Are Born-Oppenheimer energies analytic functions of nuclear positions?

Question

I am looking for references to bibliography that explores the smoothness and analyticity of eigenvalues and eigenfunctions (and matrix elements in general) of a hamiltonian that depends on some parameter.

Consider, for example, the original setting of the Born-Oppenheimer approximation, to molecular dynamics, where the nuclear wavefunction is momentarily ignored and the hamiltonian becomes parametrized by the positions $\mathbf{R}_m$ of the nuclei, $$ \hat{H}(\mathbf{R}_m)=-\sum_{i=1}^N \frac{\hbar^2}{2m}\nabla^2_i+\sum_{i>j}\frac{e^2}{|\mathbf{r}_i-\mathbf{r}_j|}-\sum_{i,m}\frac{Z_m e^2}{|\mathbf{r}_i-\mathbf{R}_m|}. $$ The energies $E_n(\mathbf{R}_m)$ then become functions of all the nuclear coordinates and therefore make up the energy landscape which governs the nuclear wavefunctions' evolution. Since the original appearance of the $\mathbf{R}_m$ is in the analytic (well, meromorphic) functions $\frac{1}{|\mathbf{r}_i-\mathbf{R}_m|}$, I would expect further dependence on the $\mathbf{R}_m$ to be meromorphic (and would definitely expect physical meaning from poles and branch cuts).

What I am looking for is references to bibliography that will establish or disprove results of this type in as general a setting as possible. In particular, given a hamiltonian that depends on a set of parameters $z_1,\ldots,z_m$ in a suitably defined analytic way, I would like to see results establishing the analyticity of matrix elements (and thus, for example, of eigenvalues) involving the eigenvectors of the hamiltonian. I would also be interested in knowing what quantities can be extended analytically to the complex plane.

Any and all pointers will be deeply appreciated.

Answer 1

Suppose that for all $z$ in some open set $Z$ of complex numbers containing $z_0$, the Hamiltonian $H(z)$ is a compact perturbation of the self-adjoint $H(z_0)$ depending analytically on $z$. Then, for every simple eigenvalue $E_0$ of $H(z_0)$ and associated normalized eigenstate $\psi_0$, there exist a complex neighborhood $N$ of $z_0$ and unique functions $E(z)$ and $\psi(z)$, defined and analytic on $N$, such that $E(z_0)=E_0$, $\psi(z_0)=\psi_0$, and $H(z)\psi(z)=E(z)\psi(z)$ and $\psi_0^*\psi(z)=1$ for all $z\in N$.

The proof is essentially the inverse function theorem in a Banach space for the resulting nonlinear system, combined with the spectral theorem applied to $H(z_0)$. I guess you can find the relevant background results (if not a perturbation statement similar to the above) in the old book by Kato.

No assumption is needed that $H(z)$ is self-adjoint (would not be the case for all $z\in Z$ unless $H(z)$ is constant). Of course the eigenvalues will generally move into the complex domain if $z_0$ was real but $z$ is complex.

Weakening the assumptions will require stronger (so-called ''hard'') forms of the inverse function theorem, which generally take a lot of technicality to state and verify.