Point interactions for 3D sub-Laplacians

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Abstract

In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q0M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0(M{q0}) is essentially self-adjoint in L2(M). A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold N, whose associated Laplace-Beltrami operator acting on C0(N{q0}) is never essentially self-adjoint in L2(N), if dimN3. We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.

Introduction

Let (M,g) be a Riemannian manifold endowed with a smooth volume ω (one can think, e.g., of the Riemannian volume). The associated Laplace operator is the operator on L2(M,ω) acting on C0(M) and defined by Δω=divω. Here, C0(M) is the space of compactly supported smooth functions on M, and divω denotes the divergence w.r.t. the measure ω and ∇ is the Riemannian gradient. A fundamental issue is the essential self-adjointness of Δω, i.e., whether it admits a unique self-adjoint extension in L2(M,ω). Indeed, the essential self-adjointness of Δω implies the well-posedness in L2(M,ω) of the Cauchy problems for the heat and Schrödinger equations, that read, respectively,{tϕ=Δωϕ,ϕ|t=0=ϕ0L2(M,ω),{itψ=Δωψ,ψ|t=0=ψ0L2(M,ω). Roughly speaking, when Δω is not essentially self-adjoint, the above Cauchy problems are not well-defined without additional requirements, as for instance boundary conditions on ∂M.

The self-adjointness of Δω is related to geometric properties of (M,g), as is evident from the following classical result.

Theorem 1.1

Let (M,g) be a Riemannian manifold that is complete as metric space, and let ω be any smooth volume on M. Then, Δω is essentially self-adjoint in L2(M,ω).

This result is due to Gaffney [18] when ω is the Riemannian volume. A simpler argument, which generalizes to arbitrary smooth measures, is given by Strichartz [34].

A simple way to obtain non-complete Riemannian manifolds from a given complete one (M,g), is by removing a point q0M. Considering Δω on M{q0} yields the pointed Laplace operator Δ˚ω. We have the following.

Theorem 1.2

Let (M,g) be a Riemannian manifold that is complete as metric space, and let ω be any smooth volume on M. Let Δ˚ω be the pointed Laplace operator at q0M. Then Δ˚ω is essentially self-adjoint in L2(M,ω) if and only if n4.

The above result for the Euclidean space endowed with the Lebesgue measure is a consequence of [29, Ex. 4, p. 160], while the case of Riemannian manifolds where ω is the Riemannian volume is treated in [12]. Similar arguments can be applied when ω is an arbitrary smooth volume.

Theorem 1.2 is relevant in physics. Indeed, in non-relativistic quantum mechanics, self-adjoint extensions of the pointed Laplace operator can be used to construct potentials concentrated at a point, the so-called point interactions, as, e.g.,{itψ=(Δω+αδq0)ψ,αR,ψ(0,q)=ψ0(q). Here, δq0 is a Dirac-like potential representing a point interaction. Dirac δ and δ are widely used in modeling of quantum systems, since Fermi's paper [13] up to contemporary applications [6], [1], [5]. In this language, Theorem 1.2 can be interpreted as the fact that point interactions do not occur in dimension 4 and higher or, equivalently, that single points are seen by Laplace operators only in dimension less or equal than 3.

In this paper we study the essential self-adjointness of sub-Laplacians, i.e., the generalization of the Riemannian Laplace operators to sub-Riemannian manifolds. Let us briefly introduce this setting. We refer to [2], [25] for a more detailed treatment.

A sub-Riemannian structure on a smooth manifold M is given by a family of smooth vector fields {X1,,Xm}Vec(M) satisfying the Hörmander condition. Namely, let D=span{X1,,Xm}, pose D1=D and recursively define Ds=Ds1+[D,Ds1], sN, s2. This defines the flag D1DsVec(M). Letting Dqs={X(q)|XDs}, s1, the Hörmander condition then amounts to the requirement that for any qM there exists r=r(q) such that Dqr=TqM. A sub-Riemannian manifold is then defined as the pair (M,{X1,,Xm}). With abuse of notation, we will sometimes denote it by M.

