An asymptotic expansion for energy eigenvalues of anharmonic oscillators

In the present contribution, we derive an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V(x)=κx^2q+ωx²,q=2,3,…$V\left(x\right)=\kappa x^{2q}+\omega x^2,q=2,3,\dots$ as the energy level n$n$ approaches infinity. The asymptotic expansion is obtained using the WKB theory and series reversion. Furthermore, we construct an algorithm for computing the coefficients of the asymptotic expansion for quartic anharmonic oscillators, leading to an efficient and accurate computation of the energy values for n≥6$n\ge 6$.     

On the energy spectra of one-dimensional anharmonic oscillators

In this paper we present explicit and simple analytical formulae for the energy eigenvaluesE _n (λ) of one-dimensional anharmonic oscillators characterized by the potentials 1/2mω ² x ²+λx ^2α withα=2, 3 and 4. A simple intuitive criterion supplemented by the requirement of correct asymptotic behaviour, has been employed in arriving at the formulae. Our energy values over a wide range ofn andλ are in good agreement with the numerical values computed by earlier workers through very elaborate techniques. To our knowledge this is the first time that formulae of such wide validity have been given. The results for pure power oscillators are trivially obtained by going over to theω→0 limit. Approximate analytic expressions for the low order even moments ofx are also given.     

Finite discontinuities in the energy eigenvalue spectra of anharmonic oscillators

The existence of finite discontinuities in the energy eigenvalue spectra of certain multiterm potentials when their coupling parameters attain suitably chosen limiting values has been reported in the literature. We show that such discontinuities are also characteristic of such well-known systems as generalized anharmonic oscillators and the doubly anharmonic oscillator in one dimension. The present study strengthens the general conjecture that eigenvalue spectra are likely to display discontinuities in situations where a potential undergoes an abrupt change in shape with smooth variation of its coupling parameters.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

The path-integral technique and the large-quantum-number behaviour of energy eigenvalues of coupled anharmonic oscillators

Using the path-integral technique, we derive here the large-quantum-number behaviour of energy eigenvalues of a coupled anharmonic oscillator. The result clearly shows the behaviour of the leading and the next to leading term with respect to the parameters contained in the coupled anharmonic part of the potential.     

Equidistant spectra of anharmonic oscillators

Some representative potentials of the anharmonic-oscillator type are constructed. Some corresponding spectra-shift operators are also constructed. These operators are a natural generalization of Fok creation and annihilation operators. The Schrodinger problem for these potentials leads to an equidistant energy spectrum for all excited states, which are separated from the ground state by an energy gap. The general properties of the dynamic system generated by spectral-shift operators of third degree are analyzed. Several examples of such anharmonic oscillators are discussed. The relationship between the eigenvectors of the Schrodinger problem and a certain type of nonclassical orthogonal polynomials is established.     

Perturbation theory of odd anharmonic oscillators

We study the perturbation theory forH=p ²+x ²+x ²ⁿ⁺¹,n=1, 2, .... It is proved that when Im0,H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schrödinger perturbation expansion, and admits an analytic continuation to Im=0 where it can be interpreted as a resonance of the problem.Partially supported by G.N.F.M., C.N.R.     

On the large order expansion for the anharmonic oscillators

Using functional approaches, we investigate the large-K behaviour of theK ^th coefficientE _k in the perturbation expansion for the ground-state energyE(g) of the generalized anharmonic oscillatorX ^2N with internalO(_n)-symmetry. We establish the equivalence between the pure functional approach and the method of Collins-Soper at any order in 1 /K. For that purpose, we first develop an algebraic treatment of perturbation series and prove a theorem on Borelsummable functions. Finally, we compute analytically the 1 /K and 1 /K ² corrections to the leading term forN=2.     

A class of exact solutions for doubly anharmonic oscillators

The solution of a difference equation in the form of an infinite continued fraction is used to obtain a class of exact solutions for the eigenfunctions and eigenvalues of doubly anharmonic oscillators described by potentials of the type (1/2)²x²+(1/4)x⁴+(1/6)x⁶, n>0, provided certain constraints on the couplings are satisfied. The class is denumerably infinite but not complete.     

Equipartition thresholds in chains of anharmonic oscillators

We perform a detailed numerical study of the transition to equipartition in the Fermi-Pasta-Ulam quartic model and in a class of potentials of given symmetry using the normalized spectral entropy as a probe. We show that the typical time scale for the equipartition of energy among Fourier modes grows linearly with system size: this is the time scale associated with the smallest frequency present in the system. We obtain two different scaling behaviors, either with energy or with energy density, depending on the scaling of the initial condition with system size. These different scaling behaviors can be understood by a simple argument, based on the Chirikov overlap criterion. Some aspects of the universality of this result are investigated: symmetric potentials show a similar transition, regulated by the same time scale.     

Glassy dynamics in strongly anharmonic chains of oscillators

We review the mechanism for transport in strongly anharmonic chains of oscillators near the atomic limit where all oscillators are decoupled. In this regime, the motion of most oscillators remains close to integrable, i.e. quasi-periodic, on very long time scales, while a few chaotic spots move very slowly and redistribute the energy across the system. The material acquires several characteristic properties of dynamical glasses: intermittency, jamming, and a drastic reduction of the mobility as a function of the thermodynamical parameters. We consider both classical and quantum systems, though with more emphasis on the former, and we discuss also the connections with quenched disordered systems, which display a similar physics to a large extent.     

A simple generation of exactly solvable anharmonic oscillators

An elementary finite difference algorithm shortens the Darboux method, permitting an easy generation of families of anharmonic potentials almost isospectral to the harmonic oscillator. Against common belief, it is possible to associate a SUSY partner to a given Hamiltonian H using a factorization energy greater than the ground state energy of H. The explicit 3-SUSY partners of the oscillator potential are found and discussed.     

An improved variation-perturbation technique for the partition function of anharmonic oscillators

A new factorization of the hamiltonian is applied to statistical-variation-perturbation theory. The zero order variational hamiltonian already contains the diagonal perturbation part. The partition function (PF) of the quartic anharmonic oscillator is in very good agreement with numerical results.     

Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators

We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from below. All transformations are carried out non-perturbatively so that all Hamiltonians in this process are spectrally exactly equivalent in the sense that they have identical instantaneous energy eigenvalue spectra.     

Energy eigenvalues for anharmonic and double-well oscillators with even power polynomial potential

Energies for different states of anharmonic and double-well oscillators described by V(x) = Σ_{j = 1}^s(a_2j/2j)x^2j have been determined using the renormalized hypervirial-Padé scheme. A comparison of the results with available exact values shows that the method is quite successful for anharmonic oscillators but fails in the case of double-well oscillators. Also included is the discussion of the dependence of the energy on various parameters.     

Stationary anharmonic oscillators in the particle-in-a-box basis: Near-exact results

Variational studies employing the basis functions of the quantal particle-in-a-box model are shown to lead to accurate estimates of eigenvalues, various expectation values and eigenfunctions of the stationary anharmonic oscillator problem. Calculations involve bothzx ^2α and (x ²⁺ zx ^2α)-type oscillators, withα=2, 3 and 4, both in weak and strong coupling regime. Apart from its recommendable computational simplicity, convergence of the present recipe has also been demonstrated to be quite fast. Results for the first ten states are reported. A few goodness tests for the approximate wavefunctions and consistency requirements for some properties are also performed.