Semiclassical spectral series of the Schrödinger operator with a delta potential on a straight line and on a sphere

We describe the spectral series of the Schrödinger operator $H=-(h^2/2) \Delta+V(x)+\alpha\delta(x-x_0)$, $\alpha\in\mathbb R$, with a delta potential on the real line and on the three- and two-dimensional standard spheres in the semiclassical limit as $h\to0$. We consider a smooth potential $V(x)$ such that $\lim_{|x|\to\infty}V(x)=+\infty$ in the first case and $V(x)=0$ in the last two cases. In the semiclassical limit in each case, we describe the classical trajectories corresponding to the quantum problem with a delta potential.