On a sub-Riemannian manifold the distance between two points q1,q2M is defined byd(q1,q2)=inf{01i=1mui(t)2dt|γ:[0,1]M,γ˙(t)=i=1mui(t)Xi(γ(t)),γ(0)=q0,γ(1)=q1,uiL1([0,1],R),i=1,,m}. Owing to the Rashevskii-Chow theorem [2], (M,d) is a metric space inducing on M its original topology. The set of vector fields {X1,,Xm} is called a generating frame and it is a generalization of Riemannian orthonormal frames. As for the latter, there are different choices of generating frames giving rise to the same metric space (M,d), which is the true intrinsic object. For an equivalent definition of sub-Riemannian manifold that does not employ generating frames, see, e.g., [2].

The above definition includes several geometric structures [2]. Indeed, letting k(q)=dim(Dq), it holds that:

  • If k()n, one obtains a Riemannian structure.

  • If k()k<n, one obtains a classical sub-Riemannian structure. In this case, we will identify DVec(M) with the vector distribution qMDqTM.

  • if k() is not constant, one obtains a so-called rank-varying sub-Riemannian structure. This includes what are usually called almost-Riemannian structures [4], [2].

Motivated by the above observations, we say that a sub-Riemannian structure is genuine if k(q)<n for all qM.

Remark 1.3

In the first two cases above, if k()m the family of (linearly independent) vector fields {X1,,Xm} is a global orthonormal frame for the (sub-)Riemannian structure. Observe that, due to topological restrictions, such a frame does not always exist. However, if k() is locally constant around q0M, there always exists a local orthonormal frame1 around q0.

In this paper a particular role is played by 3-dimensional structures.

Definition 1.4

Consider a genuine sub-Riemannian structure on a 3-dimensional manifold M. We say that qM is a contact point if Dq2=TqM. If every point of M is contact, we say that the structure is a 3-dimensional contact structure.

In other words, in the genuine 3-dimensional case, a contact point is a point in which the full tangent space is generated by the vector fields X1,,Xm and their first Lie brackets. Since M is 3-dimensional, contact points coincide with what in the literature are called regular points.

Let {X1,,Xm} be a generating frame for the sub-Riemannian structure on M. Given a smooth volume ω the associated sub-Laplacian acting on C0(M) is defined as Δω=divω where divω is computed with respect to the volume ω and ∇ is the sub-Riemannian gradient, whose expression isϕ=i=1mXi(ϕ)Xi,ϕC(M). Such an operator is intrinsic in the sense that it does not depend on the particular choice of generating frame. We have then,Δω=i=1mXi2+(divωXi)Xi. Notice the presence of the “sum of squares” of the vector fields of the generating frame plus some first order terms guaranteeing the symmetry of Δω w.r.t. the volume ω.

As a consequence of Hörmander condition, Δω is hypoelliptic [22], and we have the following generalization of Theorem 1.1.

Theorem 1.5 Strichartz, [33]

Let M be a sub-Riemannian manifold that is complete as a metric space, and let ω be any smooth volume on M. Then, Δω is essentially self-adjoint on L2(M,ω).

The main object of interest in this paper is the pointed sub-Laplacian Δ˚ω at a point q0M. Similarly to the Riemannian case, this is defined as the sub-Laplacian Δω on M{q0}.

One of the main features of sub-Riemannian manifolds, is the existence of several natural notions of dimension. Although for Riemannian manifolds these are all coinciding, this is not the case in genuine sub-Riemannian manifolds. For instance, in the case of a classical sub-Riemannian manifold, some relevant dimensions are:

  • the dimension of the space of admissible velocities k,

  • the topological dimension n,

  • the Hausdorff dimension Q of the metric space (M,d),

where k<n<Q, see [24].2 It is then a natural question to understand which of these dimensions are relevant for essential self-adjointness of the pointed sub-Laplacian. In particular, since in a 3D contact sub-Riemannian manifold we have k=2, n=3, Q=4, in view of Theorem 1.2, we focus on pointed sub-Laplacians at contact points of genuine 3D sub-Riemannian manifolds. For these structures we prove the following.

Theorem 1.6

Let M be a genuine 3-dimensional sub-Riemannian manifold that is complete as metric space, and let ω be any smooth volume on M. Let q0M be a contact point, and Δ˚ω be the pointed sub-Laplacian at q0. Then Δ˚ω, with domain C0(M{q0}), is essentially self-adjoint in L2(M,ω).

