Spectral Estimates, Contractions and Hypercontractivity

Spectral Estimates, Contractions and Hypercontractivity

Emanuel Milman1
Abstract

Sharp comparison theorems are derived for all eigenvalues of the weighted Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to -exponential measures, unit-balls of , one-dimensional spaces and Riemannian submersions. Our main tool is a general Contraction Principle for “eigenvalues” on arbitrary metric-measure spaces. Motivated by Caffarelli’s Contraction Theorem, we put forth several conjectures pertaining to the existence of contractions from the canonical sphere (and Gaussian space) to weighted-manifolds of appropriate topological type having (generalized) Ricci curvature positively bounded below; these conjectures are consistent with all known isoperimetric, heat-kernel and Sobolev-type properties of such spaces, and would imply sharp conjectural spectral estimates. While we do not resolve these conjectures for the individual eigenvalues, we verify their Weyl asymptotic distribution in the compact and non-compact settings, obtain non-asymptotic estimates using the Cwikel–Lieb–Rozenblum inequality, and estimate the trace of the associated heat-kernel assuming that the associated heat semi-group is hypercontractive. As a side note, an interesting trichotomy for the heat-kernel is obtained.

11footnotetext: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel. Supported by ISF (grant no. 900/10), BSF (grant no. 2010288) and Marie-Curie Actions (grant no. PCIG10-GA-2011-304066). The research leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 637851). Email: emilman@tx.technion.ac.il.

1 Introduction

A weighted-manifold is a triplet , where is a complete smooth -dimensional Riemannian manifold, endowed with a measure having smooth positive density with respect to the Riemannian volume measure . The manifold can be compact or non-compact, but for simplicity we assume it is without boundary. In addition, there is no restriction on the total mass of the measure . The associated weighted Laplacian is defined as:

so that the usual integration by parts formula is satisfied with respect to :

Here denotes the Levi-Civita connection, denotes the usual Laplace-Beltrami operator, and we use . One immediately sees that is a symmetric and positive semi-definite linear operator on with dense domain , the space of compactly supported smooth functions on . In fact, it is well-known (e.g. [8, Proposition 3.2.1]) that the completeness of ensures that is essentially self-adjoint on the latter domain, and so its graph-closure is its unique self-adjoint extension. We continue to denote the resulting positive semi-definite self-adjoint operator by , with corresponding domain . By the spectral theory of self-adjoint operators (see Subsection 3.2), the spectrum is a subset of . When the spectrum is discrete (such as for compact manifolds), it is composed of isolated eigenvalues of finite multiplicity which increase to infinity; we denote these by , and arrange them in non-decreasing order (repeated by multiplicity) . In the discrete case, when is connected and has finite mass, we always have . For the standard definition of when the spectrum is possibly non-discrete (as the first eigenvalues until the bottom of the essential spectrum), we refer to Subsection 3.2. In this work, we would like to investigate the spectrum of various classes of weighted-manifolds.

Definition.

The weighted-manifold satisfies the Curvature-Dimension condition , and , if:

(1.1)

as symmetric 2-tensors on . Here denotes the Ricci curvature tensor, and is called the generalized (infinite-dimensional) Ricci tensor. In this work, we restrict the Curvature-Dimension condition to connected manifolds.

The generalized Ricci tensor (1.1) was introduced by Lichnerowicz [56, 57], and extended to arbitrary generalized dimension by Bakry [5] (cf. Lott [61]). Note that in the constant density case , the generalized Ricci tensor boils down to the classical one. The Curvature-Dimension condition was introduced by Bakry and Émery in equivalent form in [7] (in the more abstract framework of diffusion generators). Its name stems from the fact that the generalized Ricci tensor incorporates information on curvature and dimension from both the geometry of and the measure , and so may be thought of as a generalized-curvature lower bound, and as a generalized-dimension upper bound. With the exception of this Introduction, we will always assume that whenever referring to the Curvature-Dimension condition, so we omit the more general definition involving an arbitrary (cf. [69]). The condition has been an object of extensive study over the last two decades (see e.g. also [73, 51, 31, 32, 85, 11, 91, 70, 69, 47, 8, 45, 67] and the references therein), especially since Perelman’s work on the Poincaré Conjecture [72], and the extension of the Curvature-Dimension condition to the metric-measure space setting by Lott–Sturm–Villani [79, 62].

