Consider the Beltrami-Laplacian $\Delta$ on $\mathbb{S}^n$ with standard metric. One can define a family of operators $A(z):H^1_0(\mathbb{S}^n)\to H^1_0(\mathbb{S}^n)$ as the following $$A(z)=\Delta+z$$ Then $A$ is holomorphic on $z$. From the Fredholm theory of elliptic operator, $A(z)$ is invertible only $z$ is not the eigenvalue of $-\Delta$ which are $k(k+n-1)$. However, it is also known that $A(z)^{-1}$ is actually meromorphic near these eigenvalues.

So my question is what is the order of the pole near each eigenvalue? Can we find Laurent series of it?

Consider the Beltrami-Laplacian $\Delta$ on $\mathbb{S}^n$ with standard metric. One can define a family of operators $A(z):H^1(\mathbb{S}^n)\to H^1(\mathbb{S}^n)$ as the following $$A(z)=\Delta+z$$ Then $A$ is holomorphic on $z$. From the Fredholm theory of elliptic operator, $A(z)$ is invertible only $z$ is not the eigenvalue of $-\Delta$ which are $k(k+n-1)$. However, it is also known that $A(z)^{-1}$ is actually meromorphic near these eigenvalues.

So my question is what is the order of the pole near each eigenvalue? Can we find Laurent series of it?