• tilde integral
• Hilbert space
• gaussian measures
• gell-mann low
• complex scaled time
• functional integral
• feynman path integral
• analytic time

In the present Thesis, we will present an extension of the infinite dimensional oscillatory integral and the relative application of this extension to a simple physical model. The extension we are considering consists in dealing with a time evolution where the time is multiplied by a complex scale, this follows along the same ideas that are employed in formally relating Euclidean an Minkowskian Feynman Path Integrals, and will be in the present setting rigorously justified. We will apply these general methods to the simple model of an Anhamonic Quantum Oscillator. This model is simple enough to allow rigorous treatment, yet it has some of the relevant features of a Quantum Field Theory, in particular it can be regarded as a zero-dimensional Quantum Field Theory, and as uch can be employed as a toy model before engaging in more sophisticated theories. The complex scale extensions of the time parameter that we are introducing will allow us, in the present context, to prove in the context of rigorous Feynman path integration a variant of the important formula commonly referred to as Gell-Mann Low formula.