On a class of anharmonic oscillators

In this work we study a class of anharmonic oscillators within the framework of the Weyl-H\"ormander calculus. The anharmonic oscillators arise from several applications in mathematical physics as natural extensions of the harmonic oscillator. A prototype is an operator on $\mathbb{R}^n$ of the form $(-\Delta)^{\ell}+|x|^{2k}$ for $k,\ell$ integers $\geq 1$. The simplest case corresponds to Hamiltonians of the form $|\xi|^2+|x|^{2k}$. Here by associating a H\"ormander metric $g$ to a given anharmonic oscillator we investigate several properties of the anharmonic oscillators. We obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers. We also study some examples of anharmonic oscillators arising from the analysis on Lie groups


Introduction
In this paper we show some spectral properties for anharmonic oscillators on R n . In the study of the Schrödinger equation i∂ t ψ = −∆ψ + V (x)ψ the analysis of energy levels E j is reduced to the corresponding eigenvalue problem for the operator −∆ + V (x). Spectral properties for the quartic oscillator V (x) = x 4 on R, have been studied by Voros in [Vor80] with methods also applicable to more general anaharmonic oscillators A 2k = − d 2 dx 2 + x 2k on R. In it Voros applied the zeta function ζ(s) of the anharmonic oscillator A 4 = C(− d 2 dx 2 +x 4 ) where C is a suitable normalisation constant related to the Gamma function. The zeta function of an operator A is defined by λ −s j , where λ j are the eigenvalues of A arranged in the increasing order. It is important to point out that the zeta function ζ(s) of the harmonic oscillator A 2 is related to the Riemann zeta function ζ R (s) by ζ(s) = (1 − 2 −s )ζ R (s).
The investigation of anharmonic oscillators is therefore not only relevant in analysis and mathematical physics, but also in the number theory. We will also see how they are closely related to some Lie groups and provide some applications to the spectral theory of some differential operators. A special case will be the Heisenberg, Engel and Cartan groups which supply a family of interesting examples. Despite the intensive research on the quartic oscillator in the last 40 years, the exact solution for the corresponding eigenvalue problem is unknown (cf. [OU11]). This fact is a further motivation for the research in approximative and qualitative methods around this problem.
The more general anharmonic oscillators appeared in the literature in the form of quartic oscillators with potentials λx 4 + x 2 2 in the works of F. T. Hioe and E. W. Montroll (cf. [HM75], [GM78]). In it the authors developed numerical calculations for the study of the corresponding energy levels. Subsequently, R. Balian, G. Parisi and A. Voros (cf. [BPV79]) started the research on quartic oscillators from the point of view of Feynman path integrals. In more recent works on anharmonic oscillators, S. Albeverio and S. Mazzucchi [AM06] considered quartic Hamiltonians with timedependent coefficients. The case of a fractional Laplacian and a quartic potential has been considered by S. Durugo and J. Lörinczi in [DL18].
A more general class of anharmonic oscillators arises in the form − d 2ℓ dx 2ℓ + x 2k + p(x) where p(x) is a polynomial of order 2k − 1 on R and with k, ℓ integers ≥ 1. The spectral asymptotics of such operators have been analysed by B. Helffer and D. Robert [HR82b], [HR82a]. In this paper we study a more general case on R n where a prototype operator is of the form (1.1) where k, ℓ are integers ≥ 1. Spectral properties for this type of operators in the case k = ℓ have been considered in [Hel84] and [Rob87]. The general setting we consider here is as follows. Let k be an integer ≥ 1, and let P 2k be the set of real-valued polynomials on R n , such that lim inf |x|→∞ p(x) |x| 2k > 0.
