scholarly journals Fast solution of Schrödinger’s equation using linear combinations of plane waves

2017 ◽  
Vol 74 (12) ◽  
pp. 3318-3327
Author(s):  
José M. Pérez-Jordá
Keyword(s):  
Plane Waves ◽  
Linear Combinations ◽  
Schrödinger’S Equation ◽  
Fast Solution ◽  
Schrödinger's Equation

BEHAVIOR OF SOUND WAVES IN SHEPHERD FLUTE ON THE BASIS OF QUANTUM THEORY OF INDIVIDUAL PHONONS

2010 ◽  
Vol 24 (17) ◽  
pp. 3439-3452
Author(s):  
SONJA KRSTIĆ ◽  
VJEKOSLAV SAJFERT ◽  
BRATISLAV TOŠIĆ
Keyword(s):  
Quantum Theory ◽  
Hyperbolic Equation ◽  
Plane Waves ◽  
Sound Waves ◽  
High Frequencies ◽  
Schrödinger’S Equation ◽  
Schrödinger's Equation ◽  
Theory And Experiment ◽  
Good Agreement

Using the linearized Hamiltonian of individual phonon, it was shown that Schrödinger's equation of individual phonon is by form identical with classical hyperbolic equation. It was also shown that damper in shepherd's flute is reflexive for high frequencies and transparent for low ones. This result was experimentally tested by authors and good agreement of theory and experiment was found. The propagation of sound in parallelopipedal and cylindrical shepherd's flute was investigated. It turned out that parallelopipedal sound propagates in z-direction, only, while in cylindrical one besides plane waves in z-direction the damped waves in x, y plane appear.

scholarly journals Orthogonal structure on a quadratic curve

2020 ◽  
Author(s):  
Sheehan Olver ◽  
Yuan Xu
Keyword(s):  
Weight Function ◽  
Orthogonal Polynomials ◽  
Extension Problem ◽  
Orthogonal Expansions ◽  
Schrödinger’S Equation ◽  
Fourier Extension ◽  
Quadratic Curve ◽  
Interpolation Of Functions ◽  
Fourier Orthogonal Expansions ◽  
Schrödinger's Equation

Abstract Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

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