The method of ascent and cos(√A²+B²)

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(A²+B²) 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 non-commutative 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.