The above result follows from Theorem 5.1, and shows that, regarding the essential self-adjointness of pointed sub-Laplacians, 3D sub-Riemannian manifolds behave like Riemannian manifolds of dimension at least 4. This suggests that the relevant dimension for self-adjointness is not the topological one, and that a more suitable candidate seems to be the Hausdorff dimension.

A crucial step in establishing Theorem 1.6 is the following corresponding result for the celebrated Heisenberg group H1.

Theorem 1.7

The operator (xy2z)2+(x+x2z)2 on C0(R3{(0,0,0)}) is essentially self-adjoint in L2(R3,dxdydz).

When q0 is not a contact point, or M is of dimension larger than 3, we conjecture that Theorem 1.6 still holds. However, our techniques are not easily extended to higher dimensions. In dimension 2, classical sub-Riemannian manifolds do not exist, while for rank varying structures we have two cases. Either the point q0 is Riemannian and then we can conclude that the pointed Laplace operator is not essentially self-adjoint; or q0 is not Riemannian and in this case we conjecture that the pointed Laplace operator is not essentially self-adjoint as well. However, the techniques necessary to study this case are very different from those developed in this paper and we do not treat this case here.

We now apply Theorem 1.6 to the Schrödinger evolution on SO(3) of a thin molecule rotating around its center of mass, described as follows. Consider a rod-shaped molecule of mass m>0, radius r>0, and length >0, as in Fig. 1. We denote by z the principal axis of the rod, and by x and y two orthogonal ones. Then, the moments of inertia of the molecule areIx=Iy=I:=m3r2+212,Iz=mr22. Letting (ωx,ωy,ωz) be the angular velocity of the molecule and Lx=Iωx, Ly=Iωy, Lz=Izωz be the corresponding angular momenta, the classical Hamiltonian isH=12(Iωx2+Iωy2+Izωz2)=12(Lx2I+Ly2I+Lz2Iz). Letting r0, while keeping and m constant, we have that Iz0, and the classical Hamiltonian readsHthin=12I(Lx2+Ly2). The corresponding Schrödinger equation isiħdψdt=Hˆthinψ,whereHˆthin=12I(Lˆx2+Lˆy2). Here, Lˆx, Lˆy, (and Lˆz) are the three angular momentum operators given by (in the following α,β,γ denote the Euler angles)Lˆx=iFx,Fx=cosαcotβα+sinαβcosαsinβγ,Lˆy=iFy,Fy=sinαcotβαcosαβsinαsinβγ,Lˆz=iFz,Fz=α. Since [Fx,Fy]=Fz we have that (SO(3),{Fx,Fy}) is a contact sub-Riemannian manifold. Moreover, being SO(3) unimodular, we have that Fx,Fy,Fz are divergence-free with respect to the Haar measure dh (see [3]) and we have that the corresponding sub-Laplacian isΔdh=Fx2+Fy2. It follows that Hˆthin=12IΔdh.

When considering the Schrödinger equation (1.9) on functions of (α,β,γ), we are describing the evolution of a thin molecule in which the thin degree of freedom (i.e., the angle α of the rod w.r.t. the z axis) is part of the configuration space. The essential self-adjointness of the pointed sub-Laplacian Δ˚dh on SO(3){(α0,β0,γ0)} given by Theorem 1.6 can be interpreted in the following way: A point interaction centered at (α0,β0,γ0) does not affect the evolution of a thin molecule.

Notice that this would not be the case if the molecule were not thin. Indeed in this case the quantum Hamiltonian would have been proportional to a left-invariant Riemannian Laplacian on SO(3), and by Theorem 1.2 the elimination of a point from the manifold crashes its essential self-adjointness.

Moreover, if the evolution of the thin molecule is considered on the 2D sphere instead than on SO(3), meaning that we are totally forgetting the thin degree of freedom, then the elimination of a point would break the essential self-adjointness of the Laplacian as well.

