On spectral asymptotic of quasi-exactly solvable quartic potential

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.

Higgs-like spectator field as the origin of structure

We show that the observed primordial perturbations can be entirely sourced by a light spectator scalar field with a quartic potential, akin to the Higgs boson, provided that the field is sufficiently displaced from vacuum during inflation. The framework relies on the indirect modulation of reheating, which is implemented without any direct coupling between the spectator field and the inflaton and does not require non-renormalisable interactions. The scenario gives rise to local non-Gaussianity with $$f_{\mathrm{NL}}\simeq 5$$ f NL ≃ 5 as the typical signal. As an example model where the indirect modulation mechanism is realised for the Higgs boson, we study the Standard Model extended with right-handed neutrinos. For the Standard Model running we find, however, that the scenario analysed does not seem to produce the observed perturbation.

$$ \mathrm{T}\overline{\mathrm{T}} $$-deformed nonlinear Schrödinger

The $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ -deformed classical Lagrangian of a 2D Lorentz invariant theory can be derived from the original one, perturbed only at first order by the bare $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ composite field, through a field-dependent change of coordinates. Considering, as an example, the nonlinear Schrödinger (NLS) model with generic potential, we apply this idea to non-relativistic models. The form of the deformed Lagrangian contains a square-root and is similar but different from that for relativistic bosons. We study the deformed bright, grey and Peregrine’s soliton solutions. Contrary to naive expectations, the $$ \mathrm{T}\overline{\mathrm{T}} $$ T T ¯ -perturbation of nonlinear Schrödinger NLS with quartic potential does not trivially emerge from a standard non-relativistic limit of the deformed sinh-Gordon field theory. The c → ∞ outcome corresponds to a different type of irrelevant deformation. We derive the corresponding Poisson bracket structure, the equations of motion and discuss various interesting aspects of this alternative type of perturbation, including links with the recent literature.

Quantum mechanical out-of-time-ordered-correlators for the anharmonic (quartic) oscillator

Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with 2〈x2〉T〈p2〉T at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.

Фундаментальный анализ сингулярных и резонансных явлений в колебательных полиадах молекулы дифторсилилена

The singular structure of the lower vibrational states of the difluorosilylene molecule (up to four quanta of total excitation) was studied by expanding the energies of each state in the series of high-order Rayleigh-Schrödinger perturbation theory and analyzing their implicit multivalued properties using the fourth degree Padé-Hermite approximants. The quartic potential energy surface in dimensionless normal coordinates was calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of the values of multivalued approximants reproduces the variational solution with high accuracy, while other values, starting from the fourth polyad, in many cases coincide with the energies of other states of the polyad. The Fermi and Darling-Dennison resonances are analyzed on the basis of the coincidence of the singular complex branch points of the approximants for interacting states inside or near a circle of unit radius on the complex plane. It was found that a pair of states can have several coinciding branch points of solutions, including those inside the unit circle. It is concluded that this approach is an effective method for determining the polyad structure of vibrational states. The calculation parameters are selected, which are necessary for the reproducibility of key results. The calculations were carried out using a software package in the Fortran language using a package of arithmetic calculations with a long mantissa of real numbers (200 digits).