ESTIMATES OF THE SPECTRUM OF

COMPACT PSEUDODIFFERENTIAL OPERATORS

IN UNBOUNDED DOMAINS

A. S. Andreev

Abstract. We study the spectrum of compact selfadjoint pseudodifferential operators with matrix symbol in unbounded sets in Euclidean space. Two-sided bounds for the spectrum are obtained by the variational method. Additional regularity assumptions on the symbol lead to a formula for the spectral asymptotics with a remainder term estimate.

The spectrum of compact selfadjoint pseudodifferential operators (ΨDO) in unbounded domains has been studied relatively little. In [1] the asymptotics of the spectrum of ΨDO with anisotropically homogeneous symbols in R^{m} was obtained. In [1] the smoothness assumptions on the symbol were minimal, there was no assumption of positive or negative definiteness, but it was assumed that the order of homogeneity l < m. A remainder term estimate was not given in [1]. The approximate spectral projection method of [2] has been used in the study of the spectrum of pencils of linear ΨDO of positive order in unbounded domains. In [2] the asymptotics of the spectrum was established with a remainder term estimate. The ΨDO in [1] and [2] have matrix symbols. In [3] and [4] a classical formula for the spectral asymptotics of scalar compact ΨDO in R^{m} was found. In this connection the symbol of the operator belongs to the Beals-Hörmander calculus, and, in general, need not possess any homogeneity properties. The assumptions in [3] and [4] imply essential restrictions of the form constant direction of the gradient of the symbol. It was shown that the remainder term of the spectral asymptotics has power order (in the spectral parameter) relative to the principal term. However, the dependence of this order on the symbol of the ΨDO was not obtained in [3, 4]. The proof is connected in an essential way with a specific problem in R^{m}.

In the present work we study the spectrum of a matrix ΨDO in an unbounded set in R^{m}. The smoothness hypotheses on the boundary of the domain coincide locally with the corresponding conditions in [5]. The symbol of the ΨDO is assumed to be a smooth matrix function. We do not make any assumption of definiteness or constancy of the multiplicity of the eigenvalues of the matrix symbol. We do not assume that the ΨDO belongs to the Beals-Hörmander class.

Two-sided bounds for the spectrum of the operator are obtained by the variational method. The proof rests mainly on local results from [5]. Under additional smoothness assumptions on the symbol these bounds yield the spectral asymptotics and the order of the remainder term.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 35S05, 47G05, 35P20.

The author is grateful to M. Z. Solomyak for helpful discussions.

§1. Statement of the main results

1. Basic notation. R^{m} denotes the Euclidean space of points x = (x_{1}, …, x_{m}); Z^{m} is the set of points with integer coordinates in R^{m}; S^{m−1} is the unit sphere in R^{m}; E^{m} denotes the dual space of R^{m} consisting of points ξ = (ξ_{1}, …, ξ_{m}); N is the set of natural numbers. For real t, [t] denotes its integer part, and t_{+} = (t + |t|)/2, t_{−} = (|t| − t)/2.

If D is a set in R^{m}, then ∑_{D}^{(k)} denotes summation over k ∈ D ∩ Z^{m}. In case D = R^{m} we omit D. Integrals without indicated domain of integration are assumed to be extended over the whole space.

For a multi-index α, |α| = α_{1} + ⋯ + α_{m}, ∂_{x}^{α} = ∂^{|α|}/∂_{x_{1}}^{α_{1}} ⋯ ∂_{x_{m}}^{α_{m}}, and similarly for ∂_{ξ}^{α}.

Let D be an open set, D ⊆ R^{m}. By L_{2}(D) we denote the space of vector valued functions u = (u_{1}, …, u_{s}) (1 ≤ s < ∞) with the standard norm. In the usual way we define the function spaces C_{0}^{∞}(D), C^{∞}(E^{m}), etc.

If T is a compact selfadjoint operator in a Hilbert space H and E_{T} (·) its spectral measure, then N_{+}(μ, T) = dim E_{T} (μ, +∞)H, N_{−}(μ, T) = dim E_{T} (−∞, −μ)H. Everywhere in the following the values of the spectral parameters λ, μ ∈ (0, 1).

For s × s matrices F = {F_{ij}} we let |F|_{s} = sup_{i,j} |F_{ij}|.

By c, C we denote various constants.

2. Let G be an unbounded set in R^{m} (the case G = R^{m} is not excluded) and let χ_{G} be the characteristic function of G. We cover G by cubes. To be precise, for k ∈ Z^{m}, Q_{k} = {x : k_{i} < x_{i} < k_{i}+1 ∀ i}. We write G_{k} = G∩Q_{k}, K = {k : G_{k} ≠ ∅}, Q = ⋃_{k∈K} \overline{Q_{k}}.

Everywhere in what follows we make the following assumption about the smoothness of the boundary of G:

(1.1)  meas{x : dist(x, ∂G_{k}) < δ} ≤ c(G)δ  ∀δ > 0, ∀k ∈ K.

3. We now consider functions that are useful in describing the symbols of ΨDO. Let f(x, ξ) : Q × E^{m} → R_{+} such that ∀x, y ∈ Q, ∀ξ ∈ E^{m}

f(x, ξ)/f(y, ξ) ≤ c_{1} (1 + |x − y|^{l_{1}}), (1.2)

f(x, ξ) ≤ c_{2} (1 + |x| + |ξ|)^{−l_{2}}, l_{2} > 0, (1.2a)

where c_{i} = c_{i}(f), l_{i} = l_{i}(f).

We write φ(x) = sup f(x, ξ), ξ ∈ E^{m}; D(λ) = {x : x ∈ Q, φ(x) > λ}, V (λ, x) = meas{ξ : f(x, ξ) > λ}.

