This article deals with nonrelativistic study of a D-dimensional superintegrable system, which generalizes the ordinary isotropic oscillator system. The coefficients for the expansion between the hyperspherical and Cartesian bases (transition matrix), and vice-versa, are found in terms of the SU(2) Clebsch--Gordan coefficients analytically continued to real values of their arguments. The diagram method, which allow one to construct a transition matrix for arbitrary dimension, is developed.
In this article we analyze the isotropic oscillator system on the two-dimensional sphere in the spherical systems of coordinates. The expansion coefficients for transitions between three spherical bases of the oscillator are calculated. It is shown that these coefficients are expressed through the Clebsch-Gordan coefficients for SU(2) group analytically continued to real values of their argument.
The Schrödinger equation is thoroughly analysed for the isotropic oscillator in the three-dimensional space of constant positive curvature in the spherical and cylindrical systems of coordinates. The expansion coefficients between the spherical and cylindrical bases of the oscillator are calculated. It is shown that the relevant coefficients are expressed through the generalised hypergeometric functions $_4F_3$ of the unit argument or $6_j$ Racah symbols extended over their indices to the region of real values. Limiting transitions to a free motion and flat space are considered in detail. Elli…
This article deals with a nonrelativistic quantum mechanical study of a dynamical system which generalizes the isotropic harmonic oscillator system in three dimensions. The problem of interbasis expansions of the wavefunctions is completely solved. A connection between the generalized oscillator system (projected on the z-line) and the Morse system (in one dimension) is discussed.
This article deals with a quantum-mechanical system which generalizes the ordinary isotropic harmonic oscillator system. We give the coefficients connecting the polar and Cartesian bases for D=2 and the coefficients connecting the Cartesian and cylindrical bases as well as the cylindrical and spherical bases for D=3. These interbasis expansion coefficients are found to be analytic continuations to real values of their arguments of the Clebsch-Gordan coefficients for the group SU(2). For D=2, the superintegrable character for the generalized oscillator system is investigated from the points of …
This work is devoted to the investigation of the quantum mechanical systems on the two dimensional hyperboloid which admit separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C.P.Boyer, E.G.Kalnins and P.Winternitz, which haven't yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.
This paper deals with the dynamical system that generalizes the MIC-Kepler system. It is shown that the Schrödinger equation for this generalized MIC-Kepler system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic and spheroidal) are studied in detail. It is found that the coefficients for this expansion of the parabolic basis in terms of the spherical basis, and vice-versa, can be expresses through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their argument…
The concept of a generalized nonanalytic expansion which involves nonanalytic combinations of exponentials, logarithms and powers of a coupling is introduced and its use illustrated in various areas of physics. Dispersion relations for the resonance energies of odd anharmonic oscillators are discussed, and higher-order formulas are presented for cubic and quartic potentials.
In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variationa…
We introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one-dimensional anharmonic oscillators. We present several methods of analytically fixing the nonlinear variational parameter specified by the domain of the trigonometric basis functions. We show that the optimized parameter enables us to determine the energy spectrum to an arbitrary accuracy. Also, using the harmonic oscillator basis functions, we indicate that the resulting optimal frequency agrees with the one obtained by the principle of the minimal sensitivity.
For the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$ the Perturbation Theory (PT) in powers of coupling constant $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) is developed constructively in $r$-space and in $(gr)$ space, respectively. The Riccati-Bloch (RB) equation and Generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate $r$-space and in $(gr)$-space, respectively, exploring the logarithmic derivative of wavefunction $y$. It is shown that PT in powers …
We construct Lewis-Riesenfeld invariants from two dimensional point transformations for two oscillators that are coupled to each other in space in a PT-symmetrical and time-dependent fashion. The non-Hermitian Hamiltonian of the model is conveniently expressed in terms of generators of the symplectic $sp(4)$ Lie algebra. This allows for an alternative systematic approach to find Lewis-Riesenfeld invariants leading to a set of coupled differential equations that we solve by using time-ordered exponentials. We also demonstrate that point transformations may be utilized to directly construct time…
We consider a two-body quantum system in dimension one composed by a test particle interacting with an harmonic oscillator placed at the position $a>0$. At time zero the test particle is concentrated around the position $R_0$ with average velocity $\pm v_0$ while the oscillator is in its ground state. In a suitable scaling limit, corresponding for the test particle to a semi-classical regime with small energy exchange with the oscillator, we give a complete asymptotic expansion of the wave function of the system in both cases $R_0 <a$ and $R_0 >a$.
It is proved that the divergent Rayleigh-Schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are Borel summable in the distributional sense to the resonances naturally associated with the system.
A new recursion procedure for deriving renormalized perturbation expansions for the one-dimensional anharmonic oscillator is offered. Based upon the $\hbar$-expansions and suitable quantization conditions, the recursion formulae obtained have the same simple form both for ground and excited states and can be easily applied to any renormalization scheme. As an example, the renormalized expansions for the sextic anharmonic oscillator are considered.
