High-frequency approximation for the Helmholtz equation: A probabilistic approach

We use a probabilistic representation of the Helmholtz equationʹs solution with analytical coefficients to derive a high-frequency approximation. This solution is related to the Schrِdinger equationʹs solution. For this last equation, when the potential is also an analytical function, its asymptotic behaviour when h goes to zero was studied in [2]. We present this result more precisely and apply it to the Helmholtz equation, obtaining, at least formally, when the frequency goes to infinity the expression for the first term of the asymptotic expansion for the wave equation.