Solvable Systems Research Articles

A class of globally solvable Markovian quadratic BSDE systems and applications

We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games and martingales on Riemannian manifolds.

On a two-dimensional solvable system of difference equations

On a two-dimensional solvable system of difference equations

Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra $\mathfrak{so}(4,\mathbb {C})$ to itself. In 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between B\^ocher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all B\^ocher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric B\^ocher contractions.

Open source Matrix Product States: Opening ways to simulate entangled many-body quantum systems in one dimension

Numerical simulations are a powerful tool to study quantum systems beyond exactly solvable systems lacking an analytic expression. For one-dimensional entangled quantum systems, tensor network methods, amongst them Matrix Product States (MPSs), have attracted interest from different fields of quantum physics ranging from solid state systems to quantum simulators and quantum computing. Our open source MPS code provides the community with a toolset to analyze the statics and dynamics of one-dimensional quantum systems. Here, we present our open source library, Open Source Matrix Product States (OSMPS), of MPS methods implemented in Python and Fortran2003. The library includes tools for ground state calculation and excited states via the variational ansatz. We also support ground states for infinite systems with translational invariance. Dynamics are simulated with different algorithms, including three algorithms with support for long-range interactions. Convenient features include built-in support for fermionic systems and number conservation with rotational U(1) and discrete Z2 symmetries for finite systems, as well as data parallelism with MPI. We explain the principles and techniques used in this library along with examples of how to efficiently use the general interfaces to analyze the Ising and Bose–Hubbard models. This description includes the preparation of simulations as well as dispatching and post-processing of them. Program summaryProgram title: Open Source Matrix Product States (OSMPS), v2.0Program Files doi:http://dx.doi.org/10.17632/vxm2mcmk4v.1Licensing provisions: GNU GPL v3Programming language: Python, Fortran2003, MPI for parallel computingCompilers (Fortran): gfortran, ifort, g95Dependencies: The minimal requirements in addition to the Fortran compiler are BLAS, LAPACK, ARPACK, python, numpy, scipy. Additional packages for plotting include matplotlib, dvipng, and LATEX packages. The Expokit package, available at the homepage http://www.maths.uq.edu.au/expokit/, is required to use the Local Runge–Kutta time evolution.Supplementary material: We provide programs to reproduce selected figures in the Appendices.Nature of problem: Solving the ground state and dynamics of a many-body entangled quantum system is a challenging problem; the Hilbert space grows exponentially with system size. Complete diagonalization of the Hilbert space to floating point precision is limited to less than forty qubits.Solution method: Matrix Product States in one spatial dimension overcome the exponentially growing Hilbert space by truncating the least important parts of it. The error can be well controlled. Local neighboring sites are variationally optimized in order to minimize the energy of the complete system. We can target the ground state and low lying excited states. Moreover, we offer various methods to solve the time evolution following the many-body Schrödinger equation. These methods include e.g. theSuzuki–Trotter decompositions using local propagators or the Krylov method, both approximating the propagator on the complete Hilbert space.

General k-Dimensional Solvable Systems of Difference Equations

The solvability of a k-dimensional system of difference equations of interest, which extends several recently studied ones, is investigated. A general sufficient condition for the solvability of the system is given, considerably extending some recent results in the literature.

Gapless Symmetry-Protected Topological Order

We introduce exactly solvable gapless quantum systems in $d$ dimensions that support symmetry protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points or phases that can be interpreted as critical condensates of domain walls "decorated" with dimension $(d-1)$ SPT systems. Using a combination of field theory and exact lattice results, we argue that such gapless SPT systems have symmetry-protected topological edge modes that can be either gapless or symmetry-broken, leading to unusual surface critical properties. Despite the absence of a bulk gap, these edge modes are robust against arbitrary symmetry-preserving local perturbations near the edges. In two dimensions, we construct wavefunctions that can also be interpreted as unusual quantum critical points with diffusive scaling in the bulk but ballistic edge dynamics.

Hamiltonian formulation of systems with balanced loss–gain and exactly solvable models

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occurs in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion are constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered.

On the solutions of two systems of quaternion matrix equations

ABSTRACTWe study the system of quaternion generalized Sylvester matrix equations , and . We establish necessary and sufficient conditions for the system to be solvable, and when such conditions are met, we present the general solution to the system. We also investigate the number of solutions a solvable system could have. Our techniques entail the investigation of the quaternion matrix system and in X and Y with the constraint that X and Y have a common submatrix. For such systems, we establish a characterization of solvable systems and provide their general solution. We also study in detail the number of solutions a solvable system could have and present necessary and sufficient conditions for a solvable system to have a unique solution. We finally illustrate our techniques in solving the generalized Sylvester matrix equations by providing an example.