We consider operators of the form (1.2) where q ∈ P 2ℓ , p ∈ P 2k , and k, ℓ are integers ≥ 1. We note that if p ∈ P 2k then p(x) > −p 0 for all x ∈ R n and for some p 0 > 0. Hence p(x) + p 0 > 0. Analogously, by taking a q 0 > 0 such that q(ξ) + q 0 > 0 for all ξ ∈ R n , we associate to the operator T a Hörmander metric g given by g = g (p,q) = dx 2 (p 0 + q 0 + p(x) + q(ξ)) 1 k + dξ 2 (p 0 + q 0 + p(x) + q(ξ)) 1 ℓ . (1.3) We analyse then the operator T within the framework of Hörmander's S(m, g) classes, the topic of Section 3. However, in Section 4 we give some simplifications from the presentational point of view, showing that the Weyl-Hörmander classes corresponding to the metric (1.3) for the operators (1.2) can be described, independently of the lower order terms, by the following symbol classes: for m ∈ R, and k, ℓ integers ≥ 1, the class Σ m k,ℓ consists of all smooth functions a ∈ C ∞ (R n × R n ) such that holds for all x, ξ ∈ R n , see Definition 4.1 and the subsequent statements. Such a description also allows one to make use of these classes and their applications without any profound knowledge of the Weyl-Hörmander theory. For example, for k = l = 1, one recovers the well-known family of Shubin classes associated to the harmonic oscillator.
Regarding the spectral analysis of the anharmonic oscillators, we will obtain the Schatten-von Neumann properties for the negative powers of such operators in the setting of S(m, g) classes. The investigation on such properties within these classes started with Hörmander [Hör79]. Here we will apply a result by Buzano and Toft [BT10] for our analysis. Other works on Schatten-von Neumann classes within the Weyl-Hörmander calculus can be found in [Tof06], [Tof08]. See also [DR14b], [DR17], [DR14a] for several symbolic and kernel criteria on different types of domains.

Preliminaries
In this section we first briefly review some basic elements of the Weyl-Hörmander calculus. For a comprehensive study on this important theory we refer the reader to [Hör85], [Ler10], [BL89]. Second, we recall basic properties of Schatten-von Neumann classes.
The Kohn-Nirenberg and Weyl quantizations are recalled below: Definition 2.1. For a = a(x, ξ) ∈ S ′ (R n × R n ) (x ∈ R n and ξ ∈ R n ), we define the Kohn-Nirenberg quantization as the operator a(x, D) : S(R n ) → S(R n ) given by The Weyl quantisation of a(x, ξ), is given by the operator defined by a w : S(R n ) → S(R n ) defined by The Weyl quantization has fundamental relations with the symplectic structure of R n × R n = T * R n . One of those already arises from the symbol of the composition.
The operation # becomes useful in order to describe the composition a w • b ω , indeed one has a w • b ω = (a#b) ω .
Both quantizations above can be seen as particular cases of the more general form where t is a real parameter (t ∈ R).
We shall now recall the definition of Hörmander metrics on the phase space.
Definition 2.3. For X ∈ R n × R n let g X (·) be a positive definite quadratic form on R n × R n . We say that g(·) is a Hörmander's metric if the following three conditions are satisfied: (1) Continuity or slowness-There exist a constant C > 0 such that We say that g satisfies the uncertainty principle if for all X, T ∈ R n × R n .
(3) Temperateness-We say that g is temperate if there exist C > 0 and J ∈ N such that Let g be a Hörmander's metric. The uncertainty parameter or the Planck function associated to g is defined by and it is clear that h g (X) = (λ g (X)) −1 . The uncertainty principle can be translated then into the condition h g (X) ≤ 1.
Remark 2.4. (i) For a split metric g, i.e. for a metric of the type where a i (X) and b i (X) are positive functions, one can prove that (ii) A special case of (i) is the one of symmetrically split metric i.e. of the type The metric (1.3) is an example of a such type.
(iii) If g is a split metric one can prove the following formula for λ g from Definition 2.3, The classical weight ξ m is generalised in the following way for a corresponding Hörmander metric.
Definition 2.5. Let M : R n × R n → (0, ∞) be a function. We say that M is g-continuous if there existsC > 0 such that ≤C.
Definition 2.6. Let M : R n × R n → (0, ∞) be a function. We say that M is g-temperate if there existC > 0 and N ∈ N such that We will say that M is a g-weight if it is g-continuous and g-temperate.