Further, U(λ, x) is a function defined for λ ∈ (0, 1), x ∈ Q. We assume that ∀t ∈ (0, 1)

U(λ, x) ≤ U(tλ, x) ≤ c_{3}t^{−ω}U(λ, x) ∀x ∈ Q; (1.3)

1 ≤ U(λ, x) ≤ c_{4}λ^{−l_{3}} ∀x ∈ D(λ); (1.3a)

c_{i} = c_{i}(U), l_{i} = l_{i}(U), ω = ω(U) > 0. The concrete values of the constants c_{i}, l_{i} are not essential in the following.

By definition the function f(x, ξ) belongs to the class Λ_{ω}(U) if the conditions (1.2)–(1.3a) hold and

(1.4)  V (λ, x) ≤ U(λ, x)  ∀x ∈ Q.

4. We shall describe the symbol of the ΨDO. Let the s × s matrix function A = A(x, y, ξ) ∈ C^{∞}(Q × Q × E^{m}), A^{∗}(x, y, ξ) = A(y, x, ξ). The eigenvalues of the matrix A(x, x, ξ) are denoted by a_{j}(x, ξ); a_{j+1}(x, ξ) ≥ a_{j}(x, ξ), j = 1, …, s − 1; and

v_{±}(λ, x) = ∑_{j=1}^{s} meas{ξ : ±a_{j}(x, ξ) > λ}.

By definition the matrix function A belongs to the class Σ_{ρ}(f, U), if the following conditions are satisfied:

|(∂_{ξ}^{α}∂_{x}^{β}∂_{y}^{γ}A)(x, y, ξ)|_{s} ≤ c(A, α, β, γ)(f(x, ξ) + f(y, ξ))(1 + |ξ|)^{−ρ|α|} (1.5)

∀x, y ∈ Q, ∀ξ ∈ E^{m}, ∀α, β, γ; f ∈ Λ_{ω}(U), ρ > 0; ∀t ∈ (0, \frac{1}{2}), ∀x ∈ Q,

v_{±}((1 − t)λ, x) ≤ v_{±}(λ, x) + c_{1}(A)tU(λ, x) + c_{2}(A). (1.6)

Condition (1.5) guarantees pseudo-locality of the ΨDO. The class Σ_{ρ}(f, U) may contain symbols, that do not satisfy Beals-Hörmander type inequalities. As an example, consider for m = s = 1 the symbol A(x, y, ξ) = (ξ^{2} + x^{2} + 1)^{−1}(ξ^{2} + y^{2} + 1)^{−1} sin x sin y. The function A ∈ Σ_{ρ}(f, U) for f(x, ξ) = (ξ^{2} + x^{2} + 1)^{−2}, U(λ, x) = λ^{−1/4}, ρ = 1.

5. Let the function A ∈ Σ_{ρ}(f, U), f ∈ Λ_{ω}(U). For a function u ∈ C_{0}^{∞}(G) we write

(⟨A⟩u)(x) = (2π)^{−m} ∫ dξ ∫ A(x, y, ξ)u(y) exp(i(x − y)ξ) dy, x ∈ R^{m};

(A^{(0)}u)(x) = (χ_{G} ⟨A⟩u)(x).

The operator A^{(0)} is extended by continuity to a compact selfadjoint operator A in the space L_{2}(G). Spectral bounds for the ΨDO A form the main contents of this work.

We shall write:

J_{±}(μ) = (2π)^{−m} ∫_{G} v_{±}(μ, x) dx, I_{a}(λ) = ∫_{D(λ)} U^{a}(λ, x) dx,

a ≥ 0, (clearly, I_{0}(λ) = meas D(λ)); σ_{0} = m(m + ρ)^{−1}, σ = m(ω + 1)^{2}(m(ω + 1)^{2} + ωρ)^{−1}. The quantity U(λ, x) is subject to the following conditions:

(1.7) I_{0}(λ) ≤ cλ^{−q_{0}}, I_{1}(λ) ≤ cλ^{−q}, I_{σ_{0}}(λ) ≤ cλ^{−p_{0}}, I_{σ}(λ) ≤ cλ^{−p}, c = c(U).

Moreover, we may assume that q_{0} ≤ p_{0} ≤ p ≤ q (cf. (1.3a)).

Theorem 1. Let A ∈ Σ_{ρ}(f, U), f ∈ Λ_{ω}(U), and let conditions (1.1) and (1.7) hold. Then for d = max(p, p_{0}(1 + p_{0})^{−1}(1 + q)), ∀ϵ > 0

(1.8)  |N_{±}(μ, A) − J_{±}(μ)| ≤ c(A, ϵ)μ^{−d−ϵ}.

Under the conditions p < q, J_{±}(μ) ≥ cμ^{−q} (c > 0) the estimate (1.8) isolates the principal term of the asymptotics of N_{±}(μ, A) (d < q). The smallest of the possible exponents d in (1.8) is obtained by the optimal choice of the function f(x, ξ) and the quantity U(λ, x) for a given symbol A(x, y, ξ).

§2. Proof of Theorem 1

1. We shall write χ_{k} for the characteristic function of the set G_{k}, and ∥ · ∥_{k} for the norm of the space L_{2}(Q_{k}), k ∈ K. Further, K(λ) = {k : k ∈ K, Q_{k} ∩ D(λ) ≠ ∅}. If k ∈ K, x ∈ Q_{k}, then in view of (1.2) φ(x) ≥ cλ, c = c(f) > 0. Therefore

(2.1)  Q_{k} ⊆ D(cλ), card K(λ) ≤ meas D(cλ).

We denote by Γ the set of points r ∈ Z^{m}, such that the cubes Q_{r} touch the cube Q_{0}, but do not coincide with it, and by Ω the set of points r ∈ Z^{m}, such that dist(Q_{r}, Q_{0}) ≥ 1.

2. Let k ∈ K, k + r ∈ K. In the cubes Q_{k}, Q_{k+r} we select points x_{k}, y_{k+r} in an arbitrary way. In view of conditions (1.5) and (1.2) ∀x ∈ Q_{k}, ∀y ∈ Q_{k+r} (2.2) |(∂_{ξ}^{α}∂_{x}^{β}∂_{y}^{γ}A)(x, y, ξ)|_{s} ≤ c(A, α, β, γ)(f(x_{k}, ξ) + f(y_{k+r}, ξ))(1 + |ξ|)^{−ρ|α|} ∀α, β, γ.