1.1 Spectrum Comparison for Positively Curved Weighted-Manifolds

Let denote the -dimensional Gaussian probability measure with covariance , namely , where is a normalization constant. When , we simply write for the standard -dimensional Gaussian measure. It is well known [52, 8] that the one-dimensional Gaussian space serves as a model comparison space for numerous functional inequalities (such as isoperimetric [9], log-Sobolev [7] and spectral-gap [8]), for the class of connected weighted-manifolds satisfying with (“positively curved weighted-manifolds”). The starting point of this work was to explore the possibility that these classical comparison properties also extend to all higher-order eigenvalues of . Contrary to many functional inequalities, which remain invariant under tensorization, thus implying that the comparison space may be chosen to be one-dimensional, the spectrum tensorization property naturally forces us to compare to the -dimensional space having the same (topological) dimension. Note that positively curved weighted-manifolds always have discrete spectrum, since they necessarily satisfy a log-Sobolev inequality by the Bakry–Émery criterion [7], and the latter is known to imply (in our finite-dimensional setting!) discreteness of spectrum (see e.g. [63, 88, 30]). In addition, if is positively curved then necessarily has finite total-mass [8, Theorem 3.2.7].

Question 1 (Spectral Comparison Question).

Given an -dimensional connected weighted-manifold satisfying with , does it hold that:

At first, this question may seem extremely bold and at the same time classical and well-studied. As for the latter impression, we are not aware of any previous instances of Question 1. The former impression perhaps stems from the extensive body of work in trying to just provide sharp lower and upper bounds on the first eigenvalue gap under various conditions (e.g. [42, 2, 4, 12, 3]), or various other conjectured lower bounds on the entire spectrum, such as Polya’s conjecture (see e.g. [28, 54, 48]).

Unfortunately, one cannot expect to have a positive answer to the above question in general, at least not for the first eigenvalues. The easiest counterexample is given by the canonical -sphere, rescaled to have Ricci curvature equal to (times the metric), so that it satisfies ; its -th eigenvalue (given by a linear function on the sphere’s canonical embedding in ) is equal to , whereas the corresponding eigenvalue for the -dimensional Gaussian space is already equal to (see Subsection 2.2 for more details).

Nevertheless, we can show:

Theorem 1.1 (Spectral Comparison for Positively Curved ).

Question 1 has a positive answer for any Euclidean space satisfying with .

In view of the above counterexample and theorem, and for reasons which will become more apparent later on, it is plausible that some topological restrictions must be enforced to obtain a positive answer to Question 1. The simplest one is to assume that is diffeomorphic to Euclidean space. We tentatively formulate this as:

Conjecture (Spectral Comparison Conjecture for Positively Curved ).

Question 1 has a positive answer for any satisfying with .

See Section 6 for a more refined version of this conjecture. Clearly, the case boils down to Theorem 1.1, so the conjecture pertains to the range . We take this opportunity to also mention the work of Ledoux [50] (cf. Bakry and Bentaleb [6]), who showed how information on higher order iterated carré-du-champ operators (so called operators) may be used to obtain higher-order eigenvalue estimates for the generator ; however, here we only assume the condition, which amounts to information on only (see [8] for more on -Calculus).

1.2 Spectrum Comparison for Additional Spaces

Our method of proof of Theorem 1.1, described in the next subsection, is very general, and in particular also yields the following additional results:

Theorem 1.2.

Let denote a Euclidean weighted-manifold where is a probability measure satisfying . Then:

Theorem 1.3.

Let denote a probability measure on for . Let denote a second probability measure on , and assume that is convex and unconditional, meaning that . Then:

Theorem 1.4.

Assume , and let denote the unit ball of , rescaled to have volume 1; the uniform measure on is denoted by . Then:

Theorem 1.5.

Given a weighted-manifold with a probability measure, denote its density by , by its cumulative distribution function, and by its one-sided flat isoperimetric profile. Let denote two such measures. Then:

Theorem 1.6.

Let denote two weighted-manifolds, and let denote a Riemannian submersion pushing forward onto up to a finite constant. Then:

In particular, this holds for , the corresponding Riemannian volume measures, if both manifolds are connected and the submersion’s fibers are minimal and compact.
In particular, this holds for any finite-sheeted Riemannian covering map between two connected manifolds.