It is shown that for the one-dimensional quantum anharmonic oscillator with potential $V(x)= x^2+g^2 x^4$ the Perturbation Theory (PT) in powers of $g^2$ (weak coupling regime) and the semiclassical expansion in powers of $\hbar$ for energies coincide. It is related to the fact that the dynamics in $x$-space and in $(gx)$-space corresponds to the same energy spectrum with effective coupling constant $\hbar g^2$. Two equations, which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, are derived. The PT in $g^2$ for the logari…
As an application of a recently developed variational perturbation theory we find the first 22 terms of the convergent strong-coupling series expansion for the ground state energy of the quartic anharmonic oscillator.
We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form $V(x)=(ix)^{2N+1}+bix$ does not agree with an earlier WKB expression for $V(x)=-(ix)^{2N+1}$ the case $b=0$. The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.
We prove a powerful scaling property for the extremality condition in the recently developed variational perturbation theory which converts divergent perturbation expansions into exponentially fast convergent ones. The proof is given for the energy eigenvalues of an anharmonic oscillator with an arbitrary $x^p$-potential. The scaling property greatly increases the accuracy of the results.
This paper considers one-mode open quantum harmonic oscillators with a pair of conjugate position and momentum variables driven by vacuum bosonic fields according to a linear quantum stochastic differential equation. Such systems model cavity resonators in quantum optical experiments. Assuming that the quadratic Hamiltonian of the oscillator is specified by a positive definite energy matrix, we consider a modified version of the quantum Karhunen-Loeve expansion of the system variables proposed recently. The expansion employs eigenvalues and eigenfunctions of the two-point commutator kernel for…
The divergent series for a function defined through Lapalce integral and the ground state energy of the quartic anharmonic oscillator to large orders are studied to test the generalized binomial transform which is the renamed version of $δ$-expansion proposed recently. We show that, by the use of the generalized binomial transform, the values of functions in the limit of zero of an argument is approximately computable from the series expansion around the infinity of the same argument. In the Laplace integral, we investigate the subject in detail with the aid of Mellin transform. In the anharmo…
We use the optimized trigonometric finite basis method to find energy eigenvalues and eigenfunctions of the time-independent Schrodinger equation with high accuracy. We apply this method to the quartic anharmonic oscillator and the harmonic oscillator perturbed by a trigonometric anharmonic term as not exactly solvable cases and obtain the nearly exact solutions.
We study the quantum mechanical motion in the $x^2y^2$ potentials with $n=2,3$, which arise in the spatially homogeneous limit of the Yang-Mills (YM) equations. These systems show strong stochasticity in the classical limit ($\hbar = 0$) and exhibit a quantum mechanical confinement feature. We calculate the partition function $Z(t)$ going beyond the Thomas-Fermi (TF) approximation by means of the semiclassical expansion using the Wigner-Kirkwood (WK) method. We derive a novel compact form of the differential equation for the WK function. After separating the motion in the channels of the equip…
The explicit semiclassical treatment of the logarithmic perturbation theory for the bound-state problem for the spherical anharmonic oscillator is developed. Based upon the $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the quartic anharmonic oscillator are considered.
Functional renormalization group methods formulated in the real-time formalism are applied to the $O(N)$ symmetric quantum anharmonic oscillator, considered as a $0+1$ dimensional quantum field-theoric model, in the next-to-leading order of the gradient expansion of the one- and two-particle irreducible effective action. The infrared scaling laws and the sensitivity-matrix analysis show the existence of only a single, symmetric phase. The field-independent term of the wavefunction renormalization turned out to be negligible, but its field-dependent piece is noticeable. It is shown that the inf…
We apply the connected moments expansion to simple quantum--mechanical examples and show that under some conditions the main equations of the approach are no longer valid. In particular we consider two--level systems, the harmonic oscillator and the pure quartic oscillator.
Magic numbers predicted by a 3-dimensional q-deformed harmonic oscillator with Uq(3)>SOq(3) symmetry are compared to experimental data for atomic clusters of alkali metals (Li, Na, K, Rb, Cs), noble metals (Cu, Ag, Au), divalent metals (Zn, Cd), and trivalent metals (Al, In), as well as to theoretical predictions of jellium models, Woods-Saxon and wine bottle potentials, and to the classification scheme using the 3n+l pseudo quantum number. In alkali metal clusters and noble metal clusters the 3-dimensional q-deformed harmonic oscillator correctly predicts all experimentally observed magic num…
Canonical quantization has taught us great things. A common example is that of the harmonic oscillator, which is like swinging a ball on a string back and forth. However, the half-harmonic oscillator blocks the ball at the bottom and then it quickly bounces backwards. This second model cannot be correctly solved using canonical quantization. Now, there is an expansion of quantization, called affine quantization, that can correctly solve the half-harmonic oscillator, and offers correct solutions to a grand collection of other problems, which even reaches to field theory and gravity. This paper …
The Schrödinger equation for the four-dimensional double singular oscillator is separable in Eulerian, doble polar and spheroidal coordinates in ${\rm I R}^4$. It is shown that the coefficients for the expansion of double polar basis in terms of the Eulerian basis can be expressed through the Clebsch-Gordan coefficients of the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the spheroidal basis in terms of the Eulerian and double polar bases are proved to satisfy three-term recursion relations.