We write φ_{k} = φ(x_{k}).

Consider the following operators in the space L_{2}(G): A_{k} = χ_{k}⟨A⟩χ_{k}, T_{kr} = Re(χ_{k}⟨A⟩χ_{k+r}) (cf. no. 5 of §1).

Lemma 1. For all ϵ > 0 we have

|N_{±}(λ, A_{k}) − (2π)^{−m} ∫_{G_{k}} v_{±}(λ, x) dx| ≤ c(A, ϵ)λ^{−ϵ}U(λ, x_{k}); (2.3)

N_{±}(λ, A_{k}) ≤ [C (V (wλ, x_{k}) + (φ_{k}^{−1}λ)^{−ϵ})], C = C(A, ϵ), w = w(A, ϵ) > 0. (2.3a)

If r ∈ Γ, ϵ > 0, then

N_{±}(λ, T_{kr}) ≤ c(A, ϵ)λ^{−ϵ}U^{σ_{0}}(λ, x_{k}); (2.4)

N_{±}(λ, T_{kr}) ≤ [C (V (wλ, x_{k}) + (φ_{k}^{−1}λ)^{−ϵ})]. (2.4a)

Proof. If r ∈ Γ, k ∈ K, k + r ∈ K, then in view of (2.2), (1.2) ∀x ∈ Q_{k}, ∀y ∈ Q_{k+r}

(2.5)  |(∂_{ξ}^{α}∂_{x}^{β}∂_{y}^{γ}A)(x, y, ξ)|_{s} ≤ c(A, α, β, γ)f(x_{k}, ξ)(1 + |ξ|)^{−ρ|α|} ∀α, β, γ.

Let U(λ, x) = V (λ, x). Then in view of (1.3), (1.6) (2.6)

\begin{matrix}V (tλ, x_{k}) ≤ c(f)t^{−ω}V (λ, x_{k}) ∀t ∈ (0, 1); \\ v_{±}((1 − t)λ, x) ≤ v_{±}(λ, x) + c_{1}(A)tV (λ, x_{k}) + c_{2}(A) ∀x ∈ Q_{k}, ∀t ∈ (0, \frac{1}{2}). \end{matrix}

In view of (2.5), (2.6) and (1.1) inequalities (2.3)–(2.4a) follow from results in [5] (Lemmas 3,4,6, Theorem 1).

Let U(λ, x) ≥ V (λ, x). In this case we repeat the proofs of [5] replacing V (λ, x_{k}) by U(λ, x_{k}) in all remainder terms and using (1.3), (1.6). This leads to the bounds (2.3), (2.4). The proof of (2.3a), (2.4a) can be repeated in its entirety. □

Corollary. Let r ∈ Γ, k ∈ K, k +r ∈ K. Then there is a constant c^{∗} = c^{∗}(A) > 0 such that, if x_{k} ∉ D(c^{∗}λ), then N_{±}(λ, A_{k}) = N_{±}(λ, T_{kr}) = 0.

This statement follows from (2.3a), (2.4a) for any ϵ > 0.

Lemma 2. Let k ∈ K, k + r ∈ K, r ∈ Ω. Then ∀N, M ∈ N, M < 2ρN − m,

(2.7)  N_{±}(λ, T_{kr}) ≤ [c(A, N, M)(|r|^{2N}(φ_{k} + φ_{k+r})^{−1}λ)^{mM^{−1}}].

Proof. Let u ∈ L_{2}(G_{k}). For x ∈ Q_{k+r}, |α| ≤ M

(∂_{x}^{α}⟨A⟩χ_{k}u)(x) = ∫_{G_{k}} A_{α}(x, y)u(y) dy,
A_{α}(x, y) = c_{α} ∫ exp(i(x − y)ξ)|x − y|^{−2N}ξ^{α} dξ ((−Δ_{ξ})^{N}A)(x, y, ξ).

From estimate (2.2) follows that for y ∈ Q_{k}, x ∈ Q_{k+r}

|A_{α}(x, y)|_{s} ≤ c(A, N, M)(φ_{k} + φ_{k+r})|r|^{−2N}.

Therefore (2.8)

|(∂_{x}^{α}⟨A⟩χ_{k}u)(x)| ≤ c(A, N, M)(φ_{k} + φ_{k+r})|r|^{−2N}∥u∥_{k} ∀x ∈ Q_{k+r}, |α| ≤ M.

This inequality is true also with k and k + r interchanged.

We denote by W_{2}^{M}(Q_{k}) the Sobolev space with standard norm ∥ · ∥_{(M,k)} and scalar product (·, ·)_{(M,k)}, and D_{(M)} is the operator corresponding to the quadratic form ∥ · ∥_{k}^{2} in the space W_{2}^{M}(Q_{k}). It is well known that (cf., for example, [6])

(2.9)  N_{±}(μ, D_{(M)}) ≤ c(m, M)μ^{−m(2M)^{−1}}.

Let v = T_{kr}u, u ∈ L_{2}(Q_{k} ∪ Q_{k+r}). In view of (2.8) (2.10)

\frac{∥v∥_{k}^{2} + ∥v∥_{k+r}^{2}}{∥u∥_{k}^{2} + ∥u∥_{k+r}^{2}} ≤ c(A, N, M)\frac{φ_{k}^{2} + φ_{k+r}^{2}}{|r|^{4N}} \frac{(D_{(M)}v, v)_{(M,k)} + (D_{(M)}v, v)_{(M,k+r)}}{∥v∥_{(M,k)}^{2} + ∥v∥_{(M,k+r)}^{2}} .