We refer to Subsection 4.4 for missing definitions. The “in particular” part of Theorem 1.6 is certainly not new (at least when the manifolds are compact, see e.g. [23, Section 3]). The eigenvalues of in Theorem 1.4 are with respect to Neumann boundary conditions.

1.3 Contracting and Lipschitz Maps

Let denote a Borel map between two metric-measure spaces. The map is said to push-forward the probability measure onto , denoted , if . To treat the case when may have different or infinite total mass, we will say that pushes forward onto up to a finite constant, if there exists so that pushes forward onto . The map is called -Lipschitz () if:

The map is called a contraction if it is Lipschitz with constant .

All of our spectrum comparison theorems are consequences of the following:

Theorem 1.7 (Contraction Principle).

Let denote an -Lipschitz map between two (complete) weighted-manifolds pushing-forward onto up to a finite constant. Then:

In particular, if has discrete spectrum, then so does .

In fact, an analogous result holds for compact weighted-manifolds with either Dirichlet or Neumann boundary conditions, see Subsection 3.4. We note that even in the classical non-weighted setting, the contraction principle is easily seen to be completely false if we omit the assumption that pushes forward the first volume measure onto the second (up to a finite constant); moreover, in that case, even if is known to be bi-Lipschitz, the resulting spectrum comparison would depend exponentially on the underlying dimension , which is often useless for applications.

While the derivation of Theorem 1.7 is straightforward, we have not encountered an application of contracting maps for spectrum comparison elsewhere. To see Theorem 1.7, it is easy to verify that on we have:

(1.2)

where and denote the push-forward and pull-back maps between and induced by . Theorem 1.7 subsequently follows by the min-max principle and a density argument. A slightly delicate point is that we do not assume injectivity of (which is useful for some of the applications above), and so the min-max argument should be carefully checked. To better appreciate the above stated comparison, the reader may wish to try and explicitly write out and compare the differential operators appearing in (1.2) using the change-of-variables formula relating , and . In Section 3, we develop an abstract argument for spectrum comparison in the general framework of metric-measure spaces.

A few words are in order regarding previous approaches towards spectrum comparison between differential operators on Riemannian manifolds (and more generally, linear operators on Hilbert spaces). The closest general argument we have found in the literature is the so-called Kato’s inequality and its generalizations (see [41, 78, 15, 16, 14, 24, 22, 23] and the references therein), which under certain conditions permit comparing the trace of the associated heat semi-groups, heat-kernels, and even the heat semi-group and resolvent operators themselves in the sense of domination of positivity preserving operators. However, these results typically do not involve the individual eigenvalues (cf. [16, III.6]), and in the few cases that do, the conclusion is in the opposite direction to the one appearing in this work (in an attempt to obtain spectral lower bounds on the source manifold by mapping it onto a simpler one). We also mention two additional classical methods of obtaining estimates on the growth and number of negative eigenvalues of a Schrödinger operator – the Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities [60, 33, 58, 75] (see also [48] and the references therein), the latter of which we will in fact employ in this work as well (see Subsection 2.5).

Back to the Contraction Principle. A celebrated contraction property was discovered by L. Caffarelli in [27]:

Theorem (Caffarelli’s Contraction Theorem).

Let satisfy with . Then there exists a map pushing forward onto up to a finite constant which contracts Euclidean distance.

Together with the Contraction Principle, this immediately yields Theorem 1.1. Caffarelli proved the above result for the Brenier Optimal-Transport map [84], which uniquely (up to a null-set deformation) minimizes the -averaged transport distance among all maps pushing forward onto . Subsequently in [44], Young-Heon Kim and the author gave an alternative proof and extended Caffarelli’s theorem using a (seemingly) different map involving a naturally associated heat-flow, which together with the Contraction Principle immediately yields Theorem 1.3. Similarly, the existence of contracting and Lipschitz maps due to Kolesnikov [46], Latała–Wojtaszczyk [49] and Bobkov–Houdré [20] yield Theorems 1.2, 1.4 and 1.5, respectively; details are provided in Section 4.

