The Birman-Schwinger principle says that if $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and we consider the operator $H=-\Delta-V$ for a positive potential $V$, then, for any $\lambda>0$, the number of eigenvalues at most $-\lambda$ of this operator is the same as the number of eigenvalues at least 1 of the operator


$$ K\_\lambda=V^{1/2}(-\Delta+\lambda)^{-1}V^{1/2}.$$


Can you suggest any source where this result is proved for $\lambda=0$?


Many thanks!



Birman-Schwinger Principle
This runs into obvious technical issues. For example, $K\_0$ will not be bounded (let alone compact) even for very nice $V$. So one also has to think about what exactly one wants to prove.


[This paper](https://arxiv.org/abs/0911.2134) discusses these issues. In particular, the Birman-Schwinger principle for $\lambda=0$ is stated in equation (1.6).


