## Abstract

We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non-trivial way. These systems depend on a parameter \(t \in [0,1]\) and exhibit different behaviors according to its value. For a certain range of values, the system is semitoric, and we compute some of its symplectic invariants. Even though these invariants have been known for almost a decade, this is to our knowledge the first example of their computation in the case of a non-toric semitoric system on a compact manifold. (The only invariant of toric systems is the image of the momentum map.) In the second part of the paper, we quantize this system, compute its joint spectrum and describe how to use this joint spectrum to recover information about the symplectic invariants.

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## Acknowledgements

YLF was supported by the European Research Council Advanced Grant 338809. Part of this work was done during a visit of YLF to AP at University of California San Diego in April 2016, and he would like to thank the members of this university for their hospitality. AP is supported by NSF CAREER Grant DMS-1518420. He also received support from Severo Ochoa Program (Grant No. Severo Ochoa Group Grant Sev-2011-0087) at ICMAT in Spain. Part of this work was carried out at ICMAT and Universidad Complutense de Madrid. We thank Joseph Palmer for useful comments, and Jaume Alonso and Holger Dullin for pointing out a mistake in the computation of the coefficient \(a_1\) in the Taylor series invariant (Proposition 3.12) in an earlier version of this manuscript. We would also like to thank two anonymous referees for their very useful comments.

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## Appendices

### Appendix A: Proofs of Technical Results

### Proof of Lemma 3.3

A standard computation shows that the symplectic form becomes

Moreover, we deduce from Eq. (7) that

Since *J* is real-valued, \(\frac{\partial J}{\partial {\bar{z}}} = \overline{\frac{\partial J}{\partial z}}\) and \(\frac{\partial J}{\partial {\bar{w}}} = \overline{\frac{\partial J}{\partial w}}\), and we obtain the desired result. Furthermore,

which yields, after simplification

A similar computation shows that

and we conclude by using the same argument as above. \(\square \)

### Proof of Lemma 3.4

Let \(\lambda _1(z,w), \lambda _2(z,w)\) be as in the previous lemma, and let \((z,w) \in \Lambda _0 \setminus \{(0,0)\}\). Since \(z \ne 0\), we get, by multiplying both the numerator and the denominator in \(\lambda _1\) by \({\bar{z}}\), that

But the second equation in (8) yields \({\bar{z}} {\bar{w}} = - zw - |w|^2 + |z|^2 |w|^2\), hence

The first equation in (8) allows us to further simplify this expression and to obtain

Now, since \(w \ne 0\), we have that

The second equation in (8) gives

and thus, using the first one again, we finally obtain that \( \lambda _2(z,w) = -\frac{w^2}{R_1 {\bar{z}}}\). \(\square \)

### Proof of Proposition 3.7

We start by diagonalizing *A* over \({\mathbb {C}} \). With our choice of parameters, it follows from Eq. (3) that

*A* has eigenvalues \(\lambda _1 = \frac{3+i}{10}, {\bar{\lambda }}_1, \lambda _2 = - \lambda _1, {\bar{\lambda }}_2\) with respective eigenvectors \(X_1, \overline{X_1}, X_2, \overline{X_2}\) where

Let \(E_{\lambda }\) be the eigenspace associated with \(\lambda \), and let \(F = E_{\lambda _1} \oplus E_{{\bar{\lambda }}_1}\) and \(G = E_{\lambda _2} \oplus E_{{\bar{\lambda }}_2}\); then \(T_{m_0}M = F \oplus G\) and a real basis of *F* is given by

There exists a unique basis \((Z_1,Z_2)\) of *G* such that \((Y_1,Y_2,Z_1,Z_2)\) is a symplectic basis of \(T_{m_0}M\) [see for instance (Meyer et al. 2009, Lemma 3.2.3)]; in the latter, *A* will take the form displayed in Eq. (12). Since

there exists \(a,b,c,d \in {\mathbb {R}} \) such that

Recall that \(\omega _{m_0} = - \mathrm{d}x_1 \wedge \mathrm{d}y_1 + \frac{5}{2} \mathrm{d}x_2 \wedge \mathrm{d}y_2\). Hence, we need to solve the system

