Square-integrable Hausdorff-based measures and their products: examples on some fractals and Quantum applications

In this work, we significantly enlarge the usual space of normalizable states of Quantum systems with m degrees of freedom. Specifically, we move in the (relativistic) Schrödinger setting of wave mechanics. In the chosen standard setting, usual quantum states belong to the Hilbert space $L^2(E;C)$, of complex Lebesgue square-integrable functions (equivalence classes) defined upon a convenient m-dimensional Euclidean space E. By our approach, we will extend the collection of possible E-based quantum states to a new class of measures, upon the Euclidean space E, that shall contain the classic Lebesgue-based function space L2, as well as the space of square integrable discrete measures and, more generally, all spaces $L^2_C(E; h_d)$ of complex functions that are square integrable with respect to the normalized Hausdorff measures $h_d$ on E. We, then, construct some examples on classic fractals, we define a natural product between two square integrable measures and we show some applications to Quantum Mechanics.