The statement of the lemma follows from (2.10) and (2.9). □

3. We represent the operator A in the following form:

(2.11)  \begin{matrix}A = Ã + T_{Γ} + T_{Ω}, \\ Ã = ∑_{K}^{(k)} ⊕ A_{k}, T_{Γ} = ∑_{Γ}^{(r)}∑_{K}^{(k)} ⊕ T_{kr}, T_{Ω} = ∑_{Ω}^{(r)}∑_{K}^{(k)} ⊕ T_{kr}. \end{matrix}

(If k + r ∉ K, then the corresponding operator T_{kr} = 0.)

Lemma 3. For all ϵ > 0 we have

(2.12)  |N_{±}(λ, Ã) − J_{±}(λ)| ≤ c(A, ϵ)λ^{−p−ϵ}.

Proof. We bound the spectrum of Ã from above. We note that in (2.3)–(2.4a) the choice of the points x_{k} ∈ Q_{k} is arbitrary. In particular, if k ∉ K(c^{∗}λ) (see the corollary of Lemma 1), then we may assume that in (2.3a) x_{k} ∉ D(c^{∗}λ), so that N_{±}(λ, A_{k}) = 0. Therefore,

N_{±}(λ, Ã) = ∑_{K}^{(k)}N_{±}(λ, A_{k}) = ∑_{K(c^{∗}λ)}^{(k)}N_{±}(λ, A_{k}).

By virtue of (2.3) ∀ϵ > 0

N_{±}(λ, Ã) ≤ (2π)^{−m}∑_{K(c^{∗}λ)}^{(k)} ∫_{G_{k}} v_{±}(λ, x) dx + c(A, ϵ)λ^{−ϵ}∑_{K(c^{∗}λ)}^{(k)}U^{σ}(λ, x_{k}).

Using (2.1) and with appropriately chosen points x_{k} ∈ Q_{k}, we conclude that

(2.13)  ∑_{K(c^{∗}λ)}^{(k)}U^{σ}(λ, x_{k}) ≤ I_{σ}(cλ), c = c(A, f) > 0.

Hence, taking condition (1.7) into account, ∀ϵ > 0

(2.14)  N_{±}(λ, Ã) ≤ J_{±}(λ) + c_{1}(A, ϵ)λ^{−p−ϵ}.

We now bound the quantity N_{±}(λ, Ã) from below. From (1.5) follows that ∀x ∈ D(Cλ) v_{±}(λ, x) = 0, C = C(A) > 0. Therefore

(2π)^{−m}∑_{K(Cλ)}^{(k)} ∫_{G_{k}} v_{±}(λ, x) dx = J_{±}(λ).

Moreover, N_{±}(λ, Ã) ≥ ∑_{K(Cλ)}^{(k)}N_{±}(λ, A_{k}). Using (2.3) we obtain that ∀ϵ > 0

N_{±}(λ, Ã) ≥ J_{±}(λ) − c(A, ϵ)∑_{K(Cλ)}^{(k)}U^{σ}(λ, x_{k})λ^{−ϵ}.

Analogously to (2.13), (2.14) we arrive at the estimate

(2.15)  N_{±}(λ, Ã) ≥ J_{±}(λ) − c_{2}(A, ϵ)λ^{−p−ϵ}  ∀ϵ > 0.

The statement of the lemma follows from (2.14) and (2.15). □

Lemma 4. For all ϵ > 0 we have

(2.16)  N_{±}(λ, T_{Γ}) ≤ c(A, ϵ)λ^{−p_{0}−ϵ}.

Proof. By Weyl’s inequality

N_{±}(λ, T_{Γ}) ≤ ∑_{Γ}^{(r)}∑_{K}^{(k)}N_{±}(c_{m}λ, T_{kr}), c_{m} = (card Γ)^{−1}.

The estimate of ∑_{K}^{(k)}N_{±}(c_{m}λ, T_{kr}), r ∈ Γ, reiterates the estimate from above of N_{±}(λ, Ã) (cf. the proof of Lemma 3). Instead of inequality (2.3) one uses (2.4). This proves the lemma. □

Lemma 5. For all ϵ > 0 we have

(2.17)  N_{±}(λ, T_{Ω}) ≤ c(A, ϵ)λ^{−q_{0}−ϵ}.

Proof. Using Weyl’s inequality we obtain that

(2.18) N_{±}(λ, T_{Ω}) ≤ ∑_{Ω}^{(r)}∑_{K}^{(k)}N_{±}(c_{m}^{∗}|r|^{−m−1}λ, T_{kr}), c_{m}^{∗} = (∑_{Ω}^{(r)}|r|^{−m−1})^{−1} .

In view of estimate (2.7) and condition (1.2)

(2.19) N_{±}(c_{m}^{∗}|r|^{−m−1}λ, T_{kr}) ≤ [c(A, N, M)|r|^{−mM^{−1}(2N−l_{1}−m−1)}(φ_{k}λ)^{−mM^{−1}}],

M < 2ρN − m. Here we put N = [2^{−1}((M + m)(1 + ρ^{−1}) + l_{1} + 3)]. The right side of (2.19) equals zero ∀k ∉ D(cλ), c = c(A, M) > 0. Hence from (2.18), (2.19) follows that

N_{±}(λ, T_{Ω}) ≤ c(A, M) (∑_{Ω}^{(r)}|r|^{−m−κ}) ∑_{D(cλ)}^{(k)}(φ_{k}λ)^{−mM^{−1}}, κ > 0.

Applying this bound for “large” M we arrive at

N_{±}(λ, T_{Ω}) ≤ C(A, ϵ) card K(cλ) ∀ϵ > 0, c = c(A, ϵ).

The statement of the lemma now follows from (2.1) and (1.7). □

4. We complete the proof of Theorem 1. In view of (2.11) ∀t ∈ (0, \frac{1}{2})

(2.20)  N_{±}(μ, A) ≤ N_{±}((1 − t)μ, Ã) + N_{±}(tμ, A_{Γ} + A_{Ω}).

