Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Point interactions for 3D sub-Laplacians
Introduction
Let 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 acting on and defined by . Here, is the space of compactly supported smooth functions on M, and denotes the divergence w.r.t. the measure ω and ∇ is the Riemannian gradient. A fundamental issue is the essential self-adjointness of , i.e., whether it admits a unique self-adjoint extension in . Indeed, the essential self-adjointness of implies the well-posedness in of the Cauchy problems for the heat and Schrödinger equations, that read, respectively, Roughly speaking, when 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 is related to geometric properties of , as is evident from the following classical result. Theorem 1.1 Let be a Riemannian manifold that is complete as metric space, and let ω be any smooth volume on M. Then, is essentially self-adjoint in .
A simple way to obtain non-complete Riemannian manifolds from a given complete one , is by removing a point . Considering on yields the pointed Laplace operator . We have the following. Theorem 1.2 Let be a Riemannian manifold that is complete as metric space, and let ω be any smooth volume on M. Let be the pointed Laplace operator at . Then is essentially self-adjoint in if and only if .
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., Here, is a Dirac-like potential representing a point interaction. Dirac δ and 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 satisfying the Hörmander condition. Namely, let , pose and recursively define , , . This defines the flag . Letting , , the Hörmander condition then amounts to the requirement that for any there exists such that . A sub-Riemannian manifold is then defined as the pair . With abuse of notation, we will sometimes denote it by M.
On a sub-Riemannian manifold the distance between two points is defined by Owing to the Rashevskii-Chow theorem [2], is a metric space inducing on M its original topology. The set of vector fields 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 , 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 , it holds that:
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If , one obtains a Riemannian structure.
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If , one obtains a classical sub-Riemannian structure. In this case, we will identify with the vector distribution .
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if is not constant, one obtains a so-called rank-varying sub-Riemannian structure. This includes what are usually called almost-Riemannian structures [4], [2].
Remark 1.3 In the first two cases above, if the family of (linearly independent) vector fields 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 is locally constant around , there always exists a local orthonormal frame1 around .
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 is a contact point if . If every point of M is contact, we say that the structure is a 3-dimensional contact structure.
Let be a generating frame for the sub-Riemannian structure on M. Given a smooth volume ω the associated sub-Laplacian acting on is defined as where is computed with respect to the volume ω and ∇ is the sub-Riemannian gradient, whose expression is Such an operator is intrinsic in the sense that it does not depend on the particular choice of generating frame. We have then, 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 w.r.t. the volume ω.
As a consequence of Hörmander condition, 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, is essentially self-adjoint on .
The main object of interest in this paper is the pointed sub-Laplacian at a point . Similarly to the Riemannian case, this is defined as the sub-Laplacian on .
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:
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the dimension of the space of admissible velocities k,
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the topological dimension n,
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the Hausdorff dimension Q of the metric space ,
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 be a contact point, and be the pointed sub-Laplacian at . Then , with domain , is essentially self-adjoint in .
A crucial step in establishing Theorem 1.6 is the following corresponding result for the celebrated Heisenberg group . Theorem 1.7 The operator on is essentially self-adjoint in .
When 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 is Riemannian and then we can conclude that the pointed Laplace operator is not essentially self-adjoint; or 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 of a thin molecule rotating around its center of mass, described as follows. Consider a rod-shaped molecule of mass , radius , and length , 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 are Letting be the angular velocity of the molecule and , , be the corresponding angular momenta, the classical Hamiltonian is Letting , while keeping ℓ and m constant, we have that , and the classical Hamiltonian reads The corresponding Schrödinger equation is Here, , , (and ) are the three angular momentum operators given by (in the following denote the Euler angles) Since we have that is a contact sub-Riemannian manifold. Moreover, being unimodular, we have that are divergence-free with respect to the Haar measure dh (see [3]) and we have that the corresponding sub-Laplacian is It follows that .
When considering the Schrödinger equation (1.9) on functions of , 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 on given by Theorem 1.6 can be interpreted in the following way: A point interaction centered at 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 , 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 , 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 solutions of the equation , 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 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
The Heisenberg group is the nilpotent Lie group on associated with the non-commutative group law The associated Haar measure, i.e., the only (up to multiplicative constant) left-invariant measure on , is the standard Lebesgue measure of . One can check that is unimodular, that is, this measure is also right-invariant.
A basis for the Lie algebra of left-invariant vector fields is given by
Sub-Riemannian Sobolev spaces
Let M be sub-Riemannian manifold with local generating family , endowed with a smooth and positive measure ω. We denote by (or ) the complex Hilbert space of (equivalence classes of) functions with scalar product where the bar denotes the complex conjugation. The corresponding norm is denoted by . Similarly, (or ) is the complex Hilbert space of sections of the complexified tangent bundle , 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 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 be a regular point, and be a local orthonormal frame for the sub-Riemannian structure in , . (See Remark 1.3.) By (1.5), we have
The purpose of this section is to prove the following. Theorem 5.1 The set of self-adjoint extensions of with domain coincides with the one with domain .
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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