Scattering theory is the mathematical formalism of interactions in quantum mechanics. Quantum mechanics being the physical theory that appeared at the turn of the 20th century when it became apparent that not only did light behave as a wave, but matter as well. Further, the waves describing matter were quantized. A classic example of this is the photoelectric effect, where light can only interact with electrons when the light has the same quantized energy as the electron. In this work I will derive mathematically properties of the Schroedinger operator, an unbounded operator on Hilbert space and how its spectra can include both continuous and discrete components. The solutions to the Schroedinger equation are either normalizable or not depending on whether they have eigenvalues in the continuous or discrete spectra respectively. However, the solutions/eigenfunctions for the continuous or scattered spectra are only approximate eigenfunctions, in a sense we will make explicit. Their existence is shown through a rigorous treatment of the Schroedinger operator with no potential – the free state. Next scattering processes for some simple one dimensional cases will be shown, in which the time-independent solutions are related to time-dependent solutions. This will be followed by two dimensional scattering with localized and long range potentials.
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