Analyticity of the Schrödinger propagator for the affine Laplacian and its applications to sampling and interpolating sequences

Abstract In this article, we consider the Schrödinger semigroup for the affine Laplacian. We discuss the analytic continuation of this semigroup, and characterize the image of a subspace of $$L^2(\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> under this semigroup as a weighted Bergman space. As an application, we characterize the sampling and the interpolating sequences in the newly obtained Bergman space. Consequently, we construct a new example of a frame in a subspace of $$L^2(\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .