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Generalized function solution of step potentials

Schrödinger equation is solved for step function–like potentials within the class of generalized functions. The solution functions carry the same symmetry as the potential. It is shown that the ground state energy is always equal to the minimum value of the potential. Finite square well potential has eigenfunctions both for continuous and discrete spectrum, they form a complete set, and the discrete spectrum does not depend on the well depth. A theorem on the completeness property of transformation operator is proved.