Canonical Transformation Method Research Articles

Distributed Global Optimization for a Class of Nonconvex Optimization With Coupled Constraints

This article examines the distributed nonconvex optimization problem with structured nonconvex objective functions and coupled convex inequality constraints on static networks. A distributed continuous-time primal-dual algorithm is proposed to solve the problem. We use the canonical transformation and Lagrange multiplier method to reformulate the nonconvex optimization problem as a convex–concave saddle point computation problem, which is subsequently solved by employing the projected primal-dual subgradient method. Sufficient conditions that guarantee the global optimality of the solution generated by the proposed algorithm are provided. Numerical and application examples are presented to demonstrate the proposed algorithm.

Effective Raman Hamiltonian revisited

In this paper, the effective Raman Hamiltonian is revisited. A common way to obtain the effective Raman Hamiltonian is by using the time-dependent perturbation method (TDPT) along with Fermi’s golden rule to keep the total energy and probability constant. However, for a non-resonant Raman process the obtained effective Hamiltonian is not convenient because it is not Hermitian. Hence, we present the Magnus expansion method for obtaining the effective Raman Hamiltonian, which has the advantages of being Hermitian and featuring effects absent in the TDPT effective Hamiltonian. To our knowledge, this is the first time that the Magnus expansion is utilized as an alternative method. We compare the our obtained effective Hamiltonian to that obtained from canonical transformation method. We determine the extra piece in second-order time-dependent perturbation theory which causes loss or gain of total probability.

Formation of intermediate coupling optical polarons and bipolarons in two-dimensional systems

The formation of the optical polaron and bipolaron in two-dimensional (2D) systems is studied in the intermediate electron–phonon coupling regime. The total energies of the 2D polaron and bipolaron are calculated by using the Buimistrov–Pekar method of canonical transformations. The obtained results are compared with other existing results obtained by using the Feynman path integral method and the modified Lee–Low–Pines unitary transformation method. It is shown that the electron–phonon correlation significantly reduces the total energy of the 2D polaron in comparison with the energy of the strong coupling (adiabatic) polaron. It is found that the polaron formation in 2D systems is possible when the electron–phonon coupling constant α is greater than the critical value αc≃2.94, which is much lower than a critical value of the electron–phonon coupling constant α in three-dimensional (3D) systems. The critical values of the Fröhlich coupling constant α and the ratio η=ε∞/ε0 (where ε∞ and ε0 are the high frequency and static dielectric constants, respectively), which determine the bipolaron stability region in 2D systems, are calculated numerically. It is interesting for application to the layered cuprate superconductors that the (bi)polarons are formed more easily in quasi-2D regions than in the bulk. It is argued that the high-Tc cuprate superconductivity can exist above the bulk superconducting transition temperature Tc as the persisting superfluidity of polaronic (bosonic) Cooper pairs and large bipolarons at quasi-2D grain boundaries or in the CuO2 layers above Tc.

Transport Through a Quantum Dot with Electron-Phonon Interaction

We theoretically study the electrical transport properties of a single level quantum dot connected to two normal conducting leads, which is coupled to the lattice vibrations. We determine the current through the quantum dot in two different situations: time-independent and time-averaged. In all situations we consider three cases: when there is no electron-phonon interaction, when the dot electrons interact with optical phonons or when they interact with acoustic phonons. At finite temperatures we take into account the temperature dependence of the chemical potential. We treat the electron-phonon interaction by the canonical transformation method. In the case of electron-longitudinal optical phonon interaction the spectrum contains a subpeak. In the case of electron-acoustic phonon interaction the spectrum is continuous. In the time-averaged situation many parasite peaks appear in the spectrum, due to the external time-modulation.

RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of <em>X<sub>m</sub></em> Laguerre or <em>X<sub>m</sub></em> Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

Non-deformed singular and non-singular exponential-type potentials

The canonical transformation method applied to the Schrödinger equation to transform it into a second-order differential equation of hypergeometric-type is presented. Starting from there, those exactly solvable multiparameter exponential-type (ME-T) potentials with hypergeometric wavefunctions that belong to the families of radial (singular) and one-dimensional (non-singular) potentials, are obtained. Furthermore, we show how the choice of the involved parameters leads, as particular cases, to different deformed or non-deformed potential models already used in the study of electronic properties of diatomic molecules. Also, the analysis of parameters lets us identify the couple of potential partners (singular/non-singular) that correspond to each choice of the parameters appearing in the ME-T potential. As a useful application of the proposal, the most important non-deformed exponential potential models are considered for which it can be viewed as a unified treatment with the following advantages: (1) It is not necessary to use a special method to solve the Schrödinger equation for a specific potential model because solution is obtained as particular case by the simple choice of the involved parameters; (2) The families of singular and non-singular potentials are straightforward identified; (3) The corresponding associated partners, between radial and one-dimensional non-deformed potentials, are found; (4) New potentials, as interesting alternatives for quantum applications, are obtained. In addition, from the conditions that parameters must meet to have physically acceptable solutions, we establish the requirements for the existence or not of singular/non-singular potential partners.

