Lamé polynomials, hyperelliptic reductions and Lamé band structure

The band structure of the Lamé equation, viewed as a one-dimensional Schrödinger equation with a periodic potential, is studied. At integer values of the degree parameter ℓ , the dispersion relation is reduced to the ℓ =1 dispersion relation, and a previously published ℓ =2 dispersion relation is shown to be partly incorrect. The Hermite–Krichever Ansatz, which expresses Lamé equation solutions in terms of ℓ =1 solutions, is the chief tool. It is based on a projection from a genus- ℓ hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lamé equation parameters take complex values, are investigated. If the Lamé equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lamé polynomials, i.e. band-edge solutions, are given. A table in the earlier literature is corrected.

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Computation of Energy Dispersion Relation for an Electron in a one Dimensional Periodic Potential Using Intel Visual Fortran 17.0 Update 3 for Windows

Archives of Current Research International ◽

10.9734/acri/2017/37145 ◽

2017 ◽

Vol 10 (4) ◽

pp. 1-12

Author(s):

Sa’adiyya Bature ◽

Abdulqadir Nura

Keyword(s):

Dispersion Relation ◽

Periodic Potential ◽

Energy Dispersion ◽

One Dimensional

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ENERGY BAND STRUCTURE OF AN ELECTRON IN A ONE-DIMENSIONAL PERIODIC POTENTIAL

FUDMA Journal of Sciences ◽

10.33003/fjs-2020-0402-155 ◽

2020 ◽

Vol 4 (2) ◽

pp. 420-424

Author(s):

Ibrahim Bagudo ◽

Abdullahi Tanimu

Keyword(s):

Band Structure ◽

Energy Band ◽

Periodic Potential ◽

Perfect Crystal ◽

Energy Band Structure ◽

Periodic Field ◽

Reduced Representation ◽

One Dimensional ◽

Bloch’S Theorem ◽

Work Done

It has been observed that electron in a perfect crystal moves in a spatially periodic field of force due to the ions and the averaged effect of all the electrons. This work shows the investigative work done to determine the energy band structure of an electron in a one-dimensional periodic potential. The application of the Kronig-Penney model was applied to an electron state in a delta-like potential. To fully understand the Kronig-Penney model, the concept of Bloch’s theorem was first introduced to describe the conduction of electrons in solids. It has been found that the periodic potential introduces gaps in the reduced representation with an increasing number of potential well/barrier strengths. It has been observed that the regions of non-propagating states, which give rise to energy band gaps, become larger with decreasing values.

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Dispersion relation of modulational instability for one-dimensional standing solitary waves in hot ultrarelativistic electron–positron plasmas

Pramana ◽

10.1007/s12043-020-02069-7 ◽

2021 ◽

Vol 95 (1) ◽

Author(s):

Ebrahim Heidari

Keyword(s):

Dispersion Relation ◽

Solitary Waves ◽

Modulational Instability ◽

One Dimensional ◽

Ultrarelativistic Electron ◽

Electron Positron

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On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽

10.3390/math9010078 ◽

2020 ◽

Vol 9 (1) ◽

pp. 78

Author(s):

Haifa Bin Jebreen ◽

Fairouz Tchier

Keyword(s):

Differential Equation ◽

Galerkin Method ◽

Linear Ordinary Differential Equation ◽

Time Interval ◽

Hyperbolic Partial Differential Equation ◽

Time Step ◽

Numerical Examples ◽

One Dimensional ◽

The Stability ◽

The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

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THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Nagoya Mathematical Journal ◽

10.1017/nmj.2020.37 ◽

2020 ◽

pp. 1-23

Author(s):

MICHELE BOLOGNESI ◽

NÉSTOR FERNÁNDEZ VARGAS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Hyperelliptic Curve ◽

Geometric Description ◽

Birational Model ◽

Rank 2 ◽

Semistable Vector Bundles ◽

Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

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On Approximate Large Deviations for 1D Diffusion

Georgian Mathematical Journal ◽

10.1515/gmj.2003.381 ◽

2003 ◽

Vol 10 (2) ◽

pp. 381-399

Author(s):

A. Yu. Veretennikov

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Large Deviations ◽

Stochastic Differential Equation ◽

Approximation Scheme ◽

Sufficient Conditions ◽

Rate Function ◽

Euler Approximation ◽

One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

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On the solvability and approximate solution of a one-dimensional singular problem for a p-Laplacian fractional differential equation

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.110948 ◽

2021 ◽

Vol 147 ◽

pp. 110948

Author(s):

KumSong Jong ◽

HuiChol Choi ◽

MunChol Kim ◽

KwangHyok Kim ◽

SinHyok Jo ◽

...

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Fractional Differential Equation ◽

Singular Problem ◽

One Dimensional ◽

Fractional Differential

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DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

Modern Physics Letters B ◽

10.1142/s021798490701244x ◽

2007 ◽

Vol 21 (02n03) ◽

pp. 139-154 ◽

Cited By ~ 6

Author(s):

J. H. ASAD

Keyword(s):

Differential Equation ◽

Green’S Function ◽

Recurrence Relation ◽

Green's Function ◽

Two Dimensional ◽

Dimensional Lattice ◽

One Dimensional ◽

The One ◽

Lattice Green's Function ◽

Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

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