# On the perturbation theory for spectra in quantum mechanics

### APA

Kontsevich, M. (2021). On the perturbation theory for spectra in quantum mechanics. Perimeter Institute. https://pirsa.org/21110011

### MLA

Kontsevich, Maxim. On the perturbation theory for spectra in quantum mechanics. Perimeter Institute, Nov. 12, 2021, https://pirsa.org/21110011

### BibTex

@misc{ pirsa_21110011, doi = {}, url = {https://pirsa.org/21110011}, author = {Kontsevich, Maxim}, keywords = {Mathematical physics}, language = {en}, title = {On the perturbation theory for spectra in quantum mechanics}, publisher = {Perimeter Institute}, year = {2021}, month = {nov}, note = {PIRSA:21110011 see, \url{https://pirsa.org}} }

## Abstract

Consider a polynomial differential operator in one variable, depending on a small parameter (Planck constant). Under appropriate conditions, the low-energy spectrum admits an asymptotic expansion in hbar.

I will present a way to calculate such a series via a purely "commutative problem", a mixture of variations of Hodge structures and of the Stirling formula. This result came from discussions with A.Soibelman. It seems that we obtain an explanation of an old observation by J.Zinn-Justin of the

universal appearance of Bernoulli numbers.