INFINITE MATRIX REPRESENTATIONS OF CLASSES OF PSEUDO-DIFFERENTIAL OPERATORS OTIS CHODOSH Abstract. Pseudo-differential operators on the standard torus Td = (R/2πZ)d are related to their representation as an infinite matrix acting on complex exponentials It is shown that given a continuous linear operator C ∞ (Td ) → C ∞ (Td ), it is a pseudodifferential operator of order r if and only if its coefficients as an infinite matrix acting on Fourier coefficients is rapidly decreasing away from the diagonal and bounded of order r along the diagonal with the finite difference operator in the direction of the diagonal decreasing the decay by one order. Similar results are obtained for isotropic pseudo-differential operators via their action on Hermite functions. Additionally, basic properties of pseudo-differential operators, Hermite functions, and isotropic pseudo-differential operators are given.
Contents 1. Introduction 2. Pseudo-differential Operators 3. Isotropic Calculus 4. Harmonic Oscillator and Hermite Functions 4.1. Hermite Functions 4.2. Harmonic Oscillator 4.3. Density of Hermite Functions 5. Infinite Matrix Representations 5.1. Preliminaries 5.2. Torus 5.3. Isotropic Calculus 6. Acknowledgements References
Date: June 10, 2010. 1
2 4 10 11 11 14 15 24 24 29 40 48 48
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1. Introduction In this thesis we study representations of pseudo-differential operators as infinite matrices acting on a basis for square integrable functions. Toroidal pseudo-differential operators acting on complex exponentials and isotropic pseudo-differential operators acting on Hermite functions are represented as infinite matrixes. We give necessary and sufficient conditions for a continuous operator to be a pseudodifferential operator in both cases. In the toroidal case we prove: Theorem 1.1. Given a continuous linear operator A : C ∞ (Td ) → C ∞ (Td ), A is an order r pseudo-differential operator if and only if the (A) coefficients Km,n = hAen , em iL2 (Td ) satisfies order r “symbolic” bounds, i.e. for all γ ∈ Nd0 , N ≥ 0 there is CN,γ such that (A) | 4γ Km,n | ≤ CN,γ (1 + |n − m|)−N (1 + |m| + |n|)r−γ
where 4γ is finite difference operator along the diagonal applied γk times in the k-th coordinate, as defined in Definition 5.2. We give the definition and properties of a pseudo-differential operator in Section 2, and further explain and prove this theorem in Section 5. In addition, we will prove a similar theorem for isotropic pseudodifferential operators on R. Theorem 1.2. Given a continuous linear A : S(R) → S(R), A is an isotropic pseudo-differential operator of order r if and only if the Her(A) mite coefficients Km,n = hAφn , φm iL2 (R) (where φn is the n-th Hermite function) satisfy the following property (A) | 4γ Km,n | ≤ CN,γ (1 + |n − m|)−N (1 + |m| + |n|)r/2−γ
Isotropic pseudo-differential operators are defined in Section 3 and various properties of Hermite functions are proved in Section 4. The theorem is proven in Section 5. Our results on the torus are well known, but seem to be unpublished, but we believe that the isotropic case is new. In fact, the isotropic case exhibits more interesting behavior than the toroidal case in higher dimensions, as we discuss at the end of Section 5.3. In their paper [6], Ruzhansky and Turunen quantize toroidal pseudodifferential operators. However, they only quantize in one coordinate. Namely, they show that Fourier series give a correspondence between r r symbols Sρ,δ (Td × Rd ) and Sρ,δ (Td × Zd ) where the latter space consists of functions a(x, ξ) : Td × Zd → C with the bounds sup x∈Td ,ξ∈Zd
|(1 + |ξ|)−m a(x, ξ)| < ∞
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS
3
as well as requiring the finite difference operator a(x, ξ + ek ) − a(x, ξ) to lower this decay by one order (replacing the standard requirement that differentiating in ξ lowers the order). They show that the class of operators obtained from these two are basically the same, and additionally develop other interesting elements of the calculus, including some microlocal aspects such as wave front sets. Our results are from a quite different viewpoint. Instead of quantizing the symbol in ξ, we will view the operator as an “infinite matrix” acting on “Fourier coefficients” (A) = hKen , em iL2 Km,n