Using (2.12) we conclude that ∀ϵ > 0

N_{±}((1 − t)μ, Ã) ≤ J_{±}((1 − t)μ) + c(A, ϵ)μ^{−p−ϵ}.

Note that ∀x ∉ D(cμ) v_{±}((1 − t)μ, x) = 0, c = c(A) > 0. Hence

J_{±}((1 − t)μ) = (2π)^{−m} ∫_{D(cμ)} χ_{G} (x)v_{±}((1 − t)μ, x) dx.

By (1.6) and (1.7)

∫_{D(cμ)} χ_{G} (x)v_{±}((1 − t)μ, x) dx
≤ (2π)^{m}J_{±}(μ) + c_{1}(A)t ∫_{D(cμ)} U(μ, x) dx + c_{2}(A) meas D(cμ)
≤ (2π)^{m}J_{±}(μ) + c(A)(tμ^{−q} + μ^{−q_{0}}).

Finally, recalling that q_{0} ≤ p, we obtain the following bound: ∀ϵ > 0

N_{±}((1 − t)μ, Ã) ≤ J_{±}(μ) + c(A, ϵ)(tμ^{−q} + μ^{−p}).

The quantity N_{±}(tμ, A_{Γ} + A_{Ω}) is estimated using (2.16) and (2.17). Since q_{0} ≤ p_{0}, we have

N_{±}(tμ, A_{Γ} + A_{Ω}) ≤ c(A, ϵ)t^{−p_{0}−ϵ} ∀ϵ > 0.

Hence, from (2.20) follows that

N_{±}(μ, A) ≤ J_{±}(μ) + c(A, ϵ)(tμ^{−q} + t^{−p_{0}}μ^{−p_{0}} + μ^{−p})μ^{−ϵ} ∀ϵ > 0. Minimizing the right-hand side of this inequality with respect to t, we obtain

(2.21)  N_{±}(μ, A) ≤ J_{±}(μ) + c(A, ϵ)μ^{−d−ϵ}  ∀ϵ > 0.

Further, we note that N_{±}(μ, A) ≥ N_{±}((1 + t)μ, Ã) − N_{∓}(tμ, A_{Γ} + A_{Ω}), t ∈ (0, \frac{1}{2}). Repeating the above argument we establish the following estimate:

(2.22) N_{±}(μ, A) ≥ J_{±}(μ) − c(A, ϵ)μ^{−d−ϵ} ∀ϵ > 0.

The statement of the theorem follows from (2.21) and (2.22).

§3. Quasiregular spectral asymptotics of ΨDO

1. In this paragraph the function f(x, ξ), occurring in the definition of the symbol of the ΨDO (see no. 3 of §1), will be represented as a product of two factors depending on x and ξ, respectively.

More precisely, f(x, ξ) = φ(x)g(ξ), where φ(x) : Q → R_{+} and g(ξ) : E^{m} → R_{+}. We shall write: D(λ) = {x ∈ Q : φ(x) > λ}, I_{0}(λ) = meas D(λ), v(λ) = meas{ξ : g(ξ) > λ}. The functions φ(x), g(ξ) are subject to the following conditions:

φ(x) ≤ c_{1} (1 + |x|)^{−l_{1}}; φ(x)/φ(y) ≤ c_{2} (1 + |x − y|^{l_{2}}), (3.1)

∀x, y ∈ Q, c_{i} = c_{i}(φ), l_{i} = l_{i}(φ), l_{1} > 0;

I_{0}(λ) ≤ c_{3}λ^{−q_{0}}, c_{3} = c_{3}(φ); (3.2)

g(ξ) ≤ (1 + |ξ|)^{−l_{3}}, l_{3} = l_{3}(g) > 0; (3.3)

c_{4}λ^{−ω} ≤ v(λ) ≤ c_{5}λ^{−ω}, ω = ω(g) > 0, (3.4)

c_{i} = c_{i}(g) > 0. The concrete values of the constants c_{i}, l_{i} are not essential in the following.

Clearly V (λ, x) = v(λ/φ(x)). Put U(λ, x) = V (λ, x) + 1. From (3.1)–(3.4) follows that conditions (1.2)–(1.3a) are satisfied. The function f ∈ Λ_{ω}(U).

2. Consider the ΨDO A with symbol A ∈ Σ_{ρ}(f, U) in the space L_{2}(G) (cf. nos. 4, 5 of §1).

Theorem 2. Let the conditions (1.1) and (3.1)–(3.4) hold with ω > q_{0}. Then for d_{0} = max(q_{0}(1 + ω)(1 + q_{0})^{−1}, σ_{0}ω(1 + ω)(1 + σ_{0}ω)^{−1}, σω), ∀ϵ > 0

(3.5)  |N_{±}(μ, A) − J_{±}(μ)| ≤ c(A, ϵ)μ^{−d_{0}−ϵ}.

Proof. We will show that ∀a ≥ 0

(3.6)  I_{a}(λ) ≤ c(f, a)(λ^{−aω} + λ^{−q_{0}})(1 + | ln λ|).

For n = 0, 1, … we write: D_{n}(λ) = D(2^{n}λ)\D(2^{n+1}λ). Hence D(λ) = ⋃_{n=0}^{N(λ)} D_{n}(λ), N(λ) = c(φ)(1 + | ln λ|). From (3.4) follows that U(λ, x) ≤ c(f)(φ^{ω}(x)λ^{−ω} + 1). Therefore

(3.7)  I_{a}(λ) ≤ c_{1}(a, f) (λ^{−aω} ∑_{n=0}^{N(λ)} ∫_{D_{n}(λ)} φ^{aω}(x) dx + I_{0}(λ)) .

Further, in view of (3.2), (3.8)

∫_{D_{n}(λ)} φ^{aω}(x) dx ≤ (2^{n+1}λ)^{aω}I_{0}(2^{n}λ) ≤ c(f, a, ω)(2^{n(aω−q_{0})}λ^{aω−q_{0}} + 2^{naω}λ^{aω}).

Inequality (3.6) follows from (3.7), (3.8).

