How to increase the applicability of integral transform approaches in physics?

Integral transform approaches are numerous in many fields of physics, but in most cases limited to the use of the Laplace kernel. However, it is well known that the inversion of the Laplace transform is very problematic, so that the function related to the physical observable is in most cases unaccessible. The great advantage of kernels of the bell-shaped form has been demonstrated in few-body nuclear systems. In fact the use of the Lorentz kernel has allowed us to overcome the stumbling block of the ab initio description of reactions to the full continuum of systems of more than three particles. The problem of finding kernels of similar form, applicable to many-body problems, deserves particular attention. If this search were successful, the integral transform approach might represent the only viable ab initio access to many observables that are not calculable directly.