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Self-adjoint quantization of Stäckel integrable systems

Jonathan M Kress, Vladimir Matveev

TL;DR

The paper proves that quadratic Hamiltonians in involution arising from a Stäckel system admit a self-adjoint, commuting quantization with symbols equal to the classical Hamiltonians, using a weight $\phi=\det(S)$. It extends the construction to include potentials, showing that commuting quantum Hamiltonians exist precisely when a Benenti-type condition on the potentials holds, and demonstrates multiplicative separation of variables for joint eigenfunctions. This establishes the conjecture from [BMD16], generalizing known special cases and providing a broad framework for quantum integrable systems derived from Stäckel matrices. The results yield explicit, separable quantum models tied to the geometry of the Stäckel construction and Robertson-compatibility conditions.

Abstract

We show that quadratic Hamiltonians in involution coming from a Stäckel system are quantizable, in the sense that one can construct commutative self-adjoint operators whose symbols are the quadratic Hamiltonians. Moreover, they allow multiplicative separation of variables. This proves a conjecture explicitly formulated in [3].

Self-adjoint quantization of Stäckel integrable systems

TL;DR

The paper proves that quadratic Hamiltonians in involution arising from a Stäckel system admit a self-adjoint, commuting quantization with symbols equal to the classical Hamiltonians, using a weight . It extends the construction to include potentials, showing that commuting quantum Hamiltonians exist precisely when a Benenti-type condition on the potentials holds, and demonstrates multiplicative separation of variables for joint eigenfunctions. This establishes the conjecture from [BMD16], generalizing known special cases and providing a broad framework for quantum integrable systems derived from Stäckel matrices. The results yield explicit, separable quantum models tied to the geometry of the Stäckel construction and Robertson-compatibility conditions.

Abstract

We show that quadratic Hamiltonians in involution coming from a Stäckel system are quantizable, in the sense that one can construct commutative self-adjoint operators whose symbols are the quadratic Hamiltonians. Moreover, they allow multiplicative separation of variables. This proves a conjecture explicitly formulated in [3].

Paper Structure

This paper contains 4 sections, 3 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}.
  3. Proof of Theorem \ref{['thm:2']}.
  4. Proof of Theorem \ref{['thm:3']}.

Key Result

Theorem 2

For the Stäckel integrable system constructed by an arbitrary Stäckel matrix $S$ and for the function $\phi:= \det(S)$, the second order differential operators commute.

Theorems & Definitions (5)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Theorem 5