Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3 (Q6603925)

This paper treats a natural sub-Riemannian Laplacian associated with the canonical contact structure, on the unit tangent bundle of a compact Riemannian surface. In the large eigenvalue limit, the escape of mass at infinity is studied in the cotangent space of eigenfunctions for hypoelliptic selfadjoint perturbations of this operator. Using semiclassical methods, it is shown that, in this subelliptic regime, eigenfunctions concentrate on certain quantized level sets along the geodesic flow direction and that they verify invariance properties involving both the geodesic vector field and the perturbation term. \N\NLet \((M, g)\) be a smooth, compact, oriented, and boundaryless Riemannian surface and denote by \(K(m)\) its sectional curvature at a given point \(m \in M\). The unit tangent bundle of \(M\) is defined by \N\[\N{\mathcal M} := SM = \{ q = (m, v) \in TM: \Vert v \Vert_{ g(m)} = 1 \}.\N\]\NLet \(X\) be the geodesic vector field on \(SM\), and \(V\) the vertical vector field, i.e. the vector field corresponding to the action by rotation in the fibers of \(SM\). When we define \(X_{\perp} := [ X, V]\), then the following commutation relations hold: \N\[ \N[ X, X_{\perp} ] = - K V, \quad [X, V ] = X_{\perp}, \quad \text{and}\quad [ X_{\perp}, V ] = - X,\N\]\Nwhere \(K\) is a function on \(SM\) (via pullback). The manifold \({\mathcal M}\) is naturally endowed with a Riemannian metric \(g_S\) (called the Sasaki metric) which makes \((X, X_{\perp}, V)\) into an orthonormal basis. The corresponding volume \(d \mu_L\) makes these three vector fields divergence free. Then the sub-Riemannian Laplacian associated with this geodesic frame can be defined as \N\[\N- \varDelta_{s R} := X_{\perp}^* X_* + V^* V = - X_{\perp}^2 - V^2.\N\]\NLet \(Q, W \in {\mathcal C}^{\infty} ({\mathcal M}, {\mathbb R})\). The goal of this article is to study, in the semiclassical limit \(h \to 0^+\), the eigenfunctions of the following formally selfadjoint operator: \N\[\N\hat{P}_h := - h^2 \varDelta_{s R} - i h^2 Q X - \frac{ i h^2 X(Q)}{2} + W, \quad h \in (0, 1 ].\N\]\NThe authors are interested in describing the asymptotic properties of the semiclassical eigenmodes satisfying: \N\[\N\hat{P}_h \psi_h = \Lambda_h \psi_h, \quad \Vert \psi_h \Vert_{ L^2} = 1, \quad \Lambda_h \to \Lambda_0 \in {\mathbb R} \quad (\text{as} \quad h \to 0^+). \N\]\NOne says that a probability measure \(\nu\) is a quantum limit for this spectral problem if, for every \(a \in {\mathcal C}^0 ({\mathcal M})\), \N\[\N\lim_{ h \to 0^+} \int_{ {\mathcal M} } a \vert \psi_h \vert^2 d \mu_L = \int_{ {\mathcal M}} a d \nu, \N\]\Nwhere \((\psi_n)_{ n \to 0^+}\) is a sequence satisfying (5). \par In view of describing the regularity properties of \(\nu\), one lifts the problem to the cotangent bundle \(T^* {\mathcal M}\) by introducing \N\[\Nw_h : {\mathcal C}_c^{\infty} (T^* {\mathcal M}) \ni a \mapsto \langle Op_h (a) \psi_h, \psi_h \rangle_{L^2}, \N\]\Nwhere \(Op_h (a)\) is a \(h\)-pseudodifferential operator with principal symbol \(a\) and \((\psi_h)_{ h \to a^+}\) is the sequence used to generate \(\nu\). Thanks to the Calderón-Vaillancourt theorem, \((w_h)_{ h \to 0^+}\) is a bounded sequence in \({\mathcal D}' (T^* {\mathcal M})\). Hence, up to extraction, it converges to some limit \(w\) which is referred as a semiclassical measure for the sequence \((\psi_h)_{ h \to 0^+}\). The theory of semiclassical pseudodifferential operators allows to prove that any such \(w\) is a finite nonnegative measure on \(T^* {\mathcal M}\) that is supported on \N\[\N{\mathcal E}^{-1} (\Lambda_0) := \{ (q, p) \in T^* {\mathcal M}: {\mathcal E} (q, p) := H_2(q,p)^2 + H_3(q,p)^2 + W(q) = \Lambda_0 \},\N\]\Nand that satisfies the following invariance property \(\{ H_2^2 + H_3^2 + W, w \}= 0\), where \N\[\NH_2(q,p) := p(X_{\perp}), \quad \text{and} \quad H_3(q,p) := p(V).