We are now ready to define our classes of symbols.
Definition 2.7. For a Hörmander metric g and a g-weight M, we denote by S(M, g) the set of all smooth functions a on R n × R n such that for any integer k there exists C k > 0, such that for all X, T 1 , ..., T k ∈ R n × R n we have (2.5) The notation a (k) stands for the k th derivative of a and a (k) (X; T 1 , ..., T k ) denotes the k th derivative of a at X in the directions T 1 , ..., T k . For a ∈ S(M, g) we denote by a k,S(M,g) the minimum C k satisfying the above inequality. The class S(M, g) becomes a Fréchet space endowed with the family of seminorms · k,S(M,g) .
Remark 2.8. Let a Hörmander metric g be given. Instead of using the Planck function for the formulation of the statements we can equivalently employ the weight λ g defined by λ g = h −1 g . The function λ g is a g-weight for the metric g (cf. [Hör85]). Given a g-weight M, it is possible to construct an equivalent smooth weight M such that M ∈ S(M, g) (cf. [Hör85], [Ler10]). In particular, for λ g there exists an equivalent smooth weight λ g such that λ g ∈ S(λ g , g). Hence, λ g ∈ S( λ g , g). A weight M such that M ∈ S(M, g) is called regular. Thus the weight λ g and consequently the Planck function h g can be assumed to be regular.
Let 0 ≤ δ ≤ ρ ≤ 1 (δ < 1). The metric g ρ,δ is defined by where ξ := (1 + |ξ| 2 ) 1 2 . It is well known that g ρ,δ is a Hörmander metric, and that with it one recovers the S m ρ,δ classes i.e., S m ρ,δ = S( ξ m , g ρ,δ ). The uncertainty parameter λ g for g = g ρ,δ is given by In the special case ρ = 1, δ = 0, one has λ g (X) = ξ . The weight λ g can be seen as an extension of the basic one ξ , for the (ρ, δ) classes. The symbols in S(λ µ g , g) for µ ∈ R can be seen as the symbols of order µ with respect to the metric g. In particular, λ µ g is a symbol of order µ with respect to g. We now recall the definition of Sobolev spaces adapted to the Weyl-Hörmander calculus. Here we adopt the Beals's definition for simplicity in the presentation of the basic theory. Comprehensive treatments on Sobolev spaces in this setting can be found in [BC94], [Ler10].
Definition 2.9. Let g be a Hörmander metric and M a g-weight. We will call Sobolev space relative to M and it will be denoted by H(M, g), the set of tempered distributions u on R n such that a w u ∈ L 2 (R n ), ∀a ∈ S(M, g).
(2.6) Remark 2.10. We observe that the definition above requires a test over all the symbols in S(M, g). In contrast, we note that the classical Sobolev spaces H m = H( ξ m , g 1,0 ) are defined by the condition on the tempered distribution u: This means that, in the classical case the Sobolev space is defined by a fixed symbol.
The action of the Weyl quantization on the Sobolev spaces is determined by the following theorem (cf. [BC94]).
Theorem 2.11. Let g be a Hörmander metric, M and M 1 be two g-weights. For every a ∈ S(M, g), we have It is customary to identify H(1, g) (M = 1) with L 2 (cf. [BC94]): Theorem 2.12. For a Hörmander's metric g we have H(1, g) = L 2 (R n ).
