SPECTRAL ASYMPTOTICS OF

PSEUDODIFFERENTIAL OPERATORS

K. Kh. Boimatov

1. We consider the symmetric operator

(1)  display

in the space L_{2}(R_{n}) with domain of definition D(P_{0}) = C_{0}^{∞}(R_{n}).

We assume that P (x, s) ∈ C^{∞}(R_{n} × R_{n}) is a real symbol. Let the following conditions hold:

a) (1 + |s|)^{a} ≤ MP (x, s), M, a > 0,

b) |D_{x}^{α}D_{s}^{β}P (x, s)| ≤ M_{αβ}P^{1−θ|β|}(x, s) for all α, β with some θ > 0,

c) P (x, s) ≤ MP (y, s) for all x, y, s ∈ R_{n}. Under these assumptions the closure P of P_{0} is (cf. [1, § 4]) a self-adjoint operator which is semi-bounded from below. In this note we study the asymptotic behavior of the spectral function e(λ, x, y) of the operator P. The results obtained are applied to the study of the asymptotic functions for the distribution of eigenvalues of pseudodifferential operators defined in domains with finite measure. The state of the art of research on the spectral asymptotics of differential and pseudodifferential operators is presented in the works [2,3].

We use the notation

(A)  display

Let l(λ) be a positive non-decreasing function such that l(2λ) = O(1)l(λ) (λ → +∞) and for some η ∈ R_{n} the inequality

(B)  display

holds.

Theorem 1. There exist numbers N, ϵ, λ_{0} > 0, σ ∈ (0, 1) such that for λ ≥ λ_{0} we have the bound

(2)  display

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 35S05, 35P20.

2. We give a sketch of the proof. For simplicity we assume that a > 2n, 8ρn < 1 (for the general case see no. 3 below). Let θ_{i}, ψ_{ij}, φ_{ij}, x_{ij} be as in 1,2, § 4 of [1]. We put

(3)  display

where G_{ij}(z) = exp(−zP_{ij}), and P_{ij} is the pseudodifferential operator in L_{2}(R_{n}) with “constant” symbol P (x_{ij}, s).

Lemma 1. (cf. Theorem 4.1 of [1]) There exists a number κ ∈ (0, 1) such that, with arbitrary σ ∈ (κ, 1), for some c > 0 the integral representation

(4)  display

holds in the region

(C)  display

where the integral is taken over a rectilinear segment, and the operator function Φ(z) satisfies the bound |Φ(z)| = O(| Re z|^{−κ}) in the region W_{σ,c}.

From the conditions a > 2n, 8ρn < 1 is not hard to deduce (see [1, § 4]) that |(−Δ)^{n}G_{0}(z)| ≤ M| Re z|^{−1/2}. Hence and from (4) follows the estimate

(D)  display

By a well-known criterion we conclude that the operators G_{0}(z), G(z) = exp(−zP),

(E)  display

z ∈ W_{σ,c}, are integral operators.

For an integral operator T with kernel T (x, y) we put

(F)  display

It is obvious that a(T_{2})a(T_{1}^{∗}) ≥ |(T_{1}T_{2})(x, y)|.

Let R^{∗}(z, x, y) be the kernel of the operator

(G)  display

Let the point y ∈ R_{n} be fixed. Then the expression R^{∗}(z, x, y) may be treated as an integral of the result of the action of the operator Φ^{∗}(w) (at the point x) on the function G_{0}^{∗}(z − w, ·, y). Therefore

(H)  display

Hence and from the estimate |Φ^{∗}(w)| ≤ M| Re z|^{−κ} follows that

(I)  display

Since G(z) = G(z/2)G^{∗}(z̄/2), we have

(J)  display

Thus, for the kernel F (z, x, y) of the operator F (z) = G(z) − G_{0}(z/2)G_{0}^{∗}(z̄/2) we have the bound

(K)  display

From the obtained inequalities we conclude that

(L)  display

The kernel of the operator G_{0}(z/2)G_{0}^{∗}(z̄/2) is estimated by means of formula (3), which gives explicit expressions for the operators G_{0}(z/2), G_{0}^{∗}(z̄/2). Finally, we obtain the following bound:

(M)  display

where G(z, x, y) is the kernel of the operator G(z) = exp(−zP),

(N)  display

We remark that

(O)  display

Since e(λ, x, x), θ(λ, x, x) are non-decreasing functions, if we apply the appropriate Tauberian theorem, we obtain the bound (2) for x = y. The proof of inequality (2) for x ≠ y is based on applying Tauberian theorems to the Laplace-Stieltjes transforms of the non-decreasing functions

(P)  display

where θ_{∗}(λ, x, y) = (θ(λ, x, y)+\overline{θ(λ, x, y)})/2. The Laplace-Stieltjes transform of the function θ_{∗}(λ, x, y) − θ(λ, x, y) satisfies a sufficiently good bound in some parabolic region W_{σ,c}, σ ∈ (0, 1) c > 0.

