adiabatic theorem. This classical analogue is defined if closed curves of constant action variables return to the same curves in phase space after an adaibatic evolution. Adiabatic evolution of Yang-Mills cosmology, which is described by a time-dependent quartic oscillator, is investigated. Phase properties of the Yang-Mills fields are analyzed and the corresponding Hannay’s angle is derived from a rigorous evaluation. The obtained Hannay’s angle shift is represented in terms of several observable parameters associated with such an angle shift. The time evolution of Hannay’s angle in Yang-Mills cosmology is examined from an illustration plotted on the basis of numerical evaluation, and its physical nature is addressed. Hannay’s angle, together with its quantum counterpart Berry’s phase, plays a pivotal role in conceptual understanding of several cosmological problems and indeed can be used as a supplementary probe for cosmic inflation.

Convergent Power Series for Boundary Value Problems and Eigenproblems with Application to Atmospheric and Oceanic Tides

It is usually impossible to apply power series to solve boundary value and eigenvalue problems because the radius of convergence does not extend to the boundaries. Hough functions, which solve a second order eigenproblem, are always entire functions or can be computed wholly in terms of entire functions. The eigenvalues ϵ are approximated by the zeros of the highest computed power series coefficient, which is always a polynomial in ϵ. The rate of convergence with series truncation N is proportional to exp(-μN) for some positive constant A brief table contains the entire Maple code for the algorithm. The scheme can also be extended to eigenproblems on an unbounded domain as here illustrated by the quantum harmonic oscillator and quartic oscillator. Computer algebra is very useful; the power series method can also be implemented in a stable, purely numerical fashion by applying trigonometric interpolation on a disk in the complex x-plane and Chebyshev interpolation on a real interval in e. Power series are briefly compared with Chebyshev, Fourier, and spherical harmonic spectral methods, which are mostly superior. The success of power series raises an interesting history-of-science issue: Why was there so little progress in solving Laplace's tidal equations in the nineteenth century?

Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize Award

The 2016 Snook Prize has been awarded to Diego Tapias, Alessandro Bravetti, and David Sanders for their paper -- Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat. They introduced a relatively stiff hyperbolic tangent thermostat force and successfully tested its ability to reproduce Gibbs' canonical distribution for the harmonic oscillator, the quartic oscillator, and the Mexican Hat potentials. Their work constitutes an effective response to the 2016 Ian Snook Prize Award goal -- Finding ergodic algorithms for Gibbs' canonical ensemble using a single thermostat variable. We confirm their work here and highlight an interesting feature of the Mexican Hat problem when it is solved with an adaptive integrator.

Accurate numerical JWKB calculations for the quartic oscillator

Accurate results for WKB and quantum mechanical approaches to the quartic oscillator are obtained by renormalized perturbation theory. The WKB estimate of \(\uppsi ^{2}\)(0) agrees closely with that obtained using a finite difference method previously published in this journal.

Oscilador quártico e funções elípticas de Jacobi

Tipicamente, pequenas oscilações em torno de um ponto de equilíbrio estável são movimentos harmônicos simples com frequência determinada pelo valor da segunda derivada da energia potencial no ponto de equilíbrio. Há casos anômalos, no entanto, em que a segunda derivada é nula e o termo de quarta ordem domina a expansão da energia potencial nas vizinhanças do mínimo. Neste caso, as pequenas oscilações são descritas pelo oscilador quártico. Por meio de um exemplo simples, em que ocorre a referida anomalia, discutimos o oscilador quártico e mostramos como sua equação de movimento pode ser resolvida em termos de funções elípticas de Jacobi. O exemplo oferece a oportunidade de familiarizar estudantes de graduação com essas funções importantes que não constam no currículo padrão dos cursos de física.