We shall now recall some basic properties of Schatten-von Neumann classes. Let H be a complex separable Hilbert space endowed with an inner product denoted by (·, ·), and let T : H → H be a linear compact operator. If we denote by T * : H → H the adjoint of T , then the linear operator (T * T ) 1 2 : H → H is positive and compact. Let (ψ k ) k be an orthonormal basis for H consisting of eigenvectors of |T | = (T * T ) 1 2 , and let s k (T ) be the eigenvalue corresponding to the eigenvector ψ k , k = 1, 2, . . . . The non-negative numbers s k (T ), k = 1, 2, . . . , are called the singular values of then the linear operator T : H → H is said to be in the trace class S 1 . It can be shown that S 1 (H) is a Banach space in which the norm · S 1 is given by where {φ k : k = 1, 2, . . . } is any orthonormal basis for H. If the singular values are square-summable, T is called a Hilbert-Schmidt operator. It is clear that every trace class operator is a Hilbert-Schmidt operator. More generally, if 0 < p < ∞ and the sequence of singular values is p-summable, then T is said to belong to the Schatten class S p (H), and it is well known that each S p (H) is an ideal in L(H). If 1 ≤ p < ∞, a norm is associated to S p (H) by If 1 ≤ p < ∞ the class S p (H) becomes a Banach space endowed by the norm T Sp . If p = ∞ we define S ∞ (H) as the class of bounded linear operators on H, with T S∞ := T op , the operator norm. In the case 0 < p < 1 the quantity T Sp only defines a quasinorm, and S p (H) is also complete.
The Schatten classes are nested, with and satisfy the important multiplication property (cf. [Hor50], [Sim79], [GK69]) We will apply (2.8) for factorising our operators T in the form T = AB with A ∈ S p and B ∈ S q , and from this we deduce that T ∈ S r . A basic introduction to the study of the trace class is included in the book [Lax02] by Peter Lax. For the basic theory of Schatten classes we refer the reader to [GK69], [RS75], [Sim79], [Sch70].

Anharmonic oscillators
In this section we specifically begin the study of our anharmonic oscillators. We first introduce a special class of polynomials on R n which determine the operators that we will consider. For an integer k ≥ 1 we define: (3.1) We now take q = q(ξ) ∈ P 2ℓ , p = p(x) ∈ P 2k , where k, ℓ are integers ≥ 1. We consider operators of the form T = q(D) + p(x).
(3.2) We observe that since p ∈ P 2k then p(x) > −p 0 for all x ∈ R n and for some p 0 > 0.
where k, ℓ are integers ≥ 1. We recall that spectral properties for operators of the form − d 2ℓ dx 2ℓ + x 2k + p 1 (x), where p 1 is a suitable polynomial of order 2k − 1 have been studied by Helffer and Robert (cf. [HR82b], [HR82a]).
We associate to the operator T = q(D) + p(x), the following metric It is clear that in the definition of g (p,q) we can assume p 0 ≥ 1 obtaining an equivalent metric.
We note that if k = ℓ in A, the metric g is equivalent to which corresponds to the symplectic metric defining the Shubin classes. However, the general case here is more delicate.
We now start by showing how the metric g (p,q) is constructed. In the following theorem, the membership a ∈ S(L, G) should be understood in the sense that the inequality (2.5) holds for a Riemannian metric G on the phase-space and a strictly positive function L on the phase-space, i.e. independent of the fact if whether or not G is Hörmander metric and L just a G-weight. These facts will be shown afterwards.
Theorem 3.1. Let g = g (p,q) be the metric defined by (3.4). Then Proof. In order to prove Theorem 3.1 we first make some observations on the construction of the metric (3.4): Stage 1 (A = (−∆) ℓ + |x| 2k ). We consider the prototype case of an operator of the form A = (−∆) ℓ + |x| 2k . In this case the metric (3.4) is equivalent to We start by seeing how the metric (3.6) arises from the operator A. The symbol of A is (2π) 2ℓ |ξ| 2ℓ + |x| 2k , after some rescaling and in order to simplify the calculations we consider instead the symbol In order to obtain the metric (3.6) from the analysis of the derivatives of σ(x, ξ) we will denote the metric we are searching for by g and we note that for 1 ≤ j ≤ 2ℓ: On the other hand, we want to obtain an estimation of the form where s, t have to be found in order to get an inequality of the form (2.5) for a suitable coefficient of a quadratic form in dξ 2 . The values of s, t will lead to (3.6).
Equivalently, we should get for 0 ≤ j ≤ 2ℓ: The inequality (3.8) lead us to two conditions (for large |x|, |ξ|): From (i) we obtain t ≤ 1. In order to get (ii) we will find 1 < p, q < ∞ where 1 p + 1 q = 1 and such that from which we get the conditions (2ℓ − j)q ≤ 2ℓ and jsp ≤ 2k.
