Minimum uncertainty wavelets in SUSY quantum mechanics, the theory of coherent states, the theory of strings, and the fermionic harmonic oscillator

The SUSY properties and string aspects of the recently derived "constrained minimum Heisenberg uncertainty" (mu)-wavelets were considered. Several types of raising and lowering operators, which play a fundamental role in the SUSY structure of these wavelets were analyzed. Compared with the harmonic oscillator, the (mu)-wavelets naturally manifest the SUSY properties. Using the results, we construct a supercoherent theory of these wavelets. In addition, the (mu)-wavelets and harmonic oscillator belong to different classes in string theory. The results should be of interest for the supersymmetric theory of quantum fields and string theory.; The exact multi-dimensional solutions of the recently derived "minimum uncertainty" (mu)-wavelets are considered. We derive isotropic non-Cartesian multi-dimensional solutions using the three different methods; the solution of differential equations, Fourier transformation, and the creation and annihilation operators. We derive multi-dimensional ALDAF (Associate Laguerre Distributed Approximating Functionals) which is a collection of mu-wavelets. ALDAF and its non-Cartesian solutions are good approximation tools for physics and other sciences. Some examples of 1 dimension, 2 dimensions, and 3 dimensions are given. The raising and lowering operators also work for the non-Cartesian coordinate system of mu wavelets.; We obtain the multi-dimensional harmonic oscillator solution to compare with multi-dimensional mu-wavelets and provide the foundation for analyzing the SUSY structure in multi-dimensional harmonic oscillators.; We find that in our general superpotential model fermionic solutions of the harmonic oscillator are generated in terms of the well-behaved Hermitian polynomial solutions. Like the conventional SUSY quantum mechanics, the symmetry of the fermionic state is always opposite from the symmetry of the bosonic state. Thus, if for any state the boson has symmetric function, then the fermionic state is an anti-symmetric function and vice versa.