On the origin of the gravitational quantization: The Titius--Bode Law

Action at distance in Newtonian physics is replaced by finite propagation speeds in classical post--Newtonian physics. As a result, the differential equations of motion in Newtonian physics are replaced by functional differential equations, where the delay associated with the finite propagation speed is taken into account. Newtonian equations of motion, with post--Newtonian corrections, are often used to approximate the functional differential equations. In ``On the origin of quantum mechanics'', preprint, physics/0505181, May 2005, a simple atomic model based on a functional differential equation which reproduces the quantized Bohr atomic model was presented. The unique assumption was that the electrodynamic interaction has finite propagation speed. Are the finite propagation speeds also the origin of the gravitational quantization? In this work a simple gravitational model based on a functional differential equation gives an explanation of the modified Titius--Bode law.Comment: 9 pages, 1 figure in EPS forma

Operational calculus and integral transforms for groups with finite propagation speed

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an operational calculus for integral transforms and functions of $A$ using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on $E$. Examples include the Mellin transform and the Mehler--Fock transform. The paper uses functional calculus for the cosine family $\cos( t\sqrt {\Delta})$ which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm--Liouville hypergroups with Laplace representation as well as analogues to the Kunze--Stein phenomenon in the hypergroup convolution setting.Comment: arXiv admin note: substantial text overlap with arXiv:1304.5868. Substantial revision to version

Uniform energy decay for wave equations with unbounded damping coefficients

We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence result of a weak solution. In this case we never impose strong assumptions such as compactness of the support of the initial data. This means that we never rely on the finite propagation speed property of the solution, and we try to deal with an essential unbounded coefficient case.Comment: 15 page