Non-existence of Radial Eigenfunctions for the Perturbed Heisenberg Sublaplacian

Abstract We prove uniform resolvent estimates in weighted $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -spaces for radial solutions of the sublaplacian $$\mathcal {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> on the Heisenberg group $$\mathbb {H}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . The proofs are based on the multipliers methods, and strongly rely on the use of suitable multipliers and of the associated Hardy inequalities. The constants in our inequalities are explicit and depend only on the dimension d . As application of the method, we obtain some suitable smallness and repulsivity conditions on a complex radial potential V on $$\mathbb {H}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> such that $$\mathcal {L}+V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> has no radial eigenfunctions.