\N\]\NWe emphasize that, contrary to the case of eigenvalue problems of elliptic nature, the energy layer \({\mathcal E}^{-1} (\Lambda_0)\) is not compact and there may be some escape of mass at infinity. In particular, \(w\) could be equal to 0. Due to this escape of mass at infinity, it is natural to study the measure \N\[\N\nu_{\infty} := \nu - \pi_* w,\N\]\Nwhere \(\pi : T^* {\mathcal M} \ni (q,p) \mapsto q \in {\mathcal M}\), and this is the main purpose of the present work. \N\N\textit{Y. C. De Verdière} et al. [Duke Math. J. 167, No. 1, 109--174 (2018; Zbl 1388.35137)] that \(X(\nu_{\infty}) = 0\) when \(Q \equiv 0\) and \(W \equiv 0\). The results of this work generalize the above theorem in two directions. First, the authors will provide a refined decomposition of \(\nu_{\infty}\), showing that the measure \(\nu_{\infty}\) decomposes into a discrete sum of non-negative Radon measures covering different asymptotic regimes \(h^{-1} \ll \vert X \vert \lesssim h^{-2}\) across the non-compact part of \({\mathcal E}^{-1} (\Lambda_0)\). Second, they will prove that each of these measures satisfies a new invariance property, different from each other, as soon as \(\nabla (Q)\) does not vanish. In view of formulating their results, they associate to each smooth function \(f\) on \({\mathcal M}\) a natural vector field lying in the contact plane \(D:=\mathrm{Span} (X_{\perp}, V)\) given by \(\Omega_f\) \(:=\) \(V(f) X_{\perp} - X_{\perp} (f) V\). Here is the main theorem of this paper:\N\N\textbf{Theorem.} Let \(Q, W \in {\mathcal C}^{\infty}({\mathcal M}, {\mathbb R})\) such that \(\Vert Q \Vert_{ {\mathcal C}^0} < 1\), let \(\Lambda_0 > \max_{ q \in {\mathcal M} } W(q)\) and set \N\[\NY_W := X + \Omega_{ ln (\lambda_0 - W)}.\N\]\NThen, for every \(\nu \in {\mathcal N}_{ \Lambda_0}\), the measure \(\nu_{\infty}\) decomposes as \N\[\N\nu_{\infty} = \bar{\nu}_{\infty} + \sum_{ k=0}^{\infty} (\nu_{k, \infty}^+ + \nu_{k, \infty}^-),\N\]\Nwhere \(\bar{\nu}_{\infty}\) and \(\nu_{k, \infty}^{\pm}\) are non-negative Radon measures on \({\mathcal M}\) verifying, for all \(a \in {\mathcal C}'({\mathcal M})\) and for all \(k \in {\mathbb Z}_+\), \N\[\N\int_{ {\mathcal M} } Y_W(a) d \bar{\nu}_{\infty} = 0 \quad \text{and} \quad \int_{ {\mathcal M} } Y_{W, Q, k}^{\pm} (a) d \nu_{k, \infty}^{\pm} = 0,\N\]\Nwith \(Y_{W, Q, k}^{\pm} := (\pm (2 k + 1) + Q) Y_W- \Omega_Q\). \par It is necessary to emphasize the importance for the above-mentioned theorem of considering sub-principal terms of the form \N\[\N- i h^2 Q X - i h^2 X(Q)/2\N\]\Nin the definition of \(\hat{P}_h\). Indeed, the resulting quantum limits \(\bar{\nu}_{\infty}\) and \(\nu_k^{\pm}\) each satisfy a different invariance property while, for \(Q \equiv 0\), only one invarince property for \(\nu_{\infty}\) occurs. \par Condition \(\Lambda_0 > \max_{ q \in {\mathcal M}} W(q)\) ensures that the classical forbidden region is empty. In the case \(\min W \leqslant \Lambda_0 \leqslant \max W\), the support of \(\bar{\nu}_{\infty}\) becomes confined inside the compact set \N\[\N{\mathcal M}_{ \Lambda_0, W} := \{ q \in {\mathcal M}: \quad \Lambda_0 - W \geq 0 \},\N\]\Nwhile the support of \(\nu_{k, \infty}^{\pm}\) is contained in the open subset \N\[\N{\mathcal U}_{ \Lambda_0, W} := \{ q \in {\mathcal M} : \Lambda_0 - W > 0 \}.\N\]\NThis more general situation is covered by the more precise description of semiclassical measure as well. Moreover, by working on the flat torus \(M = {\mathbb T}^2\), they show examples of sequences \((\psi_h, \Lambda_h)\) satisfying (5) for which the measures \(\bar{\nu}_{\infty}\) or \(\nu_{k, \infty}^{\pm}\) they construct carry the total mass of \(\nu\).