Global Dynamics of Pipes Conveying Pulsating Fluid in the Supercritical Regime

Global dynamics of supercritical pipes conveying pulsating fluid considering superharmonic resonance of the second mode with 1:2 internal resonance are investigated. The governing partial differential equations in the supercritical regime are obtained based on the nontrivial equilibrium configuration of the pipes conveying fluid and then transformed into a discretized nonlinear gyroscopic system via assumed modes and Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motion to the near-integrable Hamiltonian standard form. The energy-phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. The global solutions are subsequently interpreted in terms of the physical motion of such gyroscopic system. Two types of nonlinear normal modal motion and the chaotic pattern conversion between the locked simple bidirectional traveling wave motion and the complex bidirectional traveling wave motion are discussed.

Position-Dependent Mass Schrödinger Equation for the Morse Potential

The position dependent mass Schrödinger equation (PDMSE) has a wide range of quantum applications such as the study of semiconductors, quantum wells, quantum dots and impurities in crystals, among many others. On the other hand, the Morse potential is one of the most important potential models used to study the electronic properties of diatomic molecules. In this work, the solution of the effective mass one-dimensional Schrödinger equation for the Morse potential is presented. This is done by means of the canonical transformation method in algebraic form. The PDMSE is solved for any model of the proposed kinetic energy operators as for example the BenDaniel-Duke, Gora-Williams, Zhu-Kroemer or Li-Kuhn. Also, in order to solve the PDMSE with Morse potential, we consider a superpotential leading to a special form of the exactly solvable Schrödinger equation of constant mass for a class of multiparameter exponential-type potential along with a proper mass distribution. The proposed approach is general and can be applied in the search of new potentials suitable on science of materials by looking into the viable choices of the mass function.

Solvable Diffusion Models with Linear and Mean-Reverting Nonlinear Drifts

We present new extensions to a method for constructing families of solvable one-dimensional time-homogeneous diffusions whose transition densities and other related quantities are obtainable in analytically closed form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines a smooth monotonic differentiable map with nonzero derivative (diffeomorphism) and a measure change via a Doob-$h$ transform. This gives rise to new multiparameter solvable diffusions that are divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The first class of models, having linear drift and nonlinear (state-dependent) volatility functions, is useful for pricing equity and foreign exchange (FX) options in finance, while the second class of diffusions contains new models that are mean-reverting and are applicable to pricing interest-rate and other path-dependent derivatives such as volatility index (VIX) options. As specific examples of the first class of affine drift models, we present explicit results for two new families of models that arise from the squared Bessel process (the Bessel family) and the Ornstein--Uhlenbeck diffusion (the OU family). For the second class of nonlinear drift models, we give examples of solvable subfamilies called the Bessel family of mean-reverting diffusions and derive closed-form integral formulas for conditional expectations of certain functionals. In particular, we derive a new closed-form analytical formula for the Laplace transform with respect to the strike price of a standard call VIX option. We then succeed in Laplace inverting the expression to obtain numerically exact VIX call option prices with realistic implied volatilities with respect to strike and maturity. Moreover, we accurately calibrate the OU family of models to FX option market data, exhibiting pronounced implied volatility smiles across several strikes and maturities.

Quantum oscillator and Kepler–Coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions

The quantum oscillator and Kepler-Coulomb problems in d-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Kepler-Coulomb potentials in curved space. The oscillator on the sphere and the Kepler-Coulomb potential in a hyperbolic space are studied in detail and their extensions are proved to be consistent with already known ones in Euclidean space. The partnership between nonextended and extended potentials is interpreted in a deformed supersymmetric framework. Those extended potentials that are isospectral to some nonextended ones are shown to display deformed shape invariance, which in the Kepler-Coulomb case is enlarged by also translating the degree of the polynomial arising in the rational part denominator.