From (3.6) follows that conditions (1.7) are satisfied for q = ω + δ, p_{0} = max(σ_{0}ω, q_{0}) + δ, p = max(σω, q_{0}) + δ ∀δ > 0. Hence the hypotheses of Theorem 1 are satisfied and (3.5) is a special case of (1.8). □

3. Let for some point x_{0} ∈ G, and some j

(3.9) ±λ_{j}(x_{0}, x_{0}, ξ) ≥ cg(ξ), c > 0.

We construct the ball T_{δ} = {x : |x − x_{0}| < δ}, δ < dist(x_{0}, ∂G). Using (1.5) it is easy to show that, for sufficiently small δ, ±λ_{j}(x, x, ξ) ≥ cg(ξ)/2 ∀x ∈ T_{δ}. Hence v_{±}(μ, x) ≥ v(2c^{−1}μ) ∀x ∈ T_{δ}. By (3.4)

(3.10)  ∫_{T_{δ}} v_{±}(μ, x) dx ≥ c(x_{0}, A)v(μ),  c(x_{0}, A) > 0.

If condition (3.9) holds and ω > q_{0}, then the quantities N_{±}(μ) and v(μ) agree in order. In fact, we note that the function φ^{ω}(x) is integrable in Q (the proof of this differs from the proof of (3.6) only by simplifications). Hence, in view of (1.5), (3.4),

J_{±}(μ) ≤ ∫ V (c_{1}μ, x) dx ≤ c_{2} (∫_{Q} φ^{ω}(x) dx) v(μ) ≤ c_{3}v(μ), c_{i} = c_{i}(A), c_{1} > 0.

Simultaneously, by (3.10), J_{±}(μ) ≥ c_{4}v(μ), c_{4} > 0. Since d_{0} < ω, estimate (3.5) isolates the principal term of the spectral asymptotics of the operator A.

Consider the ΨDO A_{δ} with symbol A(x, y, ξ) in the space L_{2}(T_{δ}). In view of the results of [5]

N_{±}(μ, A_{δ}) ∼ (2π)^{−m} ∫_{T_{δ}} v_{±}(μ, x) dx, μ → +0.

Thus, the spectral asymptotics of the operator A and the operator A_{δ} of the regular problem agree in order. We remark that condition (3.9) is guaranteed by a sufficiently precise choice of the functions φ(x), g(ξ) for the symbol A(x, y, ξ). The inequality ω > q_{0} plays the role of a regularity condition for the spectral asymptotics of the ΨDO A.

4. We shall now apply the result of Theorem 2 to ΨDO with symbols that are anisotropically homogeneous with respect to the variable ξ.

We write: Ξ_{δ} = {ξ : |ξ| > δ}, δ > 0, and we let h(ξ) be a cut-off function: h ∈ C^{∞}(E^{m}), 0 ≤ h(ξ) ≤ 1, h(ξ) = 0, if |ξ| < 1, and h(ξ) = 1, if |ξ| > 2. Further, τ = (τ_{1}, …, τ_{m}), τ_{j} > 0, τ_{1} +⋯+τ_{m} = m, W_{τ} (t) = {t^{τ_{j}}δ_{ij}}, i, j = 1, …, m, t > 0.

Let the s × s matrix function A_{0}(x, y, ξ) be defined for x, y ∈ Q, ξ ∈ E^{m} \ {0}, with A_{0}^{∗}(x, y, ξ) = A_{0}(y, x, ξ). The function A_{0} is assumed to be τ-homogeneous, i.e., ∀t > 0 A_{0}(x, y, W_{τ} (t)ξ) = t^{−l}A_{0}(x, y, ξ), l > 0. Moreover,

|(∂_{ξ}^{α}∂_{x}^{β}∂_{y}^{γ}A_{0})(x, y, ξ)|_{s} ≤ c(A_{0}, α, β, γ, δ)(φ(x) + φ(y)) (A)

∀x, y ∈ Q, ∀ξ ∈ Ξ_{δ}, ∀δ > 0, ∀α, β, γ.

The function φ(x) satisfies conditions (3.1) and (3.2).

The eigenvalues of the matrix A_{0}(x, x, ξ) are denoted by a_{j}^{(0)}(x, ξ). Moreover,

v_{±}^{(0)}(x) = ∑_{j=1}^{s} meas{ξ : ±a_{j}^{(0)}(x, ξ) > 1}, C_{±}^{(0)} = (2π)^{−m} ∫_{G} v_{±}^{(0)}(x) dx.

We shall write ω = ml^{−1}, ρ_{0} = min_{i,j}(τ_{i}τ_{j}^{−1}), σ_{0} = m(m + ρ_{0})^{−1}, σ = (m + l)^{2}((m + l)^{2} + lρ_{0})^{−1}, d_{0} = max(q_{0}(ω + 1)(q_{0} + 1)^{−1}, σ_{0}ω(m + l)(l + mσ_{0})^{−1}, ωσ).

We consider in the space L_{2}(G) a ΨDO A with symbol A(x, y, ξ) = h(ξ)A_{0}(x, y, ξ). In the case G = R^{m}, ω > 1, the principal term of the spectral asymptotics of this operator was established in [1].

Theorem 3. Let conditions (1.1) and (A) hold with q_{0} < ω. Then ∀ϵ > 0

(3.11)  |N_{±}(μ, A) − C_{±}^{(0)}μ^{−ω}| ≤ c(A, ϵ)μ^{−d_{0}−ϵ}.

Proof. We write

g_{n}(ξ) = (∑_{j=1}^{m} |mξ_{j}|^{nτ_{j}^{−1}} )^{−n^{−1}l} , n > 0.

Clearly, ∀ξ ∈ Ξ_{δ}, ∀α, |α| < nm^{−1}

(3.12)  |(∂_{ξ}^{α}g_{n})(ξ)| ≤ c(n, τ, δ)g_{n}(ξ)(1 + |ξ|)^{ρ_{0}α}, g_{n}(ξ) ≤ c(n, τ)g_{1}(ξ).

