The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data

Details The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data

2009-09-30

arxiv.org/abs/0909.5672v5

Mathematics

Functional Analysis

Scientific paper

We set-up and solve the Cauchy problem for Schr\"odinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of operators appeared in models of deep earth seismology. We prove existence and uniqueness of Colombeau generalized solutions and analyze the relations with classical and distributional solutions. Furthermore, we provide a construction of generalized initial values that may serve as square roots of arbitrary probability measures.

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