Contracting, and more generally, Lipschitz maps between metric-measure spaces, constitute a very powerful tool for transferring isoperimetric, functional and concentration information from to . However, for these traditional applications, there are numerous other tools available, such as -Calculus, other parabolic and elliptic -methods, Optimal-Transport, Localization, etc.. (see e.g. [8, 47, 45]). As shown in this work, contracting maps also yield sharp comparison estimates for the entire spectrum, going well beyond the capability of the above mentioned alternative methods - we believe this to be a noteworthy (albeit simple) observation.

Motivated by Caffarelli’s Contraction Theorem on one hand, and the well-known comparison results between weighted-manifolds satisfying and the ( or equivalently -dimensional) Gaussian measure () on the other, we tentatively put forth the following conjecture, which by the Contraction Principle, would imply Conjecture :

Conjecture (Contraction Conjecture for Positively Curved ).

For any satisfying with , there exists a map:

pushing forward onto up to a finite constant and contracting the corresponding metrics.

See Section 6 for a more refined version of this conjecture. Conjecture is consistent with the Bakry-Ledoux isoperimetric comparison theorem [9] and the Bakry-Émery log-Sobolev inequality [7] for weighted-manifolds. Note that we have restricted the above conjecture to manifolds diffeomorphic to , as the counterexample of the canonical sphere from Subsection 1.1 shows that one cannot hope for such a map unto a general weighted-manifold satisfying . Moreover, there are topological obstructions to the existence of such a map between and , at least if we assume in addition that is one-to-one from the source onto the target manifold: indeed, Brouwer’s Invariance of Domain theorem [71] asserts that an injective, surjective and continuous map between two topological manifolds is in fact open, and hence the two manifolds must be homeomorphic.

For a further discussion and refinement of Conjectures and , we refer to Section 6.

1.4 Extensions to Positively Curved Constant-Density Manifolds

It is of course very natural to attempt extending the previous conjectures to the class of weighted-manifolds satisfying for and finite generalized dimension . Contrary to the situation with the usual functional inequalities (isoperimetric, Sobolev, spectral-gap, cf. [10, 69]), it is not so clear what would be the right (topologically -dimensional) model space for comparing the entire spectrum. However, when , which corresponds to the classical case of a complete connected Riemannian manifold, endowed with its canonical Riemannian volume measure and having Ricci curvature bounded below by (times the metric), the natural model space is simply the canonical -sphere with its metric rescaled to have . For similar topological reasons as in the previous subsection (see also the ensuing discussion), we restrict to the case when is diffeomorphic to a sphere.

Conjecture 3 (Spectral Comparison Conjecture For Positively Curved ).

For any satisfying with , we have:

Conjecture 3 is consistent with:

  • The Lichnerowicz spectral-gap estimate [55].

  • The Bérard–Gallot estimate on the trace of the heat-kernel [13]:

    (1.3)
  • It is immediate to show that it is compatible with Weyl’s asymptotic law – see Subsection 2.3.

  • We can actually show that it holds true up to a dimension independent multiplicative constant for – see Subsection 2.5.

Conjecture 3 would follow immediately from the Contraction Principle and the following previously unpublished conjecture of ours [65]:

Conjecture 4 (Contraction Conjecture for Positively Curved ).

For any satisfying with , there exists a map:

pushing forward onto up to a finite constant and contracting the corresponding metrics.

Note that a connected complete Riemannian manifold with , , is necessarily compact and has finite volume. The reader should note the apparent analogy between the latter conjecture and Caffarelli’s Contraction Theorem, in view of the definition of the generalized Ricci tensor (1.1). Conjecture 4 is consistent with:

  • The Bonnet–Meyers bound on the diameter of such manifolds [36]:

  • The Bishop–Gromov volume estimate [36]:

    which in particular implies (letting ) .

  • The Bakry–Émery log-Sobolev estimate [7] (see Section 5).

  • The sharp Sobolev inequality for spaces [8, Theorem 6.8.3].

  • The Gromov–Lévy isoperimetric inequality [38].

  • Conjecture 3 on the full spectrum, including all of the known consequences mentioned after its formulation above.

A positive answer to Conjecture 4 would thus yield a single reason to all of these classical facts (albeit only for manifolds which are diffeomorphic to a sphere). It would be very interesting to adapt and extend the Optimal-Transport or Heat-Flow approaches of Caffarelli [27] and Kim and the author [44] from the scalar setting (involving densities) to the above -tensorial setting (involving metrics) - cf. [65].