We represent the function A_{0}(x, y, ξ) as A_{0}(x, y, ξ) = A_{0}(x, y, W_{τ} (t)ξ)g_{n}(ξ), t = g_{n}^{l^{−1}}(ξ). Here W_{τ} (t)ξ ∈ Ξ_{δ}, δ = δ(τ, n) > 0. Using this representation for sufficiently large n, and inequalities (A), (3.12), we obtain

|(∂_{ξ}^{α}∂_{x}^{β}∂_{y}^{γ}A)(x, y, ξ)|_{s} ≤ c(A, α, β, γ)(φ(x) + φ(y))g_{1}(ξ)(1 + |ξ|)^{−ρ_{0}|α|} (3.13)

∀x, y ∈ Q, ∀ξ ∈ E^{m}, ∀α, β, γ.

Hence, it follows that

(3.14)  v ±^{(0)} (x) ≤ c(A)φ^{ω}(x),  x ∈ Q.

Moreover,

(3.15) v_{±}(λ, x) = v_{±}^{(0)}(x)λ^{−ω} + O(1), v_{±}(λ, x) = 0 ∀x ∉ D(wλ), w = w(A) > 0.

We put c_{τ} = max g_{1}(ξ), ξ ∈ Ξ_{1} and g(ξ) = (1 + c_{τ})^{−1}(h(ξ)g_{1}(ξ) + 1 − h(ξ)). The function g(ξ) clearly satisfies (3.3) and (3.4). Let f_{τ} (x, ξ) = φ(x)g(ξ), U_{τ} (λ, x) = c^{∗}(A)φ^{ω}(x)λ^{−ω} + 1. In view of (3.13)–(3.15) the symbol A ∈ Σ_{ρ_{0}}(f_{τ} , U_{τ}) for the appropriate value of the constant c^{∗}(A). From (3.14) and (3.15) follows that

J_{±}(μ) = C_{±}^{(0)}μ^{−ω} + R_{±}(μ),
|R_{±}(μ)| ≤ c(A) (I_{0}(wμ) + μ^{−ω} ∫_{Q\D(wμ)} φ^{ω}(x) dx) .

It is easy to see (cf. the proof of (3.6)) that

∫_{Q\D(wμ)} φ^{ω}(x) dx ≤ c(φ, w)μ^{ω−q_{0}},

whence |R_{±}(μ)| ≤ c(A)μ^{−q_{0}}. The estimate (3.11) now follows as a consequence of Theorem 2. □

5. We now consider the case, where the symbol A(x, y, ξ) is isotropically homogeneous with respect to the variable ξ, while the function φ(x) has power order: A(x, y, tξ) = t^{−l}A(x, y, ξ) ∀ξ ∈ Ξ_{2}, ∀t > 0, and φ(x) = (1 + |x|)^{−l_{0}}, l > 0, l_{0} > 0. Clearly here τ_{j} = 1 ∀j, ρ_{0} = 1. The exponent d_{0} in (3.11) is defined for σ_{0} = m(m + 1)^{−1}, σ = (m + l)^{2}((m + l)^{2} + l)^{−1}.

If G = R^{m}, then q_{0} = ml_{0}^{−1} and the regularity condition of the spectral asymptotics takes the form l_{0} > l.

Let G be a parabolic domain in R^{m}: G = {x : x_{m} > ∑_{j=1}^{m−1} x_{j}^{2}}. Then q_{0} = (2l_{0})^{−1}(m + 1). The regularity condition is improved: l_{0} > (2m)^{−1}(m + l)l. The form of the asymptotics remains the same as before with the corresponding coefficient C_{±}^{(0)} = C_{±}^{(0)}(A, G).

§4. On a class of ΨDO with nonregular spectral asymptotics

1. We now consider some compact ΨDO occurring in the study of the negative spectrum of the Schrödinger equation.

Let the functions P (x), F (x), T (x) be defined ∀x ∈ Q, and P (x) ≥ 1, F (x) ∈ R, T (x) > 0. It is assumed that F, P ∈ C^{∞}(Q), with

|(∂_{x}^{α}P)(x)| ≤ c(P, α)P (x), |(∂_{x}^{α}F)(x)| ≤ c(F, α)T (x), (4.1)

P (x) ≤ c(P)(1 + |x|)^{l_{1}}, T (x) ≤ c(T)(1 + |x|)^{l_{2}}, ∀x ∈ Q, ∀α, l_{i} = l_{i}(P, T).

We write: φ(x) = T (x)P^{−b}(x), b > 0. As before, D(λ) = {x : x ∈ Q, φ(x) > λ}, I_{0}(λ) = meas D(λ), and

(4.2)  \begin{matrix}φ(x)/φ(y) ≤ c(φ)(1 + |x − y|^{l_{3}}), φ(x) ≤ c(φ)(1 + |x|)^{−l_{4}}, \\ l_{i} = l_{i}(φ), l_{4} > 0, x, y ∈ Q; I_{0}(λ) ≤ c(φ)λ^{−q_{0}}. \end{matrix}

The concrete values of the exponents l_{i} in (4.1) and (4.2) are not essential. Let

A(x, y, ξ) = 2^{−1}(|ξ|^{2a} + P (x))^{−b/2}(F (x) + F (y))(|ξ|^{2a} + P (y))^{−b/2}, a ∈ N.

We consider a scalar ΨDO with symbol A(x, y, ξ) in the space L_{2}(G). This operator is compact in view of (4.2).