where A is an operator and en is a basis for L2 . To fix notation, we will represent the d-dimensional torus as Td = (R/2πZ)d . In addition, we will sometimes think of smooth functions on Td as 2π-periodic smooth functions on Rd . We will make frequent usage of the notion of a multiindex, which is a d-tuple of integers. Depending on the setting we will indicate if the multi-index consists of non-negative integers (i.e. is in Nd0 ), or integers of both signs (i.e. Zd ). Furthermore, we will often write Q things like xα for α ∈ Nd0 , which just means dk=1 xαk k , and similarly ∂ ∂ is just ∂xα1 ···∂x αd . One possibly confusing notation is that we will ∂xα 1
d
use |n| to represent the norm of a multi-index n, and sometimes we P will mean the `1 norm |n| = dk=1 |nk | and sometimes we will mean 1/2 P d 2 n . This should not be too confusing the `2 norm |n| = k=1 k as we will indicate which situation we are in, and the `1 norm will be used almost exclusively in the isotropic case, and the `2 norm will be used almost exclusively in the toroidal case, and when they appear in inequalities, either one is fine, as they are equivalent norms, and we will never care about the constant in any inequalities. Additionally, we will sometimes make use of the shorthand f (x) . g(x), meaning f (x) ≤ Cg(x) for some constant C. We will mimic Melrose’s unpublished notes [4] in our choice of Fourier transform constants. Namely, for a Schwartz function f ∈ S(Rd ) (that is for all multi-indexes α, β ∈ Nd0 there is a constant Cα,β,f such that β kxα ∂∂xβf k∞ ≤ Cα,β,f ) we define Z ˆ Ff (ξ) = f (ξ) = e−iξ·x f (x) dx Rd
With this convention the inverse Fourier transform is Z −1 F f (x) = eix·ξ f (ξ) d¯ξ Rd
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where we define d¯ξ = (2π)−d dξ. Finally, we will need the notion of a discrete derivative. For any function f : Zd × Zd → C we define (4ek f )(m, n) = f (m + ek , n + ek ) − f (m, n). Notice that this is a more specialized than a general discrete derivative, because we require the derivative to act in both of the indices at once. The reason for such a definition is that we will only consider differentiation “in the direction of the diagonal,” i.e. the direction in which both m and n are increasing together. 2. Pseudo-differential Operators In this section we will describe pseudo-differential operators on Rd and on manifolds. As a proper treatment of these topics would be outside the scope of this work, we will only list key results and explain certain definitions. For an introduction to pseudo-differential operators and microlocal analysis, we recommend Melrose’s unpublished notes [4], which was the main source for the material in this section. Alternative references are H¨ormander [2], particularly volume III, as well as Shubin [7]. We additionally found Joshi’s notes [3] to be a nice self contained reference at roughly the level of knowledge needed to understand the material below. Recall that a standard differential operator with constant coefficients is just the finite sum of derivatives X ∂α L= aα α ∂x |α|≤N
In particular L is a continuous linear operator on Schwartz functions L : S(Rd ) → S(Rd ). It is easy to check that by using the Fourier transform, acting by L is the same as multiplying by a polynomial on the Fourier side. Namely, for ϕ ∈ S(Rd ), it is easy to see that X F(Lϕ) = i|α| aα ξ α F(ϕ) |α|≤N
We call the polynomial in parenthesis the symbol of L. Now, taking the inverse Fourier transform, and writing everything as an integral (absorbing any constants into the polynomial), we have that a differential operator with constant coefficients can be written Z (2.1) Lϕ = ei(x−y)·ξ p(ξ)ϕ(y) dyd¯ξ Rd ×Rd
with p(x) a polynomial called, as above, the symbol of L.