Self-calibrating generalized phase-shifting interferometry of three steps by fitting modulation light to a polynomial of degree K

This paper presents a robust and non-iterative algorithm for phase retrieval in phase-shifting interferometry of three unknown and unequal phase-steps, based on the modulation light estimation by applying the least squares method (LSM). This approach is divided in three stages; first the background light and phase object are algebraically eliminated to obtain a single equation in three unknowns; second, the square of modulation light is approximated to a polynomial of degree K to build an error function, and third, the coefficients of this polynomial are estimated by applying the LSM, with which a solvable system of equations is formed to calculate the phase-steps, and after that, the phase object is possible to compute. Additional advantages of this approach are its simple computer implementation, and its capacity to support high spatial variations in the illumination and the phase object. This approach is described theoretically and verified numerically and experimentally.

Quantum Zeno and anti-Zeno effects in open quantum systems

Traditional approach on quantum Zeno effect (QZE) and quantum anti-Zeno effect (QAZE) in open quantum systems (implicitly) assumes the bath (environment) state returning to its original state after each instantaneous projective measurement on the system and thus ignores the cross-correlations of the bath operators between different Zeno intervals. However, this assumption is not generally true, especially for a bath with a considerably non-negligible memory effect and for a system repeatedly projected into an initial general superposition state. We find that in stark contrast to the result of a constant value found in the traditional approach, the scaled average decay rate in unit Zeno interval of the survival probability is generally time-dependent or has an oscillatory behavior. In the case of strong bath correlation, the transition between the QZE and QAZE depends sensitively on the number of measurements $N$. For a fixed $N$, a QZE region predicted by the tradition approach may be in fact already in the QAZE region. We illustrate our findings using an exactly solvable open qubit system model with a Lorentzian bath spectral density, which is directly related to realistic circuit cavity quantum electrodynamics systems. Thus the results and dynamics presented here can be verified by current superconducting circuit technology.

The boundary effects of transverse field Ising model

Advance in quantum simulations using trapped ions or superconducting elements allows detailed analysis of the transverse field Ising model (TFIM), which can exhibit a quantum phase transition and has been a paradigm in exactly solvable quantum systems. The Jordan–Wigner transformation maps the one-dimensional TFIM to a fermion model, but additional complications arise in finite systems and introduce a fermion-number parity constraint when periodic boundary condition is imposed. By constructing the free energy and spin correlations with the fermion-number parity constraint and comparing the results to the TFIM with open boundary condition, we show that the boundary effects can become significant for the anti-ferromagnetic TFIM with odd number of sites at low temperature.

Quasi adiabatic dynamics of energy eigenstates for a solvable quantum system at finite temperature

It is a fundamental problem to characterize nonequilibrium processes. For a moving one-dimensional potential, we explore the nonequilibrium dynamics of the initial energy eigenstates for a confined quantum system interacting with a large reservoir. For concreteness, we investigate a dragged harmonic oscillator linearly interacting with an assembly of harmonic oscillators, and explore the deviation from adiabatic processes by rigorously calculating the so-called persistent amplitude. In particular, we show that the phase of the persistent amplitude is considered to be common both for the ground and excited states. Also, we can define the quasi adiabatic processes in a well-defined double limit of small perturbation and a sufficiently long time in terms of the phase and absolute value of the persistent amplitude.

Separation equations for 2D superintegrable systems on constant curvature spaces

Second-order conformal quantum superintegrable systems in two dimensions are Laplace equations on a manifold with an added scalar potential and three independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are eight pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality.

Conservation in two-particle self-consistent extensions of dynamical mean-field theory

Extensions of dynamical-mean-field-theory (DMFT) make use of quantum impurity models as non-perturbative and exactly solvable reference systems which are essential to treat the strong electronic correlations. Through the introduction of retarded interactions on the impurity, these approximations can be made two-particle self-consistent. This is of interest for the Hubbard model, because it allows to suppress the antiferromagnetic phase transition in two-dimensions in accordance with the Mermin-Wagner theorem, and to include the effects of bosonic fluctuations. For a physically sound description of the latter, the approximation should be conserving. In this paper we show that the mutual requirements of two-particle self-consistency and conservation lead to fundamental problems. For an approximation that is two-particle self-consistent in the charge- and longitudinal spin channel, the double occupancy of the lattice and the impurity are no longer consistent when computed from single-particle properties. For the case of self-consistency in the charge- and longitudinal as well as transversal spin channels, these requirements are even mutually exclusive so that no conserving approximation can exist. We illustrate these findings for a two-particle self-consistent and conserving DMFT approximation.

