Exactly solvable two-dimensional stationary Schrödinger operators obtained by the nonlocal Darboux transformation

The Fokker–Planck equation associated with the two-dimensional stationary Schrödinger equation has the conservation law form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two-dimensional stationary Schrödinger equations. The examples of exactly solvable two-dimensional stationary Schrödinger operators with smooth potentials decaying at infinity are obtained.     

Spinor with Schrödinger symmetry and non-relativistic supersymmetry

We construct the d-dimensional “half” Schrödinger equation, which is a kind of the root of the Schrödinger equation, from the (d+1)-dimensional free Dirac equation. The solution of the “half” Schrödinger equation also satisfies the usual free Schrödinger equation. We also find that the explicit transformation laws of the Schrödinger and the half Schrödinger fields under the Schrödinger symmetry transformation are derived by starting from the Klein-Gordon equation and the Dirac equation in d+1 dimensions. We derive the 3- and 4-dimensional super-Schrödinger algebra from the superconformal algebra in 4 and 5 dimensions. The algebra is realized by introducing two complex scalar and one (complex) spinor fields and the explicit transformation properties have been found.     

Rogue wave solutions for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

A generalized Darboux transformation for the coupled cubic–quintic nonlinear Schrödinger equation is constructed by the Darboux matrix method. As applications, the Nth-order rogue wave solutions of the coupled cubic–quintic nonlinear Schrödinger equation have been obtained. In particular, the dynamics of the general first- and second-order rogue waves are discussed and illustrated through some figures.     

An alternative derivation of the pulse solutions of the KdV hierarchy

An alternative approach issues from the Appelle transformation of the Schrödinger equation. One solves the inverse problem for the transformed equation, a general solution of which is a quadratic form of two independent solutions of the primary Schrödinger equation. If the potential in the Schrödinger equation obeys one equation of the KdV hierarchy, the time derivative of this form is a linear combination of the form and its space derivative. The coefficients in the combination depend on the potential and the energy parameter of the Schrödinger equation only. This relation also determines the time dependence of the spectral data which along with the solution of the inverse problem gives the solution of the KdV equations as usual.     

Periodic Wave Solutions of Generalized Derivative Nonlinear Schrödinger Equation

A Darboux transformation of the generalized derivative nonlinear Schrodinger equation is derived. As an application, some new periodic wave solutions of the generalized derivative nonlinear Schrodinger equation are explicitly given.     

EXACT SOLUTIONS TO THE N-BODY SCHRÖDINGER EQUATION FOR THE HARMONIC OSCILLATOR

The exact solutions to the N-body Schrödinger equation for the harmonic oscillator are presented analytically. The permutational symmetry of the solutions for the identical three-body system of the harmonic oscillator are discussed in some detail.     

具有色散系数的(2+1)维非线性Schrödinger方程的有理解和空间孤子

非线性Schrödinger方程是物理学中具有广泛应用的非线性模型之一. 本文采用相似变换, 将具有色散系数的(2+1)维非线性Schrödinger方程简化成熟知的Schrödinger方程, 进而得到原方程的有理解和一些空间孤子.     

Note on the kinetic energy inequality leading to Lieb's negative ionization upper bound

The kinetic energy inequality is alternatively proved which yields Lieb's boundN < 2Z + 1 on the maximum negative ionization of an atom with nucleus chargeZ andN electrons when the kinetic energy operator is the nonrelativistic or relativistic magnetic Schrödinger operator. It is seen to follow from the free case where the vector potential vanishes. The proof applies to the Weyl quantized relativistic magnetic Schrödinger operator as well.Research partially supported by Grant-in-Aid for Scientific Research Nos. 04640141 and 05640165, Ministry of Education, Science and Culture, Japanese Government.     

A law of large numbers and a central limit theorem for the Schrödinger operator with zero-range potentials

We consider the Schrödinger operator with zero-range potentials onN points of three-dimensional space, independently chosen according to a common distributionV(x). Under some assumptions we prove that, whenN goes to infinity, the sequence converges to a Schrödinger operator with an effective potential. The fluctuations around the limit operator are explicitly characterized.     

