On Liouville’s Theorem for Biharmonic Functions

1971 ◽  
Vol 20 (1) ◽  
pp. 37-39 ◽  
Author(s):  
R. R. Huilgol
Keyword(s):  
Biharmonic Functions ◽  
Liouville’S Theorem

DOES LIOUVILLE'S THEOREM IMPLY QUANTUM MECHANICS?

1999 ◽  
Vol 13 (02) ◽  
pp. 161-189
Author(s):  
C. SYROS
Keyword(s):  
Quantum Mechanics ◽  
Radiation Spectrum ◽  
Black Body ◽  
Boltzmann Distribution ◽  
Black Body Radiation ◽  
Planck Constant ◽  
Klein Gordon ◽  
Energy Variable ◽  
Liouville’S Theorem ◽  
Liouville’S Equation

The essentials of quantum mechanics are derived from Liouville's theorem in statistical mechanics. An elementary solution, g, of Liouville's equation helps to construct a differentiable N-particle distribution function (DF), F(g), satisfying the same equation. Reality and additivity of F(g): (i) quantize the time variable; (ii) quantize the energy variable; (iii) quantize the Maxwell–Boltzmann distribution; (iv) make F(g) observable through time-elimination; (v) produce the Planck constant; (vi) yield the black-body radiation spectrum; (vii) support chronotopology introduced axiomatically; (viii) the Schrödinger and the Klein–Gordon equations follow. Hence, quantum theory appears as a corollary of Liouville's theorem. An unknown connection is found allowing the better understanding of space-times and of these theories.

scholarly journals Liouville's Theorem in the Radially Symmetric Case

2005 ◽  
Vol 48 (3) ◽  
pp. 405-408
Author(s):  
Richard Froese
Keyword(s):  
Short Proof ◽  
Symmetric Case ◽  
Ricatti Equation ◽  
Dirichlet To Neumann Map ◽  
Liouville’S Theorem

AbstractWe present a very short proof of Liouville's theorem for solutions to a non-uniformly elliptic radially symmetric equation. The proof uses the Ricatti equation satisfied by the Dirichlet to Neumann map.

Parabolicity and Existence of Dirichlet Finite Biharmonic Functions

1974 ◽  
Vol s2-8 (1) ◽  
pp. 145-148 ◽  
Author(s):  
Leo Sario ◽  
Cecilia Wang
Keyword(s):  
Biharmonic Functions
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