Global representations of the Heat and Schrödinger equation with singular potential

Abstract

We study the $n$-dimensional Schrödinger equation with singular potential $V_{\lambda }(x)=\lambda |x{|}^{-2}$. Its solution space is studied as a global representation of $\widetilde{SL(2,\mathbb{R})}\times O(n)$. A special subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. The space of $K$-finite vectors is calculated, obtaining conditions for $\lambda$ so that this space is non-empty. The direct sum of solution spaces, over such admissible values of $\lambda$ is studied as a representation of the $2n+1$-dimensional Heisenberg group.

References in corpus (2)

• The singular inverse square potential, limit cycles and self-adjoint extensions
• Singular inverse square potential in arbitrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string background

Cited by in corpus (2)

• On the space of $K$-finite solutions to intertwining differential operators
• Global Actions of the Lie Symmetries of the Nonlinear Filtration Equation