Global SL(2,R) representations of the Schrödinger equation with time-dependent potentials.

Date

2012-05

Authors

Franco, Jose A.

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Worldwide access.
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Abstract

We study the representation theory of the solution space of the one-dimensional Schrödinger equation with time-dependent potentials that possess sl₂-symmetry. We give explicit local intertwining maps to multiplier representations and show that the study of the solution space for potentials of the form V (t, x) = g₂(t)x²+g₁(t)x+g₀ (t) reduces to the study of the potential free case. We also show that the study of the time-dependent potentials of the form V (t, x) = λx⁻² + g₂(t)x² + g₀(t) reduces to the study of the potential V (t, x) = λx⁻². Therefore, we study the representation theory associated to solutions of the Schrödinger equation with this potential only. The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category.

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Keywords

Representation theory.

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