Open Problems in Integrable Systems — Quantum integrability for polynomial in momenta integrals

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(M,g)$ be a smooth Riemannian manifold, with geodesic Hamiltonian $K(x,p)=\tfrac12g^{ij}(x)p_ip_j$ on $T^*M$ and its canonical Poisson bracket. A polynomial geodesic first integral is a smooth function polynomial in the cotangent variables, satisfying $\{K,F\}=0$. Its homogeneous pieces are themselves first integrals. The Laplace--Beltrami operator is $\Delta_g=g^{ij}\nabla_i\nabla_j$. A homogeneous integral of degree $d$ is quantizable if there is a differential operator $\widehat F$ of order $d$, with principal symbol $F$ and arbitrary allowed lower-order corrections, such that \[[\Delta_g,\widehat F]=\Delta_g\widehat F-\widehat F\Delta_g=0.\] Quantization of a general polynomial integral means quantization of its homogeneous components; self-adjointness is not imposed. The principal symbol is the top-order homogeneous polynomial in the differential operator's cotangent variables.

Statement recorded in the supplied source.

Find necessary and/or sufficient conditions on $g$ ensuring that every polynomial first integral of its geodesic flow is quantizable in this sense, without restricting the integral degree or the dimension of $M$. [S2]

Short English statement

Which metrics allow every polynomial integral of geodesic motion to become an exactly commuting differential operator with the same principal symbol?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100710 (AMR-020-0710). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 7.11 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.