3. In the case, when the inequalities a > 2n, 8ρn < 1 do not hold simultaneously, we consider the operator P^{k}, where k is an even number such that ak > 2n, 8ρn < k. Let P ≥ E. Using convolution theorems we represent the operator P^{k} in the form P^{k} = P + Q, where Q is a bounded operator, and P is a pseudodifferential operator, whose Weyl symbol P(x, s) can be computed in explicit form. For the symbol P(x, s) there hold conditions analogous to a)–c), with the parameters ak, ρ/k instead of a, ρ.

The spectral functions e_{k}(λ, x, y), e(λ, x, y) of the operators P^{k}, P are connected by the relation

(Q)  display

The symbols P (x, s), P(x, s) satisfy a bound of the form

(R)  display

where M, λ_{0} > 0, 0 < ν < 1.

By studying the kernel of the operator function exp(−zP^{k}) − exp(−zP), one obtains a bound for the difference between the spectral functions of the operators P^{k}, P.

4. Let the conditions a)–c) hold for the symbol P (x, s). We consider the operator P′_{Ω} = χ_{Ω}P_{0}, D(P′_{Ω}) = C_{0}^{∞}(Ω), where the operator P_{0} is defined by formula (1), Ω ⊂ R_{n} is an open set, meas Ω < +∞, and χ_{Ω} is the characteristic function of the set Ω.

Since P_{0} is a symmetric operator, semi-bounded from below, the operator P′_{Ω} is also symmetric and semi-bounded from below.

We denote by P_{Ω} the Friedrichs extension of the operator P′_{Ω}. We put

(S)  display

We assume that for any number σ ∈ (0, 1) and for all x ∈ Ω

(T)  display

where ω_{σ}(λ) = o(1), λ → +∞. We also assume that there exists a positive nondecreasing function ψ(λ) ∈ C^{1}(0, +∞) such that

(5)  display

Theorem 2. Under the above conditions the operator P_{Ω} has discrete spectrum. For the function N(λ), the number of eigenvalues of the operator P_{Ω} not exceeding λ, we have the following asymptotic formula

(U)  display

5. We give a sketch of the proof. Applying Berezin’s theorem and Theorem 1 as in [4], we establish the estimate

(6)  display

For this it is necessary to take l(λ) = φ(λ, Ω) in Theorem 1.

In order to obtain a bound from below we fix a number ϵ > 0 and consider a compact set K_{ϵ} ⊂ Ω such that meas(Ω\K_{ϵ}) < ϵ. Let the function ζ_{ϵ}(x) ∈ C_{0}^{∞}(Ω) be non-negative and equal to unity in a neighborhood of the set K_{ϵ}, whereby ζ_{ϵ}(x) ≤ 1 for all x ∈ Ω. Let P be an operator as in no. 1. We have the identity

(V)  display

Hence follows that

(W)  display

where | · |_{2} denotes the Hilbert-Schmidt norm.

Using (4) as in [1, § 4] we establish the bound

(X)  display

with some δ ∈ (0, 1). The expression |e^{−tP_{Ω}}|_{2} = |e^{−2tP_{Ω}}|_{1}^{1/2}, where | · |_{1} denotes the kernel norm, is estimated from above by means of (5). The bounds for |ζ_{ϵ}e^{−tP} ζ_{ϵ}|_{2}, |e^{− \frac{1}{2} τP} ζ_{ϵ}|_{2} utilize the asymptotic behavior of the spectral function of the operator P (cf. Theorem 1).

Writing

(Y)  display

we finally obtain

(7)  display

where r(t, ζ_{ϵ}) = o(1), t → 0+, for any ϵ > 0. We divide both sides of (7) by V (t, ζ_{ϵ}) and pass to the limit t → 0+. As a result we obtain

(8)  display

By condition c) we have

(Z)  display

where the numbers M_{1}, M_{2} do not depend on ϵ. From condition (5) follows the bound V (t) = O(1)V (M_{2}^{−1}t).

Hence

(a)  display

where the number M_{3} does not depend on ϵ. Hence and from (8), in view of the arbitrariness of ϵ > 0, we conclude that

(b)  display

Since

(c)  display

(cf. (6)) we have ω(t) ∼ V (t), t → 0+. It remains only to apply the appropriate Tauberian theorem.

References

1. K. Kh. Boimatov, Memoirs of the Petrovskij seminar 7 (1981), 50–100.

2. M. Sh. Birman and M. Z. Solomjak, Itogi nauki i tekhniki. Ser. matem. analiz 14 (1977), 5–58.

3. S. Z. Levendorskij, Uspekhi Mat. Nauk 43 (1988), 123–157.

4. A. B. Khmel’nitskaja, Uspekhi Mat. Nauk 31 (1977), 279–280.

Mathematical Institute and Computing Center Academy of Science of the Tadzhik SSR, Dushanbe