Suppose $\alpha_s, s\in [0,1]$ is a smooth family of quadratic forms (regarded as symmetric matrices) on $\mathbb R^n$. Consider an operator: $$ d_\alpha:= \frac{d}{ds} + \alpha_s: W^{1,2}(\mathbb R, \mathbb R^n) \to L^2(\mathbb R, \mathbb R^n) $$ Denote by $\nu_s$ the negative eigenspace of the symmetric matrix $\alpha_s$. Then the reference I read suggests to extend $d_\alpha$ by adding the linear span of $e^{-s\lambda}v$ (with $\lambda$ negative eigenvalue of $\alpha_1$ and $v$ corresponding eigenvectors. That is, it gives an operator $$ \tilde d_\alpha: W^{1,2}(\mathbb R, \mathbb R^n)\oplus \nu_{\alpha_1} \to L^2(\mathbb R, \mathbb R^n) $$ My questions are about the following claims(without details in my book):
- $\tilde d_\alpha$ is surjective
- $\tilde d_\alpha$ has kernel, identified with $\nu_{\alpha_0}$
- $\dim \nu_{\alpha_0} = \mathrm{index}(d_\alpha) + \dim \nu_{\alpha_1}$
Also, the way of the extending operator looks strange, do you know its motivation? Thanks