✦ LIBER ✦

The method of ascent and cos(A2+B2)

✍ Scribed by Yakar Kannai

Publisher: Elsevier Science
Year: 2000
Tongue: French
Weight: 161 KB
Volume: 124
Category: Article
ISSN: 0007-4497
DOI: 10.1016/s0007-4497(00)01065-4

No coin nor oath required. For personal study only.

✦ Synopsis

The well-known Hadamard method of descent consists of deriving solutions of wave equations in n variables from (known) solutions in more than n variables. Here we show how to build the propagator cos( A 2 + B 2 ) out of cos(At) and cos(Bt). The formulas obtained for the case of commuting A and B may be illustrated by deriving solutions of the Cauchy problem for the n dimensional wave equation (and for the Klein-Gordon equation) from the very simple one-dimensional case. In the noncommutative case the formula involves differentiating a very high-dimensional integral to a high order. The result yields expressions for the harmonic oscillator and for certain simple hypoelliptic sum of squares operators, such as the Heisenberg Laplacian.

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