* [Li-Yau 1983\_Article](https://link.springer.com/content/pdf/10.1007/BF01213210.pdf)
* The second part of above paper used the discrete eigenvalues of $\frac{-\Delta}{q}$ where $q>0$ to proof the the number of non-positive eigenvalues of
Schrödinger operator $-\Delta+V$ can be bounded by the $L\_{\frac{n}{2}}$-norm of $V^-$.
* My question is: under what condition of $q$ can we proof the spectrum of $\frac{-\Delta}{q}$ is discrete.



On the Schrödinger equation and the eigenvalue problem
Assuming $q>0$ the Schroedinger operator $-\Delta/q$ is associated to the form $a(u,v)=\int\_{\mathbb R^n} \nabla u \cdot \nabla v$ in $L^2(\mathbb R^n, q\, dx)$. The form domain consists of all $u\in L^2(\mathbb R^n, q\, dx)$ such that $u \in \dot H^1:=\{u \in L^{2^\*}(\mathbb R^n), \nabla u \in L^2(\mathbb R^n)\} $ and the discreteness of the spectrum is equivalent to the compactness of the embedding of the form domain into $L^2(\mathbb R^n, q\, dx)$.


This follows when the map $$T:\dot H^1 \to L^2(\mathbb R^n), \quad Tu=q^{1/2}u$$ is compact, which is true whenever $q \in L^{n/2}$.


