Wave kernels with magnetic field on the hyperbolic plane and with the Morse potential on the real line

In this article we give explicit solutions for the wave equations associated to the modified Schr\"odinger operators with magnetic field on the disc and the upper half plane models of the hyperbolic plane. We show that the modified Schr\"odinger operator with magnetic field on the upper half plane model and the Schr\"odinger operator with diatomic molecular Morse potential on $\R$ are related by means of one-dimensional Fourier transform. Using this relation we give the explicit forms of the wave kernels associated to the Schr\"odinger operator with the diatomic molecular Morse potential on $\R$ in terms of the two variables confluent hypergeometric function $\Phi_1$.