Therefore, for j < 2ℓ we can take On the other hand, 1 j ≥ 1 2ℓ . Hence, we have s ≤ 2k 2ℓ = k ℓ .
We have omitted above the case j = 2ℓ, but its analysis is simpler since |∂ 2ℓ ξ i (|x| 2k + |ξ| 2ℓ )| ≤ (2ℓ)! and hence (3.8) is still valid for 0 ≤ j ≤ 2ℓ. In this way we get the coefficient of the metric for dξ 2 : A similar analysis for |∂ j x i (|x| 2k + |ξ| 2ℓ )|, and exchanging the roles of k and ℓ in the above estimations lead us to the coefficient of the metric for dx 2 : Therefore, we have obtained the metric which completes the construction of the metric g by observing that (3.11) is equivalent to (3.6). It also proves that In the search of p for the analysis of (3.8) we have used the classical Young inequality: Lemma Let 1 ≤ p < ∞ and q such that 1 p + 1 q = 1. Then for all u, v ≥ 0 one has (3.13) From (3.13) one has uv ≤ C p (u p + v q ), and in our case we have considered u = |x| js in order to get the estimate (3.9).
(3.15) The constants C abαβ can be expressed in the form C abαβ = C αβ · max{a, b, 1}, where C αβ are the same structural constants as for the membership of σ in the symbol class S(1 + |x| 2k + |ξ| 2ℓ , g (k,ℓ) ).
Stage 3 (T = q(D) + p(x); where q ∈ P 2ℓ , p ∈ P 2k , k, ℓ integers ≥ 1) In this case the symbol of T is equivalent to T (x, ξ) = q(ξ) + p(x). From the assumptions on q and p there are a, b > 0 and q 0 , p 0 > 0 such that (3.16) On the other hand, it is clear that from (3.15) and comparing the partial derivatives of q(ξ) with the ones of σ ab we have where C abαβ are as in Stage 2.
(3.17) This completes the proof of Theorem 3.1.
Remark 3.2. We note that by redefining the metric g (a,b) in (3.14) and letting we only left with the constants C αβ in the seminorms and therefore, they are only depending on α and β. A similar consequence can be deduced for the metric in Stage 3 to obtain structural constants independent of the coefficients in the polynomials, i.e., only dependent on α and β.
We now turn on to prove that g is indeed a Hörmander metric. The lemma below is useful to study the slowness property. In particular (ii) and (iii) help to reduce a proof of continuity. (ii) There exists a constant C ≥ 1 such that Proof. From the definition of the continuity property taking X = Y one sees that C ≥ 1. It is clear that (i) =⇒ (ii). We now prove (ii) =⇒ (iii). If g X (Y ) ≤ C −1 we observe that Now by replacing C by C 2 in the continuity condition we finish the proof.
Proof. As observed previously p 0 can be assumed ≥ 1 obtaining an equivalent metric. Therefore, it is clear that the uncertainty principle holds For the proof of the continuity it will be enough to consider the case of the metric g (k,ℓ) , we will use Lemma 3.3, (ii), and prove that for all t, τ ∈ R n × R n .
We observe that the proof of this is reduced to prove that Or even simpler Now, (3.20) can be concluded in a similar way as in the proof of the continuity of the classical metric g 1,0 , and therefore g (k,ℓ) is slow. To see the general case, we note that in the previous proof of slowness for g (k,ℓ) if we consider instead g p,q , the analysis of the corresponding inequalities is reduced to the terms as for g (k,ℓ) . This completes the proof of the continuity.
We now prove the temperateness. We will only consider the case g (k,ℓ) . According to (2.1), we need to prove that there exist C > 0 and N ∈ N such that for all t, τ ∈ R n .
We will only prove the first inequality since the other one have a similar argument.