Homoclinic orbits and chaos of a supercritical composite panel with free-layer damping treatment in subsonic flow

Multi-pulse homoclinic orbits and chaotic dynamics of a supercritical composite panel with free layer damping treatment in subsonic flow are investigated considering one to two internal resonance. Inviscid potential flow theory is employed to exhibit the aerodynamic pressure and Kelvin’s model is used to describe the viscoelastic property of the free damping layer. By Hamilton’s principle, the governing equation of the composite panel in the subcritical regime is derived. In the supercritical regime, the buckling configuration is solved analytically and the PDE is obtained by introducing a displacement transformation for nontrivial equilibrium configuration. Then the governing equation in the first supercritical region is transformed into a discretized nonlinear gyroscopic system via assumed modes and then Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motions to the near-integrable Hamiltonian standard forms. The Energy-Phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. The global solutions are finally interpreted in terms of the physical motion of the gyroscopic continua and the dynamical mechanism of chaotic pattern conversion between the forward traveling wave motion and the complex bidirectional traveling wave motion are discussed.

Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions

In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function; the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x2 and g(x)~x2/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.

Canonical Duality for Box Constrained Nonconvex and Nonsmooth Optimization Problems

This paper presents an application of the canonical duality theory for box constrained nonconvex and nonsmooth optimization problems. By use of the canonical dual transformation method, which is developed recently, these very difficult constrained optimization problems inRncan be converted into the canonical dual problems, which can be solved by deterministic methods. The global and local extrema can be identified by the triality theory. Some examples are listed to illustrate the applications of the theory presented in the paper.

Removal of ordering ambiguity for a class of position dependent mass quantum systems with an application to the quadratic Liénard type nonlinear oscillators

We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schrödinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering, we show that the class of systems can also be exactly solvable and is also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Liénard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

Integrable approximation of regular regions with a nonlinear resonance chain.

Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the nonintegrable system. In the second step a sequence of canonical transformations is applied to the integrable approximation to match the shape of regular tori. We demonstrate the method for the generic standard map at various parameters.

Integrable approximation of regular islands: The iterative canonical transformation method

Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. We present an iterative method to construct an integrable approximation H(reg), which resembles the regular dynamics of a given mixed system H and extends it into the chaotic region. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. This method works for strongly perturbed systems and arbitrary degrees of freedom. We apply it to the standard map and the cosine billiard.

Optical Properties of Modified Pöschl-Teller Quantum Well with Position-Dependent Mass

Optical properties of modified Poschl-Teller quantum well are studied with considering a position- dependent mass of an electron in the confinement potential. Analytical and numerical calculation are investigated based on the point canonical transformation method. Author to whom correspondence should be addressed. Researches shows that there are several theoretical analysis of the optical properties is associated with inter- subband optical transitions in nanostructures which are presented in scientific literatures. 18-21 In this paper, we attempt to investigate the analytical and numerical calcu- lation of the optical properties of modified Poschl-Teller potential (MPTP) with considering position-dependent mass effect. At first, we choose the exact solvable case and use PCTM to solve the wave equation and obtain the wave function and corresponding energy eigenstates. In addi- tion, the numerical solutions are introduced for compari- son and checked the accuracy of our calculations. Then, we use the same method to investigate optical properties of MPTP with an arbitrary position-dependent mass func- tion numerically. Model and discussion are brought in the following section, including the optical properties. Finally, some conclusions are highlighted.

Quantization of the Liénard II equation and Jacobi’s last multiplier

In this paper, the role of Jacobi’s last multiplier in mechanical systems with a position-dependent mass is unveiled. In particular, we map the Liénard II equation to a position-dependent mass system. The quantization of the Liénard II equation is then carried out using the point canonical transformation method together with the Von Roos ordering technique. Finally, we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions.

Optical properties of spherical quantum dot with position-dependent effective mass

In this paper, the effects of position-dependent effective mass on the optical properties of a three dimensional quantum dot are numerically investigated. For this purpose, by using the point canonical transformation method and numerical solution of the Schrödinger equation, the energy levels and wave functions of electrons in the confinement potential are obtained. Then the optical properties are studied by considering the compact density matrix approach based on the numerical calculation of the wave function. Our calculations were compared with the case of constant mass. Results show that the position-dependent effective mass plays an important role in the intersubband optical absorption coefficient and refractive index changes in a quantum dot.

Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials

Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type X1 exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of coordinate: e−αpxeαp=x+iα, to be both quasi- and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.