Mixed quantum mechanical periodic and potential wall problem

We consider the Schrödinger equation with an even-square integrable potential of period one on the negative real axis and a wall potential of heighta > 0 on the positive real axis. The spectrum of this Schrödinger equation is determined and it is proved that bounded solutions never exist if the energyE < a is lying in a gap of the periodic spectrum.     

Nonlinear wave mechanics,information theory,and thermodynamics

A logarithmic nonlinear term is introduced in the Schrödinger wave equation, and a physical justification and interpretation are provided within the context of information theory and thermodynamics. From the resulting nonlinear Schrödinger equation for a system at absolute temperatureT>0, the energy equivalence,kT 1n 2, of a bit of information is derived.     

Darboux transformation and exactly solvable potentials with quasi-equidistant spectrum

The Darboux transformation of order n introduced in our previous paper is applied to the harmonic-oscillator potential of the one-dimensional Schrödinger equation. New potentials that have a quasi-equidistant spectrum (i.e., an equidistant spectrum with lacunas) and admit of solution of the Schrödinger equation in elementary functions are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 3–10, August, 1995.     

Time-dependent quantum systems and chronoprojective geometry

Chronoprojective transformations in the framework of five-dimensional Schrödinger formalism are used to construct the solution of the Schrödinger equation with a time-dependent harmonic potential from the solution of a free Schrödinger equation.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Third-order constants of motion in quantum mechanics

We investigate the general form of a third-order linear differential operator that is required to commute with the Schrödinger Hamiltonian in two dimensions, and find that the third-order part must be a polynomial of third degree in the generators of the Euclidean group. Partial differential equations that the potentialV must satisfy are derived, and solved for the special cases where the Schrödinger equation separates in polar or Cartesian coordinates. The functionsV thus obtained are nonsingular, but are periodic through elliptic functions. After separation of variables, the Schrödinger equation gives Lame's equation.     

On Irreducible Second-Order Darboux Transformations

The second-order Darboux transform for a growing potential of a Schrödinger equation is discussed in detail. Restrictions on the transforming functions whose eigenvalues are greater than the energy of the ground state if the potential of the transformed Schrödinger equation is regular has been established. It has been shown that, along with the well-known elimination of two levels of the discrete spectrum, other opportunities can also be realized: (a) the transformed spectrum remains identical to the original one; (b) one level is eliminated from the spectrum; (c) one additional level is generated, and (d) two additional levels are generated.     

A quantum model of option pricing: When Black-Scholes meets Schrödinger and its semi-classical limit

The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ² with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.     

On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation

In this Letter, the generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. A generalized Darboux transformation (DT) of the GNLS equation is constructed with the help of the gauge transformation for an Ablowitz–Kaup–Newell–Segur (AKNS) type GNLS spectral problem, from which a unified formula of Nth-order rogue wave solution to the GNLS equation is given. In particular, the first and second-order rogue wave solutions to the GNLS equation are explicitly illustrated through some figures.     

Electronic energy levels of nanorings with impurities and Aharonov–Bohm effects

By modeling impurities along a nanoring as general potential forms the Schrödinger equation for ballistic electrons is shown to separate in cylindrical coordinates. We find an analytical eigenvalue equation for N delta-function-barrier impurities in the presence of magnetic flux. Previous calculations of the electronic states of a one-dimensional (1D) and two-dimensional (2D) nanoring for only one or two impurities modeled by equal square barriers is explicitly extended to three and four different or equal impurities modeled as delta-barrier, square-barrier, or delta-well potential forms. This is shown to be generalizable to any number N. Effects on the energy spectra due to magnetic flux and different kinds and numbers of impurities are compared in 1D and 2D nanorings.     

A one dimensional microscopical model for the study of the coherence in the stopping power problem. Part 1

A frictional quantum mechanical system consisting of a particle being scattered inelastically by a chain ofN infinitely heavy equidistantly spaced two-level atoms is studied in one dimension. The stationary Schrödinger equation of thisN+1 body problem is solved for high energies of the incident particle with suitable interaction potentials. The mean velocity is calculated as a function of the number of targets passed and a classical estimate of the stopping power given.