We now observe that we can reduce the proof of the first inequality to the following two inequalities: Therefore, it will be enough to prove the first inequality, which again is reduced to Now (3.21) can be obtained from the inequalities below 1 + |y| In general one can assume that the uncertainty parameter λ g of a Hörmander metric g is a g-weight as noted in Remark 2.8. However here one can deduce it directly. Indeed, the proof of Theorem 3.4 has some immediate consequences. In particular the validity of (3.19) implies that m = p 0 + q 0 + p(x) + q(ξ) is g-continuous.
Similarly, we can obtain the temperateness of p 0 + q 0 + p(x) + q(ξ) with respect to g from the proof above for the temperateness of g. Therefore p 0 + q 0 + p(x) + q(ξ) is a g-weight. Summarising we have: Proposition 3.5. Let g = g (p,q) be the metric defined by (3.4). Then p 0 + q 0 + p(x) + q(ξ) is a g-weight.
(ii) On the other hand, we can compare any of the metrics g (k,ℓ) with the metric g 1,0 in the sense of (ρ, δ) classes. Indeed, it is clear that for all k, ℓ ≥ 1.
As a consequence of the Theorem 3.1 we have the following property for the membership of more general polynomials in classes determined by the metric g defined in (3.4): Corollary 3.8. Let g = g (p,q) be the metric defined by (3.4). If p 1 : R n → C is a polynomial of order ≤ 2k and q 1 : R n → C is a polynomial of order ≤ 2ℓ, then 4. Some notes on the classes S(λ m g , g) We now make some observations and deduce some consequences formulating a setting in a more intrinsically way.
Let g = g (p,q) be defined by (3.4) with k, ℓ integers ≥ 1. Now that we know that g is a Hörmander metric, therefore there is a corresponding pseudo-differential calculus. We can define the class S(λ m g , g) for m ∈ R in an equivalent way without referring explicitly to the metric g. Indeed, we can define them in the following way: Definition 4.1. Let m ∈ R and let k, ℓ be integers ≥ 1. If a ∈ C ∞ (R n × R n ) we will say that a ∈ Σ m k,ℓ if It is clear that the definition above coincides with the corresponding one for the metric g = g (k,ℓ) defined by (3.6), i.e., the one associated to p(x) = |x| 2k , q(ξ) = |ξ| 2ℓ . Indeed we have: Proposition 4.2. Let m ∈ R and let g = g (k,ℓ) be defined by (3.4) with k, ℓ integers ≥ 1. Then Σ m k,ℓ = S(λ m g , g). This observation will be helpful to undertake some investigations on anharmonic oscillators in a simplified way without an explicit reference to the S(M, g) setting.
By associating to a symbol a ∈ Σ m k,ℓ a pseudodifferential operator a(x, D) we dispose of a pseudodifferential calculus on Σ m k,ℓ , inherited from the S(M, g) calculus.
We also observe that actually Proposition 4.2 is also valid if instead we use a metric g (p,q) with polynomials p ∈ P 2k , q ∈ P 2ℓ . Moreover we have: Corollary 4.3. Let m ∈ R and let k, ℓ be integers ≥ 1. For any p ∈ P 2k , q ∈ P 2ℓ , the classes S(λ m g (p,q) , g (p,q) ) all coincide and are equal to Σ m k,ℓ . We now formulate some few consequences for the classes Σ m k,ℓ . In particular, for the composition formula we have: The L 2 boundedness is obtained from the corresponding one for the S(1, g) class.
Theorem 4.5. Let k, ℓ be integers ≥ 1. If a ∈ Σ 0 k,ℓ , then a(x, D) extends to a bounded operator a(x, D) : Let m ∈ R and k, ℓ integers ≥ 1, we will associate a Sobolev space to the operator T = q(D) + p(x) as defined in (3.2) or equivalently to (−∆) ℓ + |x| 2k . We will call the Sobolev space of order m relative to k, ℓ and it will be denoted by H m k,ℓ (R n ), the set of tempered distributions u on R n such that (4.2) In the scale of those Sobolev spaces we have: Theorem 4.6. Let m ∈ R and k, ℓ integers ≥ 1. If a ∈ Σ m k,ℓ , then a(x, D) extends to a bounded operator for all s ∈ R.
