Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum $-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Does there exist reasonably convenient criteria on $V$ that can guarantee that the eigenvalues are all simple (i.e., $\lambda_1<\lambda_2<\cdots$)?

If we instead consider $H$ on an bounded or half-line interval $[a,b]$ (i.e, at least $a$ or $b$ finite), with Dirichlet boundary conditions, then there well-known (and relatively simple) results guaranteeing multiplicity-one spectra under very general conditions. However, I can't seem to find any information on the full-space case.

Consider a Schrödinger operator $H:=-\Delta+V$ on $\mathbb R$, where $V$ is such that $H$ has a purely discrete spectrum $-\infty<\lambda_1\leq\lambda_2\leq\cdots$ converging to $+\infty$. Do there exist reasonably convenient criteria on $V$ that can guarantee that the eigenvalues are all simple (i.e., $\lambda_1<\lambda_2<\cdots$)?

If we instead consider $H$ on an bounded or half-line interval $[a,b]$ (i.e, at least $a$ or $b$ finite), with Dirichlet boundary conditions, then there are well-known (and relatively simple) results guaranteeing multiplicity-one spectra under very general conditions. However, I can't seem to find any information on the full-space case.