We present an approximative calculation of the ground-state energy for the anisotropic anharmonic oscillator Using an instanton solution of the isotropic action $δ= 0$, we obtain the imaginary part of the ground-state energy for small negative $g$ as a series expansion in the anisotropy parameter $δ$. From this, the large-order behavior of the $g$-expansions accompanying each power of $δ$ are obtained by means of a dispersion relation in $g$. These $g$-expansions are summed by a Borel transformation, yielding an approximation to the ground-state energy for the region near the isotropic limit. …
We develop a formula for matching a Taylor series about the origin and an asymptotic exponential expansion for large values of the coordinate. We test it on the expansion of the generating functions for the moments and connected moments of the Hamiltonian operator. In the former case the formula produces the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We choose the harmonic oscillator and a strongly anharmonic oscillator as illustrative examples for numerical test. Our results reveal some features of the connected-moments expansion that were overlooked in earlier st…
We study the 1/N expansion in noncommutative quantum mechanics for the anharmonic and Coulombian potentials. The expansion for the anharmonic oscillator presented good convergence properties, but for the Coulombian potential, we found a divergent large N expansion when using the usual noncommutative generalization of the potential. We proposed a modified version of the noncommutative Coulombian potential which provides a well-behaved 1/N expansion.
A very simple procedure to calculate eigenenergies of quantum anharmonic oscillators is presented. The method, exact but for numerical computations, consists merely in requiring the vanishing of the Wronskian of two solutions which are regular respectively at the origin and at infinity. The first one can be represented by its series expansion; for the second one, an asymptotic expansion is available. The procedure is illustrated by application to quartic and sextic oscillators.
We investigate T-matrix for bound and continuous-spectrum states in the discrete oscillator representation. The investigation is carried out for a model problem - the particle in the field of a central potential. A system of linear equations is derived to determine the coefficients of the T-matrix expansion in the oscillator functions. We selected four potentials (Gaussian, exponential, Yukawa, and square-well ones) to demonstrate peculiarities of the T-matrix and its dependence on the potential shape. We also study how the T-matrix expansion coefficients depend on the parameters of the oscill…
We use a power-series expansion to calculate the eigenvalues of anharmonic oscillators bounded by two infinite walls. We show that for large finite values of the separation of the walls, the calculated eigenvalues are of the same high accuracy as the values recently obtained for the unbounded case by the inner-product quantization method. We also apply our method to the Morse potential. The eigenvalues obtained in this case are in excellent agreement with the exact values for the unbounded Morse potential.
We develop a method for the determination of thecdynamics of dissipative quantum systems in the limit of large number of quanta N, based on the 1/N-expansion of Heidmann et al. [ Opt. Commun. 54, 189 (1985) ] and the quantum-classical correspondence. Using this method, we find analytically the dynamics of nonclassical states generation in the higher-order anharmonic dissipative oscillators for an arbitrary temperature of a reservoir. We show that the quantum correction to the classical motion increases with time quadratically up to some maximal value, which is dependent on the degree of nonlin…
The sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Herm…
The explicit semiclassical treatment of logarithmic perturbation theory for the nonrelativistic bound states problem is developed. Based upon $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions for the one-dimensional anharmonic oscillator is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and exited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the harmonic oscillator perturbed by $λx^{6}$ are considered…
We apply power series expansion to symmetric multi-well oscillators bounded by two infinite walls. The spectrum and expectation values obtained are compared with available exact and approximate values for the unbounded ones. It is shown that the method is capable of producing to a high accuracy the eigenvalues, eigenfunctions, and the expectation values $<x^{2k}>$ of the corresponding unbounded ones as the separation between the two infinite walls becomes large.
The application of the optimized expansion for the quantum-mechanical propagation in the anharmonic potential $λx^4$ is discussed for real and imaginary time. The first order results in the imaginary time formalism provide approximations to the free energy and particle density which agree well with the exact results in the whole range of temperatures.
We propose a new treatment for the quantum three-body problem. It is based on an expansion of the wave function on harmonic oscillator functions with different sizes in the Jacobi coordinates. The matrix elements of the Hamiltonian can be calculated without any approximation and the precision is restricted only by the dimension of the basis. This method can be applied whatever the system under consideration. In some cases, the convergence property is greatly improved in this new scheme as compared to the old traditional method. Some numerical tricks to reduce computer time are also presented.