Of course the above theorems on the boundedness also hold for t-quantizations a t (x, D) in (4.3), and in particular for the Weyl quantization due to the corresponding general property in the S(m, g) setting on the switching between those quantizations.

Schatten-von Neumann properties
We shall now obtain some results in relation with the behaviour in Schatten-von Neumann classes for the negative powers of our anharmonic oscillators. In the context of the Weyl-Hörmander calculus it is useful to recall the following result by Buzano and Toft [BT10]. We recall that h g is the uncertainty parameter from (2.2).
Theorem 5.1. Let g be a Hörmander metric, M a g-weight and 1 ≤ r < ∞. Assume that h k 2 g M ∈ L r (R n × R n ) for some k ≥ 0, and let a ∈ S(M, g). Then Regarding the classes here introduced and associated to the anharmonic oscillators we have: Theorem 5.2. Let k, ℓ ≥ 1, q = q(ξ) ∈ P 2ℓ , p = p(x) ∈ P 2k , where k and ℓ are integers ≥ 1. Let g = g (p,q) as in (3.4) and 1 ≤ r < ∞. Then provided µ > n r . Consequently, if µ > n r and a ∈ S(λ −µ g , g), then a t (x, D) ∈ S r (L 2 (R n )), for all t ∈ R.
This concludes the proof of the theorem.
For the formulation of the following corollary we first recall some facts on the trace class. It is well known that neither the mere integrability of a kernel K on the diagonal nor the integrability of the symbol on the phase-space are sufficient to guarantee the traceability of the corresponding operator. Under some suitable conditions on the kernel or the symbol one can get such traceablity. We are going to apply some results from [Del10b], [Del10a] in the case of the Euclidean space R n .
(1) We will apply Theorem 5.2. If g = g (p,q) we note that by Remark 3.6, we have . Now the condition 2kℓµ k + ℓ > n r is equivalent to µ > (k+ℓ) 2kℓ n r , and the conclusion now follows from Theorem 5.2. For the proof of (2), first we see that the membership to the trace class follows from Theorem 5.2 with r = 1. For the trace formula we observe that for every x ∈ R n we have Now by applying Corollary 3.11 of [Del10a] we obtain the formula (5.1).
We can also collect some consequences for the singular values of operators in S(M, g) classes. For a compact operator A on a complex Hilbert space H we will denote by λ j (A) the eigenvalues of A ordered in the decreasing order, and s j the corresponding singular values.

Consequently, also
Proof. From Theorem 5.2 we know that a t (x, D) ∈ S r (L 2 (R n )), for all t ∈ R.
Hence, as it is well known one can get from the membership above that for every t ∈ R. Moreover from the Weyl inequality one has λ j (a t (x, D)) = o(j − 1 r ), as j → ∞, for every t ∈ R.
In the proof above we have applied the Weyl inequality (cf. [Wey49]) which relates the singular values s n (T ) and the eigenvalues λ n (T ) for a compact operator T on a complex separable Hilbert space: Remark 5.5. We point out that in the case of the operator T = (−∆) ℓ + |x| 2k and from from Theorem 3.2 of [BBR96] one can obtain an estimate for the eigenvalue counting function N(λ) of T . Indeed, for large λ the eigenvalue counting function N(λ) is bounded by C a(x,ξ)<λ dxdξ, where a(x, ξ) is the Weyl symbol of the partial differential operator T . By the change of variables ξ = λ 1/2k ξ ′ and x = λ 1/2ℓ x ′ , we can estimate for large λ that Now, from (5.4) we can deduce that (T + I) −µ ∈ S r (L 2 (R n )) for µ > n r , and therefore we can recover the same estimate on the decay of singular values and eigenvalues for (T + I) −µ as we get in (5.2) and (5.3). The advantage of Corollary 5.4 is that the assumption a ∈ S(λ −µ g , g) allows one to consider more general symbols.

Examples arising from the analysis on Lie groups
We now explain how some differential operators arising from the theory of Lie groups fit into the above setting. We start by fixing the notation and recall well known preliminary facts about Lie groups. We refer to [FR16] for more details on the definitions in this part.
Let G be a Lie group and let Lie(G) be the corresponding Lie algebra. If Lie(G) can be endowed with a vector space decomposition of the form where all but finitely many V j 's are zero, then we say that Lie(G) is graded. Lie(G) is called stratified if in addition to the condition (6.1), it can be generated by the first stratum V 1 , i.e., any element of Lie(G) can be written as a linear combination of commutators of elements of V 1 . In this case, if the Lie group G is in addition connected, simply connected, then G is also called graded and stratified, respectively. Finally, if p is the number of the non-zero V j 's in (6.1), then both G and Lie(G) are called stratified of step p. Following [Hör67], the set of vector fields, say {X 1 , · · · , X r }, that spans V 1 is said to be a Hörmander system. The operator for the given Hörmander system {X 1 , · · · , X r }, is the sub-Laplacian operator on G.
Sub-Laplacian operators are a very important tool in non-commutative harmonic analysis, and one can refer to [Fol75] or [FR16].
LetĜ denote the unitary dual of G, i.e., the set of equivalence classes of continuous, irreducible unitary representations of G on a Hilbert space H. For a given π ∈Ĝ, and any X ∈ Lie(G), the operator π(X) is called the infinitesimal representation associated to π. Each π(X) acts on the subspace, say H ∞ , of smooth vectors in H. The infinitesimal representation extends to the universal enveloping algebra of the Lie group G. For example we are allowed to consider the infinitesimal representation of the sub-Laplacian operator L G . Any π(X) is the global symbol of X in the sense of the definition in Chapter 5 of [FR16].
In the following we will show that the global symbol of the sub-Laplacian on the Engel and on the Cartan group, nilpotent Lie groups of 3-steps, are examples of the differential operators considered before. The above groups will be treated separately. First a brief description of the structure, and then of the dual of each group is given, leading to the formulas for the global symbol of the sub-Laplacian operators in both cases.
6.1. Engel group. Let l 4 = span{I 1 , I 2 , I 3 , I 4 } be a 3-step stratified Lie algebra, whose generators satisfy the non-zero relations: [I 1 , I 2 ] = I 3 , [I 1 , I 3 ] = I 4 . (6. 2) The corresponding Lie group, called the Engel group, and denoted by B 4 , is isomorphic to the manifold R 4 . This identification implies that the basis of l 4 (now called the canonical basis) can be given by the left invariant vector fields where x = (x 1 , x 2 , x 3 , x 4 ) ∈ R 4 , also satisfying the relations (6.2) as expected. The system of vector fields {X 1 , X 2 } is a Hörmander system and therefore gives rise to the sub-Laplacian (or the canonical sub-Laplacian) on B 4 , namely to the operator Now, given, the description of the dual of the group as Dixmier suggested in [Dix59], i.e., a family of operatorsB 4 = {π λ,µ |λ = 0, µ ∈ R} acting on L 2 (R) via, one can easily find the infinitesimal representation of l 4 associated to π λ,µ , and therefore show that the (global) symbol of the sub-Laplacian L B 4 is the family of operators acting on the Schwartz space S(R), given by For some fixed parameters λ ∈ R \ {0}, µ ∈ R, the symbol of an operator as in (6.4), is then where p ∈ P 4 , q ∈ P 2 and k = 2, ℓ = 1, n = 1.
In terms of the corresponding S(·, g (k,ℓ) ν ) classes, the constants involved in the membership to these classes are of the form C αβ ; see Remark 3.2.
In turn, we can analyse the corresponding Schatten-von Neumann classes for the negative powers of I − T . Indeed, by Corollary 5.3, since p ν ∈ P 2ℓ and q ν ∈ P 2k we have (I − T ) −γ ∈ S r (L 2 (R n )), (6.12) provided that γ > (k+ℓ)n 2kℓr .