As before, we have restricted Conjecture 4 to due to potential topological obstructions. Indeed, a map as in Conjecture 4 must be surjective, since is compact as a continuous image of a compact set, while its open complement satisfies , and hence must be empty. Consequently, if we assume in addition that is injective, Brouwer’s Invariance of Domain theorem would imply as before that is open, and hence must be homeomorphic to .

For simplicity, we have chosen not to explicitly formulate the most general possible conjectures in the above spirit. Let us only remark that if we do not insist on finding a topologically -dimensional model source space which conjecturally contracts onto -dimensional weighted-manifolds (), thereby giving up on obtaining asymptotically sharp eigenvalue estimates (per Weyl’s law) and on injectivity of the contracting map, then a reasonable choice for such a model source space, at least when is an integer, is the rescaled canonical -sphere; this would still be consistent with all known generalizations of the above properties (see [8, 79, 69] and the references therein), and contrary to the counterexample of Subsection 1.1, is easily verified for . It is also possible to consider adding the case to the above setting (under suitable modifications, replacing with ), but we do not have a clear sense of how reasonable this might be.

1.5 Comparison on Average

While we were not able to resolve Conjectures nor 3, we would still like to mention some tools for controlling the eigenvalues in some averaged sense. In Subsection 2.3, we recall Weyl’s asymptotic law for the distribution of eigenvalues in the compact case, and develop its analog in the weighted non-compact setting. However, we would like to obtain some concrete non-asymptotic estimates as well.

In Subsection 2.5, we show that Conjecture 3 is satisfied up to a dimension independent multiplicative constant for exponentially large (in the dimension) eigenvalues, by making use of the classical Cwikel–Lieb–Rozenblum inequality together with the sharp Sobolev inequality on weighted manifolds. We did not manage to verify a similar conclusion for Conjecture , perhaps because the condition does not directly feel the dimension . We therefore proceed to obtain some average estimates for the eigenvalues.

When is a probability measure, a very natural function encapsulating the growth of the eigenvalues is given by the trace of the heat semi-group :

where denotes the heat-kernel (with respect to ). It is an interesting question to establish conditions on which ensure that is trace-class, i.e. that for . In particular, upper bounds on yield lower bounds on the individual eigenvalues by the trivial estimate:

(1.4)

However, it may very well happen that the spectrum is discrete (equivalently, that increase to infinity), and yet for all . Note that will inevitably depend on the dimension , e.g. because of Weyl’s law or because of the spectrum’s tensorization property - see Section 2 for concrete examples such as for the -dimensional Gaussian space or sphere. This is in contrast to more traditional objects of study on weighted-manifolds (such as the spectral-gap or log-Sobolev constant), which are invariant under tensorization, and thus often dimension-independent.

In connection to the discussion regarding previously known estimates on the spectrum, we mention the following result of Bérard and Gallot [13, 14] (see also Besson [16, Appendix]). By employing the Gromov–Lévy isoperimetric inequality [38], these authors showed that for any connected with , , one has:

where denotes renormalized to be a probability measure. In particular, this yields (1.3), confirming Conjecture 3 in an averaged (yet strictly weaker) sense. By employing the Bakry–Ledoux isoperimetric inequality [9], it may also be possible to obtain a somewhat similar on-average confirmation of Conjecture ; this is not immediate and will be explored elsewhere. Here, we are more interested in another direction.

Upper bounds on and moreover lower bounds on under various assumptions on were obtain by F.-Y. Wang in [87, 89] (we refer to the excellent book [8] and to Section 5 for subsequent missing references and terminology, which we only mention here in passing). Whenever the space satisfies a Sobolev inequality (or equivalently a Nash inequality, or finite-dimensional log-Sobolev inequality [8, Chapter 6]), and in particular under a condition for and finite , it is well-known that is ultracontractive [8, Corollary 6.3.3.], i.e. that the heat-kernel is bounded, yielding a trivial upper-bound on . This ultracontractive case has been extensively studied in the literature, see e.g. [34, 8, 87]. The borderline case when some additional information is needed is precisely when is only hypercontractive, i.e. when satisfies a log-Sobolev inequality. In that case, and even under a weaker super-Poincaré (or -Sobolev) inequality, assuming in addition that the space satisfies for some , Wang obtained very general lower bounds on depending on concentration properties of the distance function to a given point .

