# Resonances for Schrodinger operators with radial potentials

Let $$V\in L^{\infty}(\mathbb{R}^3)$$ be a radial, compactly supported potential, and consider the Schrodinger operator $$H:=-\Delta + V$$ on $$L^2(\mathbb{R}^3)$$. Let $$\psi$$ be a resonance for $$H$$, i.e. a function $$\psi\in L^2(\mathbb{R}^3,\langle x\rangle^{-1-\varepsilon}dx)\setminus L^2(\mathbb{R}^3)$$ which satisfies $$(-\Delta + V)\psi=0$$.

Is it true that $$\psi$$ is radial? If not, is it at least true that the orthogonal projection of $$\psi$$ into the space of radial functions is a resonance?

• $\psi$ will factor into a radial function times an angular dependence, but it is not solely a function of the radial coordinate. – Carlo Beenakker Apr 1 at 11:20
• Ok thanks, but if I take the projection of $\psi$ into the space of radial function at least I get a radial function $\psi_r$ that solves $(-\Delta+V)\psi_r=0$. The point is to understand wheter we actually have $\psi_r\not\in L^2$. – Capublanca Apr 1 at 11:27
• you require $\psi\in L^2$, doesn't this imply $\psi_r\in L^2$? – Carlo Beenakker Apr 1 at 12:02
• No, I require $\psi\not\in L^2$. – Capublanca Apr 1 at 12:10

Expand your resonance in spherical harmonics: $$\psi = \sum_{\ell=0}^\infty \sum_m \psi_{\ell m}(r) Y_{\ell m}(\theta,\phi)$$. Then each coefficient satisfies the radial Schrödinger equation $$-\frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial \psi_{\ell m}}{\partial r} + \frac{\ell(\ell+1)}{r^2}\psi_{\ell m} + V(r) \psi_{\ell m} = 0 .$$ Since $$V(r)$$ is compactly supported, for sufficiently large $$r$$, you must have $$\psi_{\ell m} = A r^\ell + B r^{-\ell-1}$$. Your asymptotic condition forces $$A=0$$ for all $$\ell$$. On the other hand, the asymptotics of the remaining term mean that, for $$\ell \ge 1$$, $$\psi_{\ell m}(r) Y_{\ell m}(\theta,\phi)$$ will be in $$L^2(\mathbb{R}^3)$$. So the component of $$\psi$$ orthogonal to radial functions is necessarily in $$L^2(\mathbb{R}^3)$$.
Only the $$\ell=0$$ case escapes $$L^2(\mathbb{R}^3)$$. And of course, when $$\psi_{\ell=0}$$ is non-vanishing, it is $$O(r^{-1})$$ at infinity and hence a resonance by your definition. So the answer to your second question is Yes, but if there are any solutions at higher $$\ell$$ (they will be normalizable eigenfunctions), then mixing them with an $$\ell=0$$ resonance will give you non-radial resonances.