scholarly journals On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1657
Author(s):  
Mikhail G. Ivanov ◽  
Alexey E. Kalugin ◽  
Anna A. Ogarkova ◽  
Stanislav L. Ogarkov
Keyword(s):  
Renormalization Group ◽  
Schrödinger Equation ◽  
Functional Equations ◽  
Schrodinger Equation ◽  
Constant Field ◽  
Integration Formula ◽  
Invariant Solutions ◽  
Rigorous Derivation ◽  
Translation Invariant ◽  
Functional Renormalization Group

We consider the functional Hamilton–Jacobi (HJ) equation, which is the central equation of the holographic renormalization group (HRG), functional Schrödinger equation, and generalized Wilson–Polchinski (WP) equation, which is the central equation of the functional renormalization group (FRG). These equations are formulated in D-dimensional coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG scale, a “holographic” scalar field Λ is introduced. The extra coordinate (or scale) is obtained as the amplitude of delta-field or constant-field configurations of Λ. For all the functional equations above a rigorous derivation of corresponding integro-differential equation hierarchies for Green functions (GFs) as well as the integration formula for functionals are given. An advantage of the HJ hierarchy compared to Schrödinger or WP hierarchies is that the HJ hierarchy splits into independent equations. Using the integration formula, the functional (arbitrary configuration of Λ) solution for the translation-invariant two-particle GF is obtained. For the delta-field and the constant-field configurations of Λ, this solution is studied in detail. A separable solution for a two-particle GF is briefly discussed. Then, rigorous derivation of the quantum HJ and the continuity functional equations from the functional Schrödinger equation as well as the semiclassical approximation are given. An iterative procedure for solving the functional Schrödinger equation is suggested. Translation-invariant solutions for various GFs (both hierarchies) on delta-field configuration of Λ are obtained. In context of the continuity equation and open quantum field systems, an optical potential is briefly discussed. The mode coarse-graining growth functional for the WP action (WP functional) is analyzed. Based on this analysis, an approximation scheme is proposed for the generalized WP equation. With an optimized (Litim) regulator translation-invariant solutions for two-particle and four-particle amputated GFs from approximated WP hierarchy are found analytically. For Λ=0 these solutions are monotonic in each of the momentum variables.

scholarly journals A rigorous derivation of the Hamiltonian structure for the nonlinear Schrödinger equation

2020 ◽  
Vol 365 ◽  
pp. 107054
Author(s):  
Dana Mendelson ◽  
Andrea R. Nahmod ◽  
Nataša Pavlović ◽  
Matthew Rosenzweig ◽  
Gigliola Staffilani
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Hamiltonian Structure ◽  
Nonlinear Schrodinger Equation ◽  
Rigorous Derivation ◽  
Nonlinear Schrödinger

scholarly journals RENORMALIZATION OF FUNCTIONAL SCHRÖDINGER EQUATION BY BACKGROUND FIELD METHOD

1998 ◽  
Vol 13 (21) ◽  
pp. 1709-1717 ◽  
Author(s):  
K. ZAREMBO
Keyword(s):  
Renormalization Group ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Background Field ◽  
Field Method ◽  
Asymptotic Freedom ◽  
Yang Mills ◽  
State Wave ◽  
Background Field Method

Renormalization group transformations for Schrödinger equation are performed in both φ4 and Yang–Mills theories. The dependence of the ground state wave functional on rapidly oscillating fields is found. For Yang–Mills theory, this dependence restricts a possible form of variational ansatz compatible with asymptotic freedom.

scholarly journals RENORMALIZATION OF SCHRÖDINGER EQUATION AND WAVE FUNCTIONAL FOR RAPIDLY OSCILLATING FIELDS IN QCD

1998 ◽  
Vol 13 (22) ◽  
pp. 1795-1801 ◽  
Author(s):  
K. ZAREMBO
Keyword(s):  
Renormalization Group ◽  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Background Field ◽  
Field Method ◽  
State Wave ◽  
Background Field Method

Background field method is used to perform renormalization group transformations for Schrödinger equation in QCD. The dependence of the ground state wave functional on rapidly oscillating fields is found.

Optical soliton group invariant solutions by optimal system of Lie subalgebra with conservation laws of the resonance nonlinear Schrödinger equation

2020 ◽  
Vol 34 (35) ◽  
pp. 2050402 ◽  
Author(s):  
Vinita ◽  
Santanu Saha Ray
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Lie Symmetry ◽  
Nonlinear Schrodinger Equation ◽  
Invariant Solutions ◽  
Plasma Dynamic ◽  
One Dimensional ◽  
Nonlinear Schrödinger

In this article, the resonance nonlinear Schrödinger equation is studied, which elucidates the propagation of one-dimensional long magnetoacoustic waves in a cold plasma, dynamic of solitons and Madelung fluids in various nonlinear systems. The Lie symmetry analysis is used to achieve the invariant solution and similarity reduction of the resonance nonlinear Schrödinger equation. The infinitesimal generators, symmetry groups, commutator table and adjoint table have been obtained by the aid of invariance criterion of Lie symmetry. Also, one-dimensional system of subalgebra is constructed with the help of adjoint representation of a Lie group on its Lie algebra. By one-dimensional optimal subalgebra, the main equations are reduced to ordinary differential equations and their invariant solutions are provided. The general conservation theorem has been used to establish a set of non-local and non-trivial conservation laws.

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