We find \(a = \frac{1}{60}, b = \frac{1}{20}, c = \frac{1}{20}\) and \(d = -\frac{1}{60}\), hence

Consequently, the matrix *P* of change of basis from the basis \({\mathcal {B}}\) associated with \((x_1,y_1,x_2,y_2)\) to the basis \((Y_1,Y_2,Z_1,Z_2)\) satisfies

which yields the desired expression for the coordinates \(u_1,u_2,\xi _1,\xi _2\). Moreover, \(P^{-1}AP\) is as in Eq. (12), with \(\alpha = \frac{-3}{10}\) and \(\beta =\frac{1}{10}\); this gives

This is not satisfactory since we want the lower right coefficient in this matrix to be positive. In order to obtain a *B* satisfying this requirement, it suffices to perform the symplectic change of coordinates \((u_1,u_2,\xi _1,\xi _2) \mapsto (-\xi _1,-\xi _2, u_1, u_2)\). \(\square \)

### Appendix B: Critical Points of Corank One

We explain how to prove the claim about the critical points of corank one of \(F=(J,H_t)\) in the proof of Corollary 2.11, namely that they are non-degenerate for every \(t \in ]0,1]\). (The case \(t=0\) is clear since the system is toric up to vertical scaling.) It suffices to prove that for every *E* in the image of *J*, except the ones corresponding to critical points of corank two (which are the only ones for which \(dJ = 0\) for this system), the critical points of the restriction of \(H_t\) to the symplectic quotient \(M^{\text {red}}_E = J^{-1}(E) / S^1\) with respect to the action generated by *J* are non-degenerate, see Toth and Zelditch (2003, Definition 3) for a general explanation or Hohloch and Palmer (2018, Corollary 2.5) for the particular case of dimension 4. Coming back to our particular case, let \(E \in (-(R_1 + R_2), R_1 + R_2) \setminus \{ R_1 - R_2, R_2 - R_1 \}\); since the poles do not give rise to critical points on \(J^{-1}(E)\), we may work with cylindrical coordinates as in Sect. 3.5 [or as in Hohloch and Palmer (2018, Section 3.3)] where a similar computation is performed)

In these coordinates, \(H_t\) reads

Since \(z_2\) can be deduced from \(z_1\) on \(J^{-1}(E)\), namely

and since the action of *J* preserves the angle \(\theta = \theta _1 - \theta _2\), we can use \((z_1, \theta )\) as coordinates on \(M^{\text {red}}_E\):

The first derivatives of \(H_t\) read

Hence if \((z_1^*,\theta ^*)\) is a critical point, then necessarily \(\theta ^* \in \{0, \pi \}\), and \(\frac{\partial ^2 H_t}{\partial \theta \partial z_1}(z_1^*,\theta ^*) = 0\). Let \(\varepsilon = \cos \theta ^* \in \{-1,1\}\); then one readily checks that

We claim that this last quantity has the sign of \(-\varepsilon \); this follows from the fact that

for any \(E, z_1\) satisfying the above bounds. In order to prove this, one may check that *Q* is minimal at \((E,z_1) = (0,0)\); since \(Q(0,0) = -4 \left( R_2 + \frac{R_1^2}{R_2} \right) \) and since the function \(x > 0 \mapsto x + \frac{R_1^2}{x}\) is minimal at \(x = R_1\) with value \(2 R_1\), we obtain the desired result because \(R_2 > R_1\).

In fact, this analysis gives us the sign of the determinant of the Hessian of \(H_t\) at a critical point, so we can deduce from it that the corank one critical points are of elliptic–transverse type. Hence, if one is only interested in proving this, and not in finding a parametrization of the boundary of the image of the momentum map, this appendix constitutes a faster way to obtain Corollary 2.11.

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Le Floch, Y., Pelayo, Á. Symplectic Geometry and Spectral Properties of Classical and Quantum Coupled Angular Momenta.
*J Nonlinear Sci* **29**, 655–708 (2019). https://doi.org/10.1007/s00332-018-9501-y

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DOI: https://doi.org/10.1007/s00332-018-9501-y