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS
5
Furthermore, notice that differential operators have symbols which are polynomials of the same order of the operator. A pseudo-differential operator is a generalization of differential operators motivated by the above observation. There are various reasons that one would like to generalize the notion of differential operator. Firstly, even if a differential operator is an isomorphism S(Rd ) → S(Rd ) its inverse as an operator is never a differential operator. For example, 1 + ∆ (where ∂2 ∂2 ∆ is the (positive) Laplacian, ∆ := − ∂x 2 − · · · − ∂x2 ) is bijective as an 1 d operator on Schwartz functions, but its inverse is certainly not a differential operator. Secondly, generalizing to pseudo-differential operators certainly takes a bit of technical work, but once this is done, certain results about differential operators, such as elliptic regularity are readily obtained. Finally, pseudo-differential operators are an interesting class of operators in their own right, for example one can make sense of (1+∆)s for s ∈ R and many other such operators as pseudo-differential operators. It turns out that a good way to generalize differential operators is to replace the polynomial symbol p(ξ) in (2.1) with a function in a more general symbol class. This symbol class, which we will denote S r (Rd , Rd ) for r ∈ R, consists of functions a(x, ξ) ∈ C ∞ (Rd × Rd ) such that for α, β ∈ N0 multiindexes we have α,β −r+β ∂ 0 such that for ϕ ∈ S(Rd ) kAϕkL2 (Rd ) ≤ CkϕkL2 (Rd ) and thus A extends to a bounded operator L2 (Rd ) → L2 (Rd ). Sketch of Proof. Using principal symbol arguments as well as a technical property known as “asymptotic completeness” of the symbol spaces, one can show that for every self adjoint, elliptic A ∈ Ψr (Rd ) for r > with principal symbol [a] ∈ S m−[1] (Rd , Rd ) having a ∈ [a] with a ≥ 0,
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then A has an approximate square root, A = B 2 + C for C ∈ Ψ−∞ (Rd ) and B ∈ Ψr/2 (Rd ). Now, given this, for the above A, we have that σL (A∗ A) (where A∗ is the adjoint of A2) is bounded in ξ as it is order 0, so there is C such that C − σL (A∗ A) > 0, so by the above there is B with C − A∗ A = B ∗ B + G for G ∈ Ψ−∞ (Rd ). For ϕ ∈ S(Rd ), we thus have that applying this to ϕ, then taking the L2 inner product with ϕ and moving the adjoints over to the other side Ckϕk2L2 (Rd ) − kAϕk2L2 (Rd ) = kBϕk2L2 (Rd ) + hϕ, GϕiL2 (Rd ) It is not hard to see that G is bounded on L2 (Rd ), so rearranging, we have that kAϕk2L2 (Rd ) ≤ Ckϕk2L2 (Rd ) + |hϕ, GϕiL2 (Rd ) | ≤ C 0 kϕk2L2 (Rd ) as desired.
Finally, we describe how to put pseudo-differential operators on manifolds. We basically will require that in a local coordinate patch the operator is a pseudo-differential operator on a subset of Rd . We restrict our attention to compact manifolds M , but one could extend this to noncompact manifolds, or to operators acting on sections of vector bundles over M . Definition 2.3. Let M be a compact d-dimensional manifold. A linear operator A : C ∞ (M ) → C ∞ (M ) is a pseudo-differential operator of order r, denoted by A ∈ Ψr (M ), if for any coordinate patch W ⊂ M and coordinates F : W → Rd and ψ ∈ Cc∞ (W ) then there is a B ∈ Ψr (Rd ) whose kernel is supported in F (W ) × F (W ) such that for all u ∈ C ∞ (M ) ψA(ψu) = F ∗ (B((F −1 )∗ ψu))) where F ∗ denotes precomposition by F , i.e. (F −1 )∗ (ψu) = (ψu) ◦ F −1 . Furthermore, we demand that if supp φ ∩ supp ψ = ∅ for φ, ψ ∈ C ∞ (M ) then there is K ∈ C ∞ (M × M ) such that Z φA(ψu) = K(x, y)u(y)dV (y) M ∞
for u ∈ C (M ). 2It
is easy to see that this is a pseudo-differential operator of the same order, and in general if a(x, y, ξ) is the symbol of A, then a(y, x, ξ) is the symbol of A∗ .
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS
9
The first condition is the condition discussed above, that restricted to any coordinate chart, A is a pseudo-differential operator on a subset of Rd . It is not hard to show coordinate invariance of pseudo-differential operators by first looking at linear transformations, and then using Taylor’s theorem, so this definition is not dependent on the choice of coordinates on M . The second condition is slightly more mysterious, but it is just added to rule out operators such as A : C ∞ (S d ) → C ∞ (S d ) given by Af (x) = f (−x) which one clearly would like to avoid calling it a pseudo-differential operator on S d , but A satisfies the first condition if we take small enough coordinate charts. In fact, it is readily verified that on Rd , the Schwartz kernel of a pseudo-differential operator is smooth off of the diagonal, in this exact manner. It is not hard to see that most of the previously stated properties, such as L2 boundedness and elliptic regularity apply with little change for pseudo-differential operators on M . We will not make much use of this general definition, as we will only consider pseudo-differential operators on the torus Td and on Rd .The following calculation shows that we can take “toroidal” symbols in C ∞ (T2d × Rd ) having the same symbolic decay in ξ uniformly in x, y and for such an a(x, y, ξ) define an operator C ∞ (Td ) → C ∞ (Td ) by Z