In Section 5, we expand on the quantitative relation between hypercontractivity of the heat semi-group, the property of being trace-class (i.e. upper estimates on ), and higher-order integrability properties of the associated heat-kernel, both in general and under a condition, . Our approach closely follows Wang’s method, based on his dimension-free Harnack inequality. Our results are weaker and less general than Wang’s, but the proofs are a bit simplified, yielding estimates with concrete dimension-dependence. Finally, a general interesting trichotomy for the heat-kernel is deduced. In Section 6, we provide some concluding remarks.

Acknowledgement. I thank Franck Barthe, Mikhail Gromov, Tobias Hartnick and Michel Ledoux for their comments.

2 Eigenvalue Calculation and Asymptotics

We begin with calculating the eigenvalues or their asymptotic distribution for several notable weighted-manifolds.

2.1 Gaussian Space

It is well known that the one-dimensional Gaussian Space has simple spectrum at (so that each of the eigenvalues has multiplicity one), with the eigenfunctions of being precisely the Hermite polynomials. By the tensorization property of the spectrum, it follows that the product space has spectrum , where the sum is repeated times and is counted with multiplicity. In other words, the spectrum consists of and the multiplicity of the eigenvalue is given by . It follows that the eigenvalue counting function satisfies for :

and consequently:

(2.1)

Furthermore, we record that as we have:

(2.2)

2.2 Canonical Sphere

Let denote the -sphere with its canonical metric and volume measure, embedded as the unit-sphere in Euclidean space ; its Ricci curvature is equal to . It is well known that the eigenfunctions of the associated Laplacian are given by spherical-harmonics, i.e. the restriction of harmonic homogeneous polynomials in onto . The eigenvalue associated to harmonic polynomials of degree is [83]. Since it is well known that any homogeneous polynomial of degree can be uniquely decomposed into its harmonic components as follows:

we see that the subspace spanned by spherical harmonics of even degree at most or of odd degree at most is of dimension . Consequently, the subspace of all spherical harmonics of degree at most is of dimension , with corresponding eigenvalues being at most .

Now let us rescale the canonical sphere to have radius , so that its Ricci curvature coincides with the metric and therefore satisfies . Since the eigenvalues scale quadratically in the metric, the eigenvalue counting function consequently satisfies for all :

One could hope that the counting function of the rescaled -sphere is always dominated by that of the -dimensional Gaussian:

However, this is not the case for the first eigenvalues, and is most apparent for , i.e. linear functions on the sphere. Indeed, for all , on the rescaled sphere is equal to the eigenvalue of the last among its linear functionals, i.e. to , whereas on Gaussian space it is already equal to . This show that in general, one cannot hope for a positive answer to Question 1.

For future reference, we record that the unscaled canonical sphere satisfies for all :

(2.3)

2.3 Weyl’s asymptotic law for weighted-manifolds

When is compact, then as soon as the density of is bounded away from and , the classical Weyl law [28] for the eigenvalue asymptotics of the unweighted Laplacian applies to the weighted one , and we have as :

(2.4)

Note that by Bishop’s volume comparison theorem [36], for any (connected) with with , and so we see that Weyl’s formula (2.4) confirms Conjecture 3 in an asymptotic sense (as ):

(2.5)

When is non-compact, the situation is more delicate. As we have not found an explicit reference in the literature, we derive the asymptotics ourselves from the known results for Schrödinger operators, and for simplicity, we restrict to the Euclidean case .

Recall that . Denote by the isometric isomorphism given by the multiplication operator . Conjugating by , we obtain the Schrödinger operator given by:

with domain . This well-known procedure is a form of Doob’s h-transform (e.g. [8, Section 1.15.8],[29, Section IV]). Since and are unitarily equivalent they are both self-adjoint on their respective domains and have the same spectrum. Clearly and hence . Since it is well known (e.g. [43]) that a Schrödinger operator is essentially self-adjoint on as soon as is in and bounded from below, it follows that in such a case its unique self-adjoint extension necessarily coincides with the one described above having domain . Consequently, we may apply the known Weyl formula for eigenvalue asymptotics of self-adjoint Schrödinger operators (e.g. [40, 18, 77]), which asserts that under suitable regularity assumptions:

(2.6)

where denotes the following phase-space level set of the operator’s symbol:

More precisely (see e.g. [40, Theorem 6]), (2.6) holds under the assumptions that:

  1. is smooth and bounded below.