Solvable product-type system of difference equations with two dependent variables

It has been recently noticed that there is a finite number of two-dimensional classes of product-type systems of difference equations solvable in closed form. We present a new class of this type. A detailed analysis of the form of its solutions is given. Our results complement the previous ones on such systems and present one of the final steps in describing the forms of their solutions.

Analysis of (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation via Lax pair investigation and group transformation method

Singular manifolds method (SMM) is utilized to detect the Lax pair and Bäcklund transformation of (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. An extension of group transformation method with three parameters is exploited in transforming the Lax pair directly to a system of ordinary differential equations (ODEs). Two cases of similarity variables are considered resulting in solvable systems of ODEs. New analytical solutions are attained and graphed containing some soliton waves. Some of these solutions are applicable of plasma explosions and fluid dynamics. Both of investigation of Lax pair of (3+1)-dimensional evolution equations and three parameters group method are seldom utilized in this kind of evolution equations.

An analytical-numerical solution to assess the dynamic response of viscoelastic plates to a moving mass

• Dynamic response of viscoelastic plates to a moving mass is investigated. • The proposed method is applicable for any viscoelastic material with no restriction. • The effect of material and foundation damping is accounted for. • Different trajectories of the moving mass are examined. • The effect of convective accelerations is also taken into account in the solution. In this paper, the dynamics of a viscoelastic plate resting on a viscoelastic Winkler foundation and traversed by a moving mass is studied. The Laplace transform is employed to derive the governing equation of the problem. Thereafter, an analytical-numerical method is proposed in order to determine the dynamic response of the plate. The method is based on transforming the governing partial differential equation into a new solvable system of linear ordinary differential equations. To that extent, the proposed solution proves to be applicable to plates made of any viscoelastic material and with various boundary conditions. Moreover, the moving mass may travel at any arbitrary trajectory with no restriction. The capability of the method is demonstrated through several illustrative examples, wherein a comprehensive parametric study is performed to investigate the effect of various parameters on the response of the system. A comparison between the results obtained by the complete moving mass approach with those coming from the approximate solutions is also carried out. The comparative study suggests that neglecting the inertia of the moving mass in the problems where a relatively heavy mass is traveling at high speeds on a plate with a low level of viscosity leads, in general, to an unsafe design. Additionally, ignoring the effect of convective accelerations in the moving mass formulation results in an overly uneconomical design in most cases. However, for large values of viscosity, the accuracy of the approximate methods further improves, and those methods can be used to reasonably estimate the response.

Classes of globally solvable involutive systems

We study a linear operator associated with a real smooth closed non-exact 1-form b defined on a closed orientable surface. Locally the operator can be seen as an overdetermined system of first order linear partial differential equations. Here we present a result that completely characterizes a class of systems that are globally solvable, namely when b has rank equal to 1, in terms of a topological condition. Such a condition bears on the superlevel and sublevel sets of primitives of b. In a certain covering space, called minimal covering space, the condition is equivalent to the connectedness of the superlevel and sublevel sets of the primitives there defined (a property that frequently appears in related papers). We furthermore exhibit another class of globally solvable systems by constructing smooth closed non-exact 1-forms of arbitrary rank on surfaces of genus greater than 1 out of 1-forms which individually define globally solvable systems on tori.

Analytic solutions throughout a period doubling route to chaos.

We show examples of dynamical systems that can be solved analytically at any point along a period doubling route to chaos. Each system consists of a linear part oscillating about a set point and a nonlinear rule for regularly updating that set point. Previously it has been shown that such systems can be solved analytically even when the oscillations are chaotic. However, these solvable systems show few bifurcations, transitioning directly from a steady state to chaos. Here we show that a simple change to the rule for updating the set point allows for a greater variety of nonlinear dynamical phenomena, such as period doubling, while maintaining solvability. Two specific examples are given. The first is an oscillator whose set points are determined by a logistic map. We present analytic solutions describing an entire period doubling route to chaos. The second example is an electronic circuit. We show experimental data confirming both solvability and a period doubling route to chaos. These results suggest that analytic solutions may be a more useful tool in studying nonlinear dynamics than was previously recognized.

Spectral intertwining relations in exactly solvable quantum-mechanical systems

In exactly solvable quantum-mechanical systems, ladder and intertwining operators play a central role because, if they are found, the energy spectra can be obtained algebraically. In this paper, we propose the spectral intertwining relation as a unified relation of ladder and intertwining operators in a way that can depend on the energy eigenvalues. It is shown that the spectral intertwining relations can connect eigenfunctions of different energy eigenvalues belonging to two different Hamiltonians, which cannot be obtained by previously known structures such as shape invariance. As an application, we find new spectral intertwining operators for the Hamiltonians of the hydrogen atom and the Rosen--Morse potential.