Self-adjoint quantization of St\"ackel integrable systems

TL;DR

This paper demonstrates that quadratic Hamiltonians from Stäckel systems can be quantized into self-adjoint operators, enabling separation of variables and confirming a previously conjectured property.

Contribution

It proves that Stäckel systems' quadratic Hamiltonians are quantizable into self-adjoint operators, confirming a specific conjecture.

Findings

Quadratic Hamiltonians from Stäckel systems are quantizable.

Quantization yields commuting self-adjoint operators.

Separation of variables is possible with these operators.

Abstract

We show that quadratic Hamiltonians in involution coming from a St\"ackel 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].