Sections 2 and 3 are devoted to preliminaries on the Heisenberg group and some of the functional analytic properties of sub-Riemannian manifolds, respectively. The remaining sections contain the proof of the main result of the paper, Theorem 5.1. This is obtained by first establishing Theorem 1.7 in Section 4, which is then extended to 3D genuine sub-Riemannian manifolds in Section 5.

More precisely, the proof of Theorem 1.7 consists in first reducing the problem of essential self-adjointness to the absence of L2 solutions of the equation (Δω+i)θ=φ, where φ is a linear combination of derivatives of the Dirac delta mass at 0, see Lemma 4.1. This criterion is then verified in Section 4.2 by exploiting the non-commutative Fourier transform associated with the Heisenberg group structure. Then, in Theorem 4.4, we localize the above result, showing that the self-adjoint extensions of the pointed sub-Laplacian at 0 defined on a domain ΩH1 coincide with those of the (standard) sub-Laplacian on the same domain. The latter result is then generalized to any 3D genuine sub-Riemannian manifold via local normal forms, in Section 5.

Finally, in Appendix A, we show how a criterion for essential self-adjointness based on an Hardy inequality with constant strictly bigger than 1, exploited e.g. in [17], [26], [28], fails for the Heisenberg group. This, in particular, raises a crucial criticism against the results contained in [35], and forces us to consider the above strategy of proof for Theorem 5.1.

Section snippets

The Heisenberg group H1

The Heisenberg group H1 is the nilpotent Lie group on R3 associated with the non-commutative group law(x,y,z)(x,y,z)=(x+x,y+y,z+z+12(xyxy)). The associated Haar measure, i.e., the only (up to multiplicative constant) left-invariant measure on H1, is the standard Lebesgue measure L3 of R3. One can check that H1 is unimodular, that is, this measure is also right-invariant.

A basis for the Lie algebra of left-invariant vector fields is given byXH(x,y,z)=xy2z,YH(x,y,z)=y+x2z,ZH=z.

Sub-Riemannian Sobolev spaces

Let M be sub-Riemannian manifold with local generating family {X1,,Xm}, endowed with a smooth and positive measure ω. We denote by L2(M,ω) (or L2(M)) the complex Hilbert space of (equivalence classes of) functions u:MC with scalar product(u,v)=Muv¯dω,u,vL2(M,ω), where the bar denotes the complex conjugation. The corresponding norm is denoted by uL2(M)2=(u,u). Similarly, L2(TM,ω) (or L2(TM)) is the complex Hilbert space of sections of the complexified tangent bundle X:MTMC, with scalar

Essential self-adjointness of the Heisenberg pointed sub-Laplacian

In this section we focus on the pointed sub-Laplacian in the Heisenberg group. We start by proving Theorem 1.7 via non-commutative harmonic analysis techniques. We then conclude the section by localizing this result in Theorem 4.4. That is, we show that the self adjoint extensions of the pointed sub-Laplacian on a domain ΩH1 coincide with those of the (standard) sub-Laplacian on the same domain.

Essential self-adjointness of 3D pointed sub-Laplacians

Let M be a 3-dimensional genuine sub-Riemannian manifold, endowed with a smooth and positive measure ω. Let pM be a regular point, and {X1,X2} be a local orthonormal frame for the sub-Riemannian structure in UM, pU. (See Remark 1.3.) By (1.5), we haveΔω=X12+X22+X0,whereX0=divω(X1)X1+divω(X2)X2.

The purpose of this section is to prove the following.

Theorem 5.1

The set of self-adjoint extensions of Δω with domain C0(M{p}) coincides with the one with domain C0(M).

We remark that, since when M is

Declaration of Competing Interest

There is no competing interest.

Acknowledgements

The authors are grateful to Alessandro Teta for suggesting reference [27], that led to the strategy of proof for the essential self-adjointness of the pointed Laplacian on the Heisenberg group.

The authors acknowledge the partial support of MIUR Grant Dipartimenti di Eccellenza (2018-2022) E11G18000350001, ANR-15-CE40-0018 project SRGI - Sub-Riemannian Geometry and Interactions, ANR-17-CE40-0007 project QUACO - Contrôle quantique: systèmes d'EDPs et applications à l'IRM, G.N.A.M.P.A. project

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