  2. for some .

  3. For some we have .

Remark 2.1.

It is frequently assumed in the study of Schrödinger operators that in order to obtain a positive semi-definite operator, and this is also the standing assumption in [40]. However, if is only assumed bounded below, we can simply consider where is a constant so that ; the resulting shift in the spectrum is immaterial for the asymptotic distribution of eigenvalues, thereby justifying our slightly extended assumptions above. The assumption that is bounded below also ensures that is essentially self-adjoint on , as explained above.

Note that in typical situations (e.g. as in the next subsection):

(2.7)

Assuming w.l.o.g. that the minimum of is attained at the origin, it follows that if with , then and hence . In that case, (2.6) implies that as :

in asymptotic accordance with Theorem 1.1. An extension of this reasoning to the manifold setting would similarly asymptotically confirm Conjecture , but we do not pursue the details here.

For future reference, it will be more convenient to rewrite (2.6) as:

(2.8)

2.4 Asymptotics for the measures

Let us now calculate the asymptotic distribution of eigenvalues for the product measures , , which appear in various places in this work. We exclude the case of the exponential measure since does not have discrete spectrum (this will also be apparent from the ensuing calculations). Fixing , we have:

and:

An application of Hölder’s inequality verifies that outside the compact set when , and so it is clearly bounded below in that case. We now address the unboundedness of from below when , in tandem with the minor nuisance that and are not smooth on the coordinate hyperplanes (for any non-even ). Indeed, we may always approximate by smooth functions so that:

and so that in addition, when , , whereas when , is bounded above uniformly in . The min-max principle (recalled in Section 3) will immediately ensure that this results in a perturbation in the spectrum by a multiplicative factor of at most , which can be made arbitrarily close to . The above properties of ensure that:

uniformly in . It is then easy to see that the function has polynomial growth, and so the regularity and boundedness assumptions described above are satisfied, and the asymptotic distribution (2.6) is valid, uniformly in . As in (2.7), we conclude from (2.8) that:

where denotes the Beta function. Plugging in the known formulae for the volume of the ball and the Beta function, we finally obtain:

(2.9)

Note that for , as , this is in precise agreement with the calculation carried out for the Gaussian case (2.2). Also observe that as the function on the right-hand side explodes, in accordance with the formation of essential (non-discrete) spectrum in the limiting case . Finally observe that as , converges to the uniform measure on , and the right-hand side converges to:

in precise accordance with the classical Weyl estimate (2.4).

2.5 Cwikel–Lieb–Rozenblum inequality

Let be a connected manifold satisfying with . We have already seen in Subsection 2.3 that Weyl’s law confirms Conjecture 3 in an asymptotic sense, but without delving into its proof, it is not possible to extract from it non-asymptotic estimates on the individual eigenvalues.

However, individual (loose) estimates may be obtained from the Bérard–Gallot heat-kernel estimate mentioned in the Introduction ([13, 14], [16, Appendix]):

which confirms Conjecture 3 in an averaged sense. In particular, this may be used to prove Conjecture 3 up to a multiplicative constant (in fact, without restricting to manifolds diffeomorphic to the sphere).

In this subsection, we mention yet another method for obtaining (non-sharp but quite good) explicit estimates on the individual eigenvalues, by means of the Cwikel–Lieb–Rozenblum inequality [33, 58, 75] (see also [59, 76, 48]) for the number of negative eigenvalues of the Schrödinger operator , established independently and by different means by these three authors. Lieb’s approach [59] relied on the ultracontractivity of the associated semi-group, whereas Li and Yau provided in [54] yet another proof based on the Sobolev inequality; these two assumptions were shown to be equivalent by Varopoulos [82] (see also [8]). The CLR inequality has since been generalized to a very abstract setting, and we employ the following version by Levin and Solomyak from [53]:

Theorem (Generalized CLR inequality, Levin–Solomyak).

Let denote a -finite measure space. Let denote a self-adjoint operator on