One can hear semitoric systems
After several years of work, we can finally claim: yes, one can
hear semitoric systems!
What does this mean? Semitoric systems are one of the most simple yet
extremely rich family of completely integrable systems with two
degrees of freedom. Essentially, they are those who possess a global
$S^1$ symmetry. The real definition is a bit more technical: one requires
non-degenerate singularities, connectedness of fibers of the
‘energy-momentum’ map, and properness of the Hamiltonian function
generating the $S^1$ action.
Now suppose you have a quantum version of these: two commuting
operators $(\hat J,\hat H)$ whose underlying classical symbols form a
semitoric system $F$ on some symplectic manifold $(M,ω)$. Suppose you
“observe” the quantum system like a spectroscopist observing a
molecule: you obtain the quantum spectrum; or, more precisely, the
joint quantum spectrum of the commuting operators $(\hat J,\hat
H)$. From this data, in the semiclassical limit $ħ\to 0$, can you
“compute” $(M,ω)$ and $F$?
In this paper with
Yohann Le Floch, we prove
that the answer is yes, in a strong way: you may extract from the
spectrum, in a constructive way, a complete set of symplectic
invariants. It remains to apply the classification result obtained
with Álvaro Pelayo' to
recover $(M,ω)$ and $F$ from these invariants.
In the special case of toric systems, the result was previously
with Laurent Charles
and Álvaro Pelayo: from the joint spectrum, one can recover, in the
limit $ħ\to 0$, the image of the momentum map, which is a Delzant
polytope, and Delzant’s theorem ensures that this is enough to
characterize the system.
For more general semitoric systems, there are many more classifying
invariants than just the momentum image. In the case of systems “of
Jaynes–Cummings type” (semitoric systems with one focus-focus
singularity) we previously proved with Álvaro and Yohann the
injectivity part: if two such systems have the same joint spectrum,
then the classical systems must be isomorphic, see .
The core of our new work is to find explicit formulas for computing
all classifying invariants.
The picture above illustrates how to recover, from the joint spectrum,
one of the trickiest invariants: the twisting index combined with
one of the linear terms of the Taylor series. The blue diamonds are
the quantities obtained from the spectrum, while the brown line is the
theoretical value for the particular system under study (spin-orbit
coupling on $S^2\times S^2$). Going to the right on this graph
corresponds to the semiclassical limit $\hbar\to 0$.
It would be very interesting, now, to understand whether and how this
inverse spectral problem could fail by relaxing some of the
hypothesis in the definition of semitoric systems…