2. We shall write: ω = (2ab)^{−1}m, σ_{0} = m(m + 1)^{−1}, σ = m(ω + 1)^{2}(m(ω + 1)^{2} + ω)^{−1}. In addition, the function T (x) is subject to the following conditions:

(4.3)  \begin{matrix}∫_{D(λ)} T^{ω}(x) dx ≤ cλ^{−w}, ∫_{D(λ)} T^{σ_{0}ω}(x) dx ≤ cλ^{−r_{0}}, \\ ∫_{D(λ)} T^{σω}(x) dx ≤ cλ^{−r},  c = c(T). \end{matrix}

We write: q = max(ω + w, q_{0}), p_{0} = max(r_{0} + ωσ_{0}, q_{0}), p = max(r + ωσ, q_{0}), d = max(p, p_{0}(1 + p_{0})^{−1}(1 + q)).

Theorem 4. Let conditions (1.1) and (4.1)–(4.3) hold, and let m ≥ 2a. Then ∀ϵ > 0

(4.4)  |N_{±}(μ, A) − J_{±}(μ)| ≤ c(A, ϵ)μ^{−d−ϵ}.

Proof. Put f(x, ξ) = (|ξ|^{2a} + P (x))^{−b}T (x), U(λ, x) = c_{m}λ^{−ω}T^{ω}(x) + 1, c_{m} = m^{−1} meas S^{m−1}. Clearly, V (λ, x) = c_{m}λ^{−ω}(T^{b^{−1}}(x) − λ^{b^{−1}}P (x))_{+}^{ωb}. Condition (1.2) holds for f(x, ξ) in view of (4.2). Since (1.3)–(1.4) are also satisfied, we have f ∈ Λ_{ω}(U). From (4.1) follows that inequality (1.5) is true for ρ = 1.

Moreover, v_{±}(λ, x) = c_{m}λ^{−ω}(F_{±}^{b^{−1}}(x) − λ^{b^{−1}}P (x))_{+}^{ωb}, ωb = (2a)^{−1}m ≥ 1. It is easy to see that

(\frac{∂v_{±}}{∂λ} ) (λ, x) ≤ c(A)λ^{−ω−1}F_{±}^{ω}(x) ≤ c(A)λ^{−ω−1}T^{ω}(x),

so that (1.6) holds. Hence, the symbol A ∈ Σ_{1}(f, U). Inequality (1.7) is satisfied in view of (4.3). The bound (4.4) is a consequence of (1.8). □

3. The quantity J_{±}(μ) = c_{m}μ^{−ω}Ψ_{±}(μ), where

Ψ_{±}(μ) = ∫_{G}(F_{±}^{b^{−1}}(x) − μ^{b^{−1}}P (x))_{+}^{ωb} dx.

If the function F_{±}^{ω}(x) is integrable in G, then Ψ_{±}(μ) = c_{±}(A, G) + o(1), μ → +0. Under the condition q_{0} < ω the spectral asymptotics of the operator A becomes regular: N_{±}(μ, A) ∼ c_{±}c_{m}μ^{−ω}.

Let the function F_{±}^{ω}(x) be nonintegrable in G, and Ψ_{±}(μ) ≥ cμ^{−ω_{0}}, ω_{0} > 0, c > 0. If d < ω + ω_{0}, then the estimate (4.4) isolates the principal term of the spectral asymptotics. These asymptotics are different in order than the regular ones.

4. We consider the case of homogeneous “potentials” P (x) and F (x). More precisely, G = R^{m}, z = |x|^{−1}x ∈ S^{m−1}, P (x) = h(x)P(z)|x|^{l} + 1 − h(x/2), F (x) = h(x)F(z)|x|^{n}. Here P, F ∈ C^{∞}(S^{m−1}), with P(z) ≥ 1, F(z) ∈ R, z ∈ S^{m−1}, and h(x) is a cut-off function, analogous to the one introduced in no. 4 of §3. Concerning the orders of homogeneity l, n, we assume that l ≥ 0, n < bl, ωn > −m. As before, m ≥ 2a.

Put T (x) = h(x)|x|^{n}+1−h(x). Clearly, condition (4.1) is then satisfied, and (4.2) holds for q_{0} = m(bl−n)^{−1}. Inequality (4.3) is true for w = (m+ωn)(bl−n)^{−1}, r_{0} = (m+σ_{0}ωn)(bl −n)^{−1}, r = (m+σωn)(bl −n)^{−1}. Calculation of the quantity Ψ_{±}(μ) leads to the following result: Ψ_{±}(μ) = c_{±}(A)μ^{−ω_{0}} +O(1), ω_{0} = (m+lωb)(bl −n)^{−1} and

c_{±}(A) = b(bl − n)^{−1}B((ωn + m)(bl − n)^{−1}, ωb + 1) ∫_{S^{m−1}}(F_{±}(z)P^{−b}(z))^{ω_{0}} ds(z),

where B(·, ·) is the beta function.

Since in the case under consideration d < ω + ω_{0}, the bound (4.4) isolates the spectral asymptotics of the operator A:

N_{±}(μ, A) ∼ c_{±}(A)c_{m}μ^{−ω−ω_{0}} + O(μ^{−d−ϵ}) ∀ϵ > 0.

References

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2. S. Z. Levendorskiĭ, The method of approximate spectral projection, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), 1177–1227; Math. USSR Izv. 27 (1986).

3. M. Dauge and D. Robert, Formule de Weyl pour une classe d’opérateurs pseudodifférentiels d’ordre négatif sur L^{2}(R^{n}), C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 175–178.

4. M. Dauge and D. Robert, Weyl’s formula for a class of pseudodifferential operators with negative order on L^{2}(R^{n}), Pseudodifferential Operators (Oberwolfach 1986), Lecture Notes in Math., vol. 1256, Springer-Verlag, 1987, pp. 91–122.

5. A. S. Andreev, Asymptotics of the spectrum of compact pseudodifferential operators in a Euclidean domain, Mat. Sbornik 137 (1988), 202–223; Math. USSR Sbornik 65 (1990).

6. M. Sh. Birman and M. Z. Solomyak, Quantitative analysis in the Sobolev imbedding theorems and applications to spectral theory, X Mathematical School, Kiev, Naukova Dumka, 1974, pp. 5–189.

Translated by S. LARSSON

Higher Naval College of Radioelectronics Leningrad