scholarly journals The functional integral with unconditional Wiener measure for anharmonic oscillator

2008 ◽  
Vol 49 (11) ◽  
pp. 113505 ◽  
Author(s):  
J. Boháčik ◽  
P. Prešnajder
Keyword(s):  
Anharmonic Oscillator ◽  
Functional Integral ◽  
Wiener Measure

scholarly journals Dynamical Symmetries of the 2D Newtonian Free Fall Problem Revisited

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 27
Author(s):  
Tuong Trong Truong
Keyword(s):  
Symmetry Group ◽  
Anharmonic Oscillator ◽  
Symmetry Algebra ◽  
Functional Space ◽  
Free Fall ◽  
Dynamical Symmetry ◽  
Functional Integral ◽  
Theoretical Physics ◽  
Classical Level ◽  
Dynamical Symmetries

Among the few exactly solvable problems in theoretical physics, the 2D (two-dimensional) Newtonian free fall problem in Euclidean space is perhaps the least known as compared to the harmonic oscillator or the Kepler–Coulomb problems. The aim of this article is to revisit this problem at the classical level as well as the quantum level, with a focus on its dynamical symmetries. We show how these dynamical symmetries arise as a special limit of the dynamical symmetries of the Kepler–Coulomb problem, and how a connection to the quartic anharmonic oscillator problem, a long-standing unsolved problem in quantum mechanics, can be established. To this end, we construct the Hilbert space of states with free boundary conditions as a space of square integrable functions that have a special functional integral representation. In this functional space, the free fall dynamical symmetry algebra is shown to be isomorphic to the so-called Klink’s algebra of the quantum quartic anharmonic oscillator problem. Furthermore, this connection entails a remarkable integral identity for the quantum quartic anharmonic oscillator eigenfunctions, which implies that these eigenfunctions are in fact zonal functions of an underlying symmetry group representation. Thus, an appropriate representation theory for the 2D Newtonian free fall quantum symmetry group may potentially open the way to exactly solving the difficult quantization problem of the quartic anharmonic oscillator. Finally, the initial value problem of the acoustic Klein–Gordon equation for wave propagation in a sound duct with a varying circular section is solved as an illustration of the techniques developed here.

Polar Decomposition of Wiener Measure and Schwarzian Integrals

2020 ◽  
Vol 41 (4) ◽  
pp. 709-713
Author(s):  
E. T. Shavgulidze ◽  
N. E. Shavgulidze
Keyword(s):  
Polar Decomposition ◽  
Wiener Measure

Inverse spectral problem of an anharmonic oscillator on a half-axis with the Neumann boundary condition

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Agil K. Khanmamedov ◽  
Nigar F. Gafarova
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Inverse Spectral Problem ◽  
Anharmonic Oscillator ◽  
Neumann Boundary ◽  
Effective Algorithm ◽  
Inverse Spectral Problems ◽  
Main Equation ◽  
Inverse Spectral ◽  
Half Axis

AbstractAn anharmonic oscillator {T(q)=-\frac{d^{2}}{dx^{2}}+x^{2}+q(x)} on the half-axis {0\leq x<\infty} with the Neumann boundary condition is considered. By means of transformation operators, the direct and inverse spectral problems are studied. We obtain the main integral equations of the inverse problem and prove that the main equation is uniquely solvable. An effective algorithm for reconstruction of perturbed potential is indicated.

Fuzzy Bezier splines with application to fuzzy functional integral equations

2020 ◽  
Vol 24 (8) ◽  
pp. 6069-6084
Author(s):  
Alexandru Mihai Bica ◽  
Constantin Popescu
Keyword(s):  
Integral Equations ◽  
Functional Integral ◽  
Functional Integral Equations
Sign in / Sign up

Export Citation Format

Share Document