ei(x−y)·ξ a(x, y, ξ)f (y) dyd¯ξ
f 7→ Td ×Rd
To see this, for A ∈ Ψr (Td ), choose a bump function ϕ(x) ∈ C ∞ (R) with ϕ(x) = 1 for x ∈ (π/2, 3π/2), ϕ(x) ≥ 0 on R and supp ϕ compactly contained in [0, 2π]. Let ϕ1 (x) ≡ ϕ(x) and ϕ2 (x) ≡ 1 − ϕ(x). Also, define Ω1 = T1 \{0} and Ω2 = T1 \{π}. Now, for j = (j1 , j2 , . . . , jd ) with jk ∈ {1, 2}, we let ϕj (x) ≡ ϕj1 (x1 ) · · · ϕjd (xd ) and ΩI = Ωj1 ×· · ·×Ωjd and we see that {ϕj }j is a partition of unity for Td , subordinate to the chart {ΩI }. Note that on ΩI , there are coordinates given by the obvious embedding uI : ΩI ,→ [0, 2π]d Now, for f ∈ C ∞ (Td ) we have that (because all of the ΩI overlap) AI,J = ϕI AϕJ is pseudo-differential operator on Rd (supported on an open subset of [0, 2π]d ) pasted on the torus via the uI coordinate chart. Letting aI,J be a symbol of AI,J we have A(f ) =
X
ϕI A (ϕJ f ) =
I,J
=
∗ (uI )∗ AI,J (u−1 I ) f
I
XZ I,J
X
(0,2π)d
Z Rd
ei(uI (x)−y)·ξ aI,J (uI (x), y, ξ)f (u−1 ¯ξ I (y)) dyd
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=
Z
XZ Td
I,J
Z
ei(x−y)·ξ aI,J (x, y, ξ)f (y) dyd¯ξ
Rd
Z
ei(x−y)·ξ a(x, y, ξ)f (y) dyd¯ξ P where a(x, y, ξ) = I,J aI,J (x, y, ξ). This justifies our above claim, and from now on, we will not worry about the generalities of Definition 2.3, and will just consider “toroidal” symbols as described above. =
Td
Rd
3. Isotropic Calculus In this section, we give the definition of an isotropic pseudodifferential operator as well as properties which we will use in the sections to come. All of the material in this section, as well as a great deal more, can be found with proofs in Melrose’s [4] Chapter 4. Isotropic pseudo-differential operators are a subclass of pseudodifferential operators, having the property that the decay is “in both x and ξ simultaneously.” For a ∈ C ∞ (R2d ) we say that a is an isotropic symbol of order r if for multi-indexes α, β ∈ Nd0 we have that α+β ∂ a ≤ (1 + |x| + |ξ|)r−|α|−|β| (x, ξ) ∂xα ∂ξ β r and if this is the case, we write a ∈ Siso (R2d ). Using the inequalities
1 + |x| + |ξ| ≤ (1 + |x|)(1 + |ξ|) 1 + |x| + |ξ| ≥ (1 + |x|)t (1 + |ξ|)1−t (where in the second inequality t ∈ [0, 1]) we have that Lemma 3.1. Isotropic symbols are contained in the regular symbols in the following manner + |x|2 )r/2 S r (Rd , Rd ) r>0 (1\ r 2d 2 s/2 r−s d d S (R ) ⊂ (1 + |x| ) S (R , R ) r ≤ 0 r≤s≤0
In particular, this shows that the spaces Ψriso (Rd ) make sense. It can be shown that the composition of two isotropic pseudo-differential operators is again an isotropic pseudodifferential operator, and that the adjoint of an isotropic pseudo-differential operator is again an isotropic pseudo-differential operator of the same order (see, e.g. Melrose [4] Proposition 4.1 and Theorem 4.1). Furthermore, it can be shown that d elliptic elements of Ψriso (Rd ) have a two sided parametrix in Ψ−r iso (R ). 0 d 0 d 2 Additionally, because Ψiso (R ) ⊂ Ψ (R ), we have L boundedness of order 0 isotropic pseudo-differential operators. We will make use of
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 11
these facts when discussing the isotropic case, below. Finally, we will need that the isotropic calculus is closed under the Fourier transform c ˆ = Aˆ in the following sense: defining Aˆ by the formula Au u, it can ˆ be shown that A is a isotropic pseudodifferential operator of the same order with the symbol a ˆ(x, ξ) := −a(−ξ, −x). 4. Harmonic Oscillator and Hermite Functions In this section we define the Hermite functions and harmonic oscillator and prove various properties. 4.1. Hermite Functions. Because they are eigenfunctions of the harmonic oscillator (one of the simplest elliptic isotropic differential operators), the Hermite functions are a natural choice of basis for L2 (Rd ) if one is interested in the isotropic calculus. Here, we introduce the Hermite functions, and prove various properties that will serve useful in examining the decay of the resulting Schwartz kernel matrix. We define the Hermite polynomials in one variable in the following manner. