On the Spectral Asymptotics of Operators on Manifolds with Ends

Theorem 1. Let M be a manifold with ends of dimension n and let $$ be a positive self-adjoint operator such that m, μ > 0, m ≠ μ, with domain H^m,μ(M)↪L²(M). Then, the following Weyl formulae hold for λ → +∞:

where ε₁ = min {1/μ, n((1/m) − (1/μ))} and ε₂ = min {1/m, n((1/μ) − (1/m))}.

Definition 2. (i) A symbol p(x, ξ) belongs to the class $$ if there exist $$, i = 0,1, …, positively homogeneous functions of order m − i with respect to the variable ξ, smooth with respect to the variable x, such that, for a 0-excision function ω,

(ii) A symbol p(x, ξ) belongs to the class $$ if there exist $$, k = 0, …, positively homogeneous functions of order μ − k with respect to the variable x, smooth with respect to the variable ξ, such that, for a 0-excision function ω,

Definition 3. A symbol p(x, ξ) is SG-classical, and we write $$, if

(i) there exist $$ such that for a 0-excision function ω, $$ and

(ii) there exist $$ such that for a 0-excision function ω, $$ and We set $$.

Remark 4. The definition could be extended in a natural way from operators acting between scalars to operators acting between (distributional sections of) vector bundles. One should then use matrix-valued symbols whose entries satisfy the estimates (3).

Note that the definition of SG-classical symbol implies a condition of compatibility for the terms of the expansions with respect to x and ξ. In fact, defining $$ and $$ on $$ and $$, respectively, as

It is possibile to prove that

Moreover, the composition of two SG-classical operators is still classical. For $$ the triple σ(P) = (σ_ψ(P), σ_e(P), σ_ψe(P)) = (p_m,·, p_·,μ, p_m,μ) = (p_ψ, p_e, p_ψe) is called the principal symbol of P. The three components are also called the ψ-, e- and ψe-principal symbol, respectively. This definition keeps the usual multiplicative behaviour, that is, for any $$, $$, (r, ρ), (s, σ) ∈ ℝ², σ(RS) = σ(S)σ(T), with component-wise product in the right-hand side. We also set

for a fixed 0-excision function ω. Theorem 5 allows to express the ellipticity of SG-classical operators in terms of their principal symbol.

Theorem 5. An operator $$ is elliptic if and only if each element of the triple σ(P) is nonvanishing on its domain of definition.

Definition 6. A manifold with a cylindrical end is a triple (M, X, [f]), where M = ℳ∐_C𝒞 is a n-dimensional smooth manifold and

• (i)

  ℳ is a smooth manifold, given by ℳ = (M₀∖D) ∪ C with a n-dimensional smooth compact manifold without boundary M₀, D a closed disc of M₀, and C ⊂ D a collar neighbourhood of ∂D in M₀;

• (ii)

  𝒞 is a smooth manifold with boundary ∂𝒞 = X, with X diffeomorphic to ∂D;

• (iii)

  f : [δ_f, ∞) × 𝕊ⁿ⁻¹ → 𝒞, δ_f > 0, is a diffeomorphism, f({δ_f} × 𝕊ⁿ⁻¹) = X and f({[δ_f, δ_f + ε_f)} × 𝕊ⁿ⁻¹), ε_f > 0, is diffeomorphic to C;

• (iv)

  the symbol ∐_C means that we are gluing ℳ and 𝒞, through the identification of C and f({[δ_f, δ_f + ε_f)} × 𝕊ⁿ⁻¹);

• (v)

  the symbol [f] represents an equivalence class in the set of functions

  $$ ()

where f ~ g if and only if there exists a diffeomorphism Θ ∈ Diff(𝕊ⁿ⁻¹) such that for all ρ ≥ max {δ_f, δ_g} and γ ∈ 𝕊ⁿ⁻¹.

Definition 7. The set $$ consists of all the symbols $$ which fulfill (3) for $$ only. Moreover, the symbol a belongs to the subset $$ if it admits expansions in asymptotic sums of homogeneous symbols with respect to x and ξ as in Definitions 2 and 3, where the remainders are now given by SG-symbols of the required order on $$.

Note that, since $$ is conical, the definition of homogeneous and classical symbol on $$ makes sense. Moreover, the elements of the asymptotic expansions of the classical symbols can be extended by homogeneity to smooth functions on ℝⁿ∖{0}, which will be denoted by the same symbols. It is a fact that, given an admissible atlas $$ on M, there exists a partition of unity {θ_i} and a set of smooth functions {χ_i} which are compatible with the SG-structure of M, that is,

• (i)

  supp θ_i ⊂ Ω_i, supp χ_i ⊂ Ω_i, χ_iθ_i = θ_i, i = 1, …, N;

• (ii)

  $$ and $$ for all $$.

Moreover, θ_N and χ_N can be chosen so that $$ and $$ are homogeneous of degree 0 on U_δ. We denote by u^* the composition of u : ψ_i(Ω_i) ⊂ ℝⁿ → ℂ with the coordinate patches ψ_i, and by v_* the composition of v : Ω_i ⊂ M → ℂ with $$, i = 1, …, N. It is now possible to give the definition of SG-pseudodifferential operator on M.

Definition 8. Let M be a manifold with a cylindrical end. A linear operator $$ is an SG-pseudodifferential operator of order (m, μ) on M, and we write P ∈ L^m,μ(M), if, for any admissible atlas $$ on M with exit chart (Ω_N, ψ_N):

• (1)

  for all i = 1, …, N − 1 and any $$, there exist symbols pⁱ(x, ξ) ∈ S^m(ψ_i(Ω_i)) such that

  $$ ()

• (2)

  for any θ_N, χ_N of the type described above, there exists a symbol $$ such that

  $$ ()

• (3)

  K_P, the Schwartz kernel of P, is such that

  $$ ()
  where Δ is the diagonal of M × M and $$ with any conical neighbourhood V of the diagonal of $$.

Definition 9. Let P ∈ L^m,μ(M). P is an SG-classical operator on M, and we write $$, if $$ and the operator P, restricted to the manifold ℳ, is classical in the usual sense.

Definition 10. Let $$ and let us fix an exit map f_π. We can define local objects p_m−j,μ−i, p_·,μ−i as

Definition 11. An operator $$ is elliptic, and we write $$, if the principal part of $$ satisfies the SG-ellipticity conditions on $$ and the operator P, restricted to the manifold ℳ, is elliptic in the usual sense.

Proposition 12. The properties P ∈ L^m,μ(M) and $$, as well as the notion of SG-ellipticity, do not depend on the (admissible) atlas on M. Moreover, the local functions p_e and p_ψe give rise to invariantly defined elements of $$ and $$, respectively.

Proposition 13. Every $$, considered as an unbounded operator P : 𝒮(M) ⊂ L²(M) → L²(M), admits a unique closed extension, still denoted by P, whose domain is 𝒟(P) = H^m,μ(M).

Theorem 14 (Spectral theorem). Let $$ be regarded as a closed unbounded operator on L²(M) with dense domain H^m,μ(M). Assume also that m,   μ > 0 and P^* = P. Then

• (i)

  (λI − P) ⁻¹ is a compact operator on L²(M) for every λ ∈ ϱ(P). More precisely, (λI − P) ⁻¹ is an extension by continuity from 𝒮(M) or a restriction from 𝒮^′(M) of an operator in $$.

• (ii)

  spec (P) consists of a sequence of real isolated eigenvalues {λ_j} with finite multiplicity, clustering at infinity; the orthonormal system of eigenfunctions {e_j} _j≥1 is complete in L²(M) = H^0,0(M). Moreover, e_j ∈ 𝒮(M) for all j.

Lemma 15. $$ is an operator with kernel $$.

Theorem 16. Define $$, where θ_k and χ_k are as in Definition 8, with χ_kθ_k = θ_k, k = 1, …, N, while the A_k(t) are SG FIOs which, in the local coordinate open set U_k = ψ_k(Ω_k) and with v ∈ 𝒮(ℝⁿ), are given by

Each A_k(t) solves a local Cauchy problem (D_t − Q_k)∘A_k ∈ C^∞((−T, T), L^{−∞,−∞}(ℝⁿ)), A_k(0) = I, with Q_k = Op(q_k) and $$ local (complete) symbol of Q associated with {θ_k}, {χ_k}, with phase and amplitude functions such that

Then, V(t) satisfies and U − V ∈ C^∞((−T, T), L^{−∞,−∞}(M)).

Remark 17. Trivially, for k = 1, …, N − 1, q_k and a_k can be considered SG-classical, since, in those cases, they actually have order −∞ with respect to x, by the fact that q_k(x, ξ) vanishes for x outside a compact set.

Remark 18. Notation like b ∈ C^∞((−T, T), S^r,ρ(ℝⁿ)), B ∈ C^∞((−T, T), L^r,ρ(M)), and similar, in Theorem 16 and in the sequel, also mean that the seminorms of the involved elements in the corresponding spaces (induced, in the mentioned cases, by (3)), are uniformly bounded with respect to t ∈ (−T, T).

Theorem 19. Assume that

• (i)

  $$ is an even function satisfying ψ(0) = 1, $$, $$;

• (ii)

  N_Q(λ) is a nondecreasing function, supported in [0, +∞), continuous from the right, with polynomial growth at infinity and isolated discontinuity points of first kind {η_j}, j ∈ ℕ, such that η_j → +∞;

• (iii)

  there exists d₀ ≥ 0 such that

  $$ ()

with m ∈ (0,1), n^* = min {n, (n/m) − 1}.

Then

Remark 20. The previous statement can be modified as follows: with ψ, N_Q, and m as in Theorem 19, when

with m ∈ (0,1), then N_Q(λ) = (d₀/2π)λ^n/m + O(λ^(n/m)−1) + O(λⁿ), for λ → +∞.

Remark 21. The assumption on q⁻¹ above amounts, at most, to modifying q by adding and subtracting a compactly supported symbol, that is, an element of S^{−∞,−∞}(ℝⁿ). The corresponding solutions φ and a of the eikonal and transport equations, respectively, would then change, at most, by an element of C^∞((−T, T), S^{−∞,−∞}(ℝⁿ)), see [23, 24, 28]. It is immediate, by integration by parts with respect to t, that an integral as (46) is O(|λ|^−∞) for a ∈ C^∞((−T, T), S^{−∞,−∞}(ℝⁿ)). Then, the modified q obviously keeps the same sign everywhere.

Remark 22. The ellipticity of q yields, for λ < 0,

which, by integration by parts, implies I(λ) = O(|λ|^−∞) when λ → −∞.

Proposition 23. Let H₁ be any function in $$ such that supp H₁⊆[(2k₁) ⁻¹, 2k₁], 0 ≤ H₁ ≤ 1 and H₁ ≡ 1 on $$, where k₁ > 1 is a suitably chosen, large positive constant. Then

Proof. Write

and observe that, by A⁻¹〈x〉〈ξ〉^m ≤ q(x, ξ) ≤ A〈x〉〈ξ〉^m, x, ξ ∈ ℝⁿ, we find

Thus, if k₁ > 2AC we have |∂_tΦ(t; x, ξ; λ)| ~ λ + 〈x〉〈ξ〉 ^m on the support of 1 − H₁(〈x〉〈ξ〉 ^m/λ), and the assertion follows integrating by parts with respect to t in the first integral of (51).

Remark 24. We actually choose k₁ > 4AC > 2AC, since this will be needed in the proof of Proposition 28; see also Section C in the Appendix.

Let us now pick $$ such that 0 ≤ H₂(υ) ≤ 1, H₂(υ) = 1 for |υ | ≤ k₂ and H₂(υ) = 0 for |υ | ≥ 2k₂, where k₂ > 1 is a constant which we will choose big enough (see below). We can then write

In what follows, we will systematically use the notation S^r,ρ = S^r,ρ(y, η), y ∈ ℝ^k, η ∈ ℝ^l, to generally denote functions depending smoothly on y and η and satisfying SG-type estimates of order r, ρ in y, η. In a similar fashion, $$ will stand for some function of the same kind which, additionally, depends smoothly on t ∈ (−T, T), and, for all s ∈ ℤ₊, $$ satisfies SG-type estimates of order r, ρ in y, η, uniformly with respect to t ∈ (−T, T).

To estimate I₁(λ), we will apply the stationary phase theorem. We begin by rewriting the integral I₁(λ), using the fact that φ is solution of the eikonal equation associated with q and that q is a classical SG-symbol. Note that then $$, since

In view of the Taylor expansion of φ at t = 0, recalling the property q(x, ξ) = ω(x)q_e(x, ξ) + S^m,0(x, ξ), ω a fixed 0-excision function, we have, for some 0 < δ₁ < 1,

where the subscript e denotes the x-homogeneous (exit) principal parts of the involved symbols, which are all SG-classical and real-valued, see [28].

Proposition 25. Choosing the constants k₁, λ₀ > 1 large enough and T > 0 suitably small, one has, for any k₂ > 1 and for a certain sequence c_j, j = 0,1, …,

that is, I₁(λ) = c₀λⁿ⁻¹ + O(λⁿ⁻²), with

Proof. It is easy to see that, on the support of U₁, the phase function F₁(X, Y) admits a unique, nondegenerate, stationary point X₀ = X₀(Y) = (0, q_e(ς, ξ) ⁻¹), that is, $$ for all Y such that (X, Y) ∈ supp U₁, provided that T > 0 is chosen suitably small (see, e.g., [25, page 136]), and the Hessian $$ equals −q_e(ς, ξ) ² < 0. Moreover, the amplitude function

is compactly supported with respect to the variables X and Y and satisfies, for all $$, for all X, Y, λ ≥ λ₀. In fact,

• (1)

  $$, ς ∈ 𝕊ⁿ⁻¹, supp [H₂(|ξ|)]⊆{ξ  :  |ξ| ≤ 2k₂}, and

  $$ ()
  where λ₀ > 2k₁〈2k₂〉 ^m;

• (2)

  all the factors appearing in the expression of U₁ are uniformly bounded, together with all their X-derivatives, for X ∈ S_X = supp ψ × [ζ₀, ζ₁], Y ∈ S_Y = 𝕊ⁿ⁻¹ × {ξ:|ξ | ≤ 2k₂}, and λ ≥ λ₀.

Of course, (2) trivially holds for the cutoff functions ψ(t) and H₂(|ξ|), and for the factor ζⁿ⁻¹. Since $$, on S_X × S_Y we have, for all $$ and λ ≥ λ₀ > 1,

Moreover, since $$ is actually in $$ on S_X × S_Y, the same holds for exp (iG₁), by an application of the Faà di Bruno formula for the derivatives of compositions of functions, so also this factor fulfills the desired estimates. Finally, another straightforward computation shows that, for all γ₂ ∈ ℤ₊ and λ ≥ λ₀ > 1,

on S_X × S_Y.  The proposition is then a consequence of the stationary phase theorem (see [30, Proposition  1.2.4], [31, Theorem  7.7.6]), applied to the integral with respect to X = (t, ζ). In particular, the leading term is given by λⁿ/(2π)ⁿ⁻¹ times the integral with respect to Y of $$, that is recalling that ψ(0) = 1, a(0; x, ξ) = 1 for all x, ξ ∈ ℝⁿ.

Indeed, having chosen k₁ > 2A, λ₀ > 2k₁〈2k₂〉 ^m, (58) implies

uniformly for ς ∈ 𝕊ⁿ⁻¹, ξ ∈ supp [H₂(|ξ|)], λ ≥ λ₀. This concludes the proof.

Remark 26. Incidentally, we observe that a rough estimate of λ^n/mI₂(λ) is

An even less precise result would be the bound λ^n/m, using the convergence of the integral with respect to x in the whole ℝⁿ, given by −(n/m) + n < 0.

Lemma 27. $$ for any ζ ∈ [0, +∞), x ∈ ℝⁿ, ς ∈ 𝕊ⁿ⁻¹, λ ≥ λ₀, m ∈ (0,1), and, for all $$,

Proposition 28. If k₁, k₂, λ₀ > 1 are chosen large enough, one has

Explicitly,

Proposition 29. With the choices of T, k₁, λ₀, for any ε ∈ (0, 1/2), one can find k₂ > 1 large enough such that J₁(λ) = O(λ^−∞).

Proof. Since 0 < m < 1, in view of (3), (74), and (79), we can choose k₂ > 1 so large that, for an arbitrarily fixed ε ∈ (0, 1/2), for any λ ≥ λ₀, ζ ∈ (0, +∞) satisfying |ξ | = (λζ) ^1/m ≥ k₂,

uniformly with respect to (X, Y) ∈ S_X × S_Y⊇supp U₂(·; λ). Then, F₂ is nonstationary on supp V₁, since there we have |(ζ/ζ₀) − 1| ≥ (3/2)ε, while which implies ∂_tF₂(X, Y; λ)≻1. Observing that, on supp V₁, $$, as well as $$, the assertion follows by repeated integrations by parts with respect to t, using the operator and recalling Remark 26.

Proposition 30. With the choices of ε, T > 0, k₁, k₂, λ₀ > 1, one can assume, modulo an O(λⁿ⁻¹) term, that the integral with respect to x in J₂(λ) is extended to the set {x ∈ ℝⁿ : 〈x〉≤ϰλ}, with

Proof. Indeed if $$, we can split J₂(λ) into the sum

since the inequality $$ is true when k₂ is sufficiently large.

Observing that, on suppU₂,

switching back to the original variables, the first integral in (88) can be treated as I₁(λ), and gives, in view of Proposition 25, an O(λⁿ⁻¹) term, as stated.

Proposition 31. With ε ∈ (0, 1/2), T > 0, k₁,   k₂,   λ₀ > 1 fixed previously, $$ vanishes on supp V₂ only for $$, that is, $$ for all Y such that (X, Y; λ) ∈ supp V₂. Moreover,

holds on supp V₂.

Proof. We have to solve

(X, Y; λ) ∈ supp V₂. By (79) and (84a) and (84b), with the choices of ε, T > 0, k₁, k₂, λ₀, the coefficient of t in the second equation does not vanish at any point of supp V₂. Then t = 0, and ζ must satisfy Since, by the choice of k₂, |∂_ζG(ζ; Y; λ)| ≤ k₀ < 1, uniformly with respect to Y ∈ 𝕊ⁿ⁻¹ × {x ∈ ℝⁿ : 〈x〉≤ϰλ}, λ ≥ λ₀, G has a unique fixed point $$, smoothly depending on the parameters; see the Appendix for more details. Since we can assume that $$ and the choices of the other parameters imply, on supp V₂, So we have proved that, on supp V₂,

By (3), (92), and $$, (X, Y) ∈ S_X × S_Y⊇supp V₂(·; λ), we also find

uniformly with respect to λ ≥ λ₀. The proof is complete.

Remark 32. The choice of k₂ depends only on the properties of q and on the values of k₁ and ε; that is, we first fix k₁ > 4AC > 2AC > 2 and ε ∈ (0, 1/2), then T > 0 small enough as explained at the beginning of the proof of Proposition 25, then k₂ > 1 as explained in the proofs of Propositions 29 and 31, then, finally, $$.

Lemma 33. Assume $$, |α | > 0,

W is smooth, W_k(X, Y; λ)≺〈x〉^k, k ∈ ℤ₊, and has a SG-behaviour as the factors appearing in the expression of V₂. Then where $$ has the same SG-behaviour, support and x-order of V₂, including the powers of ζ.

Proof. By arguments similar to those used in the proof of Proposition 29, on supp W

Assume that the first condition in (97) holds. Under the hypotheses, if α₁ > 0, we can first insert $$ in the left-hand side of (98), where L_ζ = D_ζ/λ∂_ζF₂(X, Y; λ), and integrate by parts α₁ times. Similarly, if α₂ > 0, we subsequently use $$, L_t = D_t/λ∂_tF₂(X, Y; λ), and integrate by parts α₂ times. The assertion then follows, remembering that ζ-derivatives of W produce either an additional ζ⁻¹ factor or a lowering of the exponent of $$, and that $$ on supp W. The proof in the case that the second condition in (97) holds is the same, using first L_ζ and then L_t.

Proof of Proposition 28. Define,

and, for s ∈ [0,1], Remembering that $$, $$, 𝒬 is the Taylor polynomial of degree two of F₂ at $$, so that 𝒢 vanishes of order 3 at $$. Obviously, ℱ₀(X, Y; λ) = 𝒬(X, Y; λ) and ℱ₁(X, Y; λ) = F₂(X, Y; λ). Write $$, and consider the Taylor expansion of 𝒥_τ(s) of order 2𝒩 − 1, 𝒩 > 1, so that

Since

Remark 26 and Lemma 33 imply that $$, $$, s ∈ [0,1]. Indeed, it is easy to see, by direct computation, that 𝒢 can be bounded by linear combinations of expressions of the form with W_k, k ∈ ℤ₊, having the required properties. Then, the bound of 𝒢^2𝒩 will always contain a term of the type $$, which corresponds to the (minimum) value |α| = 3𝒩 in (97).

Each term $$, k = 0, …, 2𝒩 − 1, has the quadratic phase function 𝒬, which of course also satisfies

Then, denoting by Γ the Taylor expansion of 𝒢 at $$ of order 3𝒩, we observe that 𝒢^k − Γ^k can be bounded by polynomial expressions in $$ of the kind appearing in the right-hand side of (97), with |α | = 𝒩 + k (cf. the proof of Theorem 7.7.5 in [31]). Setting Lemma 33 implies

We now apply the stationary phase method to $$ and prove that

which is a consequence of with M evaluated with τ⁻¹ in place of λ. Recalling (95), it follows that the inverse matrix M⁻¹ satisfies, on supp V₂, in view of the ellipticity of the involved symbols. Then, the operators L_j,k,Y,τ, j, k ∈ ℤ₊, do not increase the x-order of the resulting function with respect to that of their arguments, (iλΓ) ^kV₂, which is the same of V₂, uniformly with respect to τ. The proof of (110) then follows by Theorem 7.6.1, the proof of Lemma 7.7.3 and formula (7.6.7) in [31]; see also [26, 32]. Indeed, by the mentioned results, since 〈x〉≺λ on supp V₂. It is then enough to sum all the expansions of $$, k = 0, …, 2𝒩 − 1, and sort the terms by decreasing exponents of λ (as in the proof of Theorem 7.7.5 in [31]) to obtain (109) with the usual expression so that, in particular, for any j ∈ ℤ₊, $$. We can then integrate 𝒥_τ(1) and its asymptotic expansions with respect to Y ∈ 𝕊ⁿ⁻¹ × {x ∈ ℝⁿ : 〈x〉 ≤ ϰλ} and find Recall that ψ(0) = 1 and a(0, x, ξ) = 1, for all x, ξ ∈ ℝⁿ. Moreover, for $$, the factors H₁, H₂, and H₃ are identically equal to 1 (see the Appendix). Then, the coefficient of the leading term in (115) is given by with M evaluated in $$. We say that To confirm this, first note that $$, λ → +∞, for any (Y; λ) belonging to the support of the integrand, see the Appendix. Moreover, the integrand is uniformly bounded by the summable function 〈x〉^−n/m, and its support is included in the set S. Then, recalling (95) and setting $$,

The second integral is always O(λ^n−(n/m)), since q_ψ(x, ς) ~ 〈x〉 implies

The first integral can be estimated as follows. Since

by the properties of $$ (see the appendix) we find since $$. By (95), we similarly have $$, so that If n > 1/(1 − m)⇔n − 1 − ((n − 1)/m) < −1, n ∈ ℕ, m ∈ (0,1), the integral in R₁ is convergent for λ → +∞ and R₁ = O(λ^−1/m). In this case, R₁ contributes an O(λ^{(n/m)−1−(1/m)}) term to the expansion of I₂(λ), which is of lower order than the O(λ^(n/m)−2) term, which is one of the remainders appearing in (77). On the other hand, if n < 1/(1 − m), the integral in R₁ is divergent, and R₁ itself is O(λ^n−(n/m)), since, trivially Finally, if n = 1/(1 − m), R₁ is O(λ^−1/mln λ), by and again contributes a term of lower order than the remainder O(λ^(n/m)−2). Similar conclusions can be obtained for the subsequent terms of the expansion of J₂(λ).

The proof is complete, combining the contributions of the remainders like R with the other terms in the expansion of J₂(λ), and remembering that

Remark 34. The same conclusions concerning the behaviour of R₁ in the final step of the proof of Proposition 28 could have been obtained studying the Taylor expansion of the extension of $$, τ = λ⁻¹, to the interval $$, similarly to [32].

Proof of Theorem 1. The statement for μ > m follows by the arguments in Section 3 and Propositions 23, 25 and 28, summing up the contribution of the local symbol on the exit chart to the contributions of the remaining local symbols, which gives the desired multiple of the integral of $$ on the cosphere bundle as coefficient of the leading term λ^n/m. The remainder has then order equal to the maximum between (n/m) − 1 and n, as claimed. The proof for μ < m is the same, by exchanging step by step the role of x and ξ.