The inverse spectral problem for quantum semitoric systems

Abstract

Given a quantum semitoric system composed of pseudodifferential operators, Berezin-Toeplitz operators, or a combination of both, we obtain explicit formulas for recovering, from the semiclassical asymptotics of the joint spectrum, all symplectic invariants of the underlying classical semitoric system. Our formulas are based on the possibility to obtain good quantum numbers for joint eigenvalues from the bare data of the joint spectrum. In the spectral region corresponding to regular values of the momentum map, the algorithms developed by Dauge, Hall and the second author [27] produce such labellings. In our proof, it was crucial to extend these algorithms to the boundary of the spectrum, which led to the new notion of asymptotic half-lattices, and to globalize the resulting labellings. Using the construction given by Pelayo and the second author in [79], our results prove that semitoric systems are completely spectrally determined in an algorithmic way~: from the joint spectrum of a quantum semitoric system one can construct a representative of the isomorphism class of the underlying classical semitoric system. In particular, this recovers the uniqueness result obtained by Pelayo and the authors in [62,61], and completes it with the explicit computation of all invariants, including the twisting index. In the cases of the spin-oscillator and the coupled angular momenta, we implement the algorithms and illustrate numerically the computation of the invariants from the joint spectrum.

Figures (32)

• Figure 1: A few representatives of the polygonal invariant for the spin-oscillator system. The polygons in the second row are obtained from those in the first row by applying the global transformation $T^{-1}$ defined in \ref{['equ:matriceT']} followed by the vertical translation by $0-1$. The polygons in the second column are obtained from those in the first column by changing the cut direction from upwards to downwards, see \ref{['equ:epsilon_etoile']}.
• Figure 2: Projection of the Hamiltonian flows of $H_r$ (in blue) and $H = J + \varepsilon H_r$ (in red) in the $(x_1,x_2)$-plane, for $\varepsilon = 0.01$. In other words, in this example $f_r(x,y) = (x,\frac{x-y}{\varepsilon})$. The projection of the flow of $H$ is the map $(t,x_1,x_2) \mapsto e^{(i+\varepsilon)t} (x_1 + i x_2)$, while the projection of the flow of $H_r$ is the map $(t,x_1,x_2) \mapsto e^t (x_1 + i x_2)$, see for instance san-focus.
• Figure 3: The privileged polygon for the spin-oscillator system.
• Figure 4: The blue dots are the joint eigenvalues of the spin-oscillator system in the region $-1 \leq x \leq 2$ for $k = 15$. The red line corresponds to the boundary of the image of the momentum map, and the black circle indicates the position of the focus-focus value.
• Figure 5: A trivial asymptotic lattice.

Theorems & Definitions (95)

• Theorem 1.1: Theorem \ref{['theo:main']}
• Corollary 1.2
• Theorem 2.1: Eliasson normal form eliasson-these
• Example 2.3
• Definition 2.4
• Definition 2.5: san-alvaro-I
• Example 2.6
• Definition 2.7
• Example 2.8
• Proposition 2.9: san-semi-global,san-daniele