The semi-classical spectrum and the Birkhoff normal form

Abstract: The purposes of this note are: 1) to propose a direct and "elementary" proof of the main result proved by Guillemin-Paul-Uribe [GPU], namely that the semi-classical spectrum near a global minimum of the classical Hamiltonian determines the whole semi-classical Birkhoff normal form (denoted the BNF) in the non-resonant case. 2) to present in the completely resonant case a similar problem.

• The paper demonstrates that in non-resonant settings, the semi-classical spectrum uniquely determines the full Birkhoff normal form.
• It provides an explicit construction that maps spectral eigenvalues to quantum BNF coefficients using symplectic transformations and power series expansions.
• The study contrasts non-resonant and resonant cases, highlighting group-theoretic obstructions that challenge the uniqueness in spectral determination.

The Semi-Classical Spectrum and Birkhoff Normal Form: A Direct Approach

Introduction and Context

This work addresses the inverse spectral problem for semi-classical Hamiltonians in the vicinity of non-degenerate critical points, focusing particularly on the relationship between the semi-classical spectrum and the associated Birkhoff normal form (BNF). The primary contribution is an explicit, elementary proof that the semi-classical spectrum near a global minimum of a classical, non-resonant Hamiltonian completely determines the semi-classical BNF, and vice versa. The analysis also contrasts the non-resonant case with the resonant setting, where uniqueness properties of the normal form break down and group-theoretic obstructions emerge.

Main Results in the Non-Resonant Case

The note considers semi-classical Hamiltonians $H$ on $\mathbb{R}^d$, viewed as Weyl quantizations of classical symbols having a non-degenerate global minimum with incommensurate frequencies $\omega_j$. In this regime, after a symplectic transformation, the quadratic part $H_0$ at the minimum $z_0$ is diagonal, and all $\omega_j > 0$ are linearly independent over $\mathbb{Q}$.

The Birkhoff normal form of $H$ is uniquely characterized as a formal series involving quantized action variables and $\hbar$-dependent corrections. The spectrum of this normal form is labelled by multi-indices corresponding to quantum excitations along each coordinate direction.

The central theorem (Theorem 1.1) establishes that, under linear independence of the frequencies, the semi-classical spectrum modulo $\mathcal{O}(\hbar)$ and the series defining the semi-classical BNF determine one another. This result provides an explicit link between spectral data and microlocal invariants in the generic, non-resonant scenario.

Technical Approach

The proof strategy systematically constructs a bijection between spectral values and multi-indices arising from eigenvalues labelled in the BNF. The main difficulty—arising from the difference between the ordering of spectrum labels ($N \in \mathbb{N}$) and the BNF's natural lattice labelling ($k \in \mathbb{Z}_+^d$)—is resolved by leveraging the independence of frequencies, guaranteeing spectral non-degeneracy and a well-defined increasing sequence.

Explicitly, the semi-classical eigenvalues possess a power series in $\hbar$ with coefficients determined by the classical minimum, the first-order correction ($E_1$), and polynomial invariants derived from the BNF. These structure the spectrum via the map between quantum numbers and energy levels.

Two approaches are provided for reconstructing frequencies $\omega_j$ from spectral data:

• Via the partition function $Z(z)$ constructed from the eigenvalues, whose poles encode the set $\{\omega_j\}$, up to permutation.
• By algorithmic elimination, recursively identifying frequencies from the gaps in the spectrum.

Once the quantum numbers and frequencies are dictated by the spectrum, further expansion coefficients—invariants in the BNF—are also determined by analyzing higher-order terms in the spectral asymptotics.

The Resonant Case: Obstructions and Group Action

In the completely resonant case ($\omega_1 = \ldots = \omega_d$), the uniqueness of the BNF fails. The spectrum clusters, and the BNF is only defined up to automorphism of the semi-classical Weyl algebra commuting with the harmonic oscillator. The group $G$ controlling these automorphisms has non-trivial structure, including a symplectic linear part (the unitary group $U(d)$) and a pseudo-differential part described via Moyal brackets.

A fundamental open question is whether the equivalence class of the quantum BNF modulo $G$ can still be recovered from the semi-classical spectrum (i.e., the positions of spectral clusters) in the resonant setting. The non-uniqueness complicates the spectral determination of microlocal invariants, and resolving this would require a deeper analysis of the interplay between group action and spectral data.

Implications and Perspective

The results provide an explicit, constructive spectral characterization of quantum Hamiltonians in the non-resonant case, sharpening the correspondence between spectral invariants and Birkhoff normal form coefficients. This strengthens foundational links between semi-classical analysis, microlocal normal forms, and spectral theory. The techniques, while tailored for non-resonant quadratic minima, indicate broader applicability, as trace formula-based methods are expected to address more general (non-degenerate, non-resonant) critical points.

On the theoretical side, this work clarifies the nature of quantum spectral rigidity versus flexibility: in the generic setting, the spectral data are maximally constraining, while resonances admit additional symmetries obstructing uniqueness. Understanding the precise nature of these obstructions, especially regarding automorphism groups of quantum normal forms, remains an open frontier.

For practical and computational applications, these results imply that, for generic systems, numerically computed low-lying eigenvalues robustly encode the full semi-classical BNF, providing an algorithmic pathway to infer spectral invariants from quantum spectra. Advances in addressing the resonant case could yield important tools for inverse spectral problems in higher symmetry settings.

Conclusion

This work presents a direct proof that the semi-classical spectrum of a non-resonant quantum Hamiltonian near a global minimum uniquely determines, and is determined by, the full Birkhoff normal form. The approach provides practical algorithms for extracting all relevant invariants from spectral data. In resonant situations, uniqueness fails due to group-theoretic ambiguities, highlighting the need for further exploration of spectral determination modulo automorphism classes. These insights advance the understanding of semi-classical inverse spectral theory and inform both theoretical analysis and computational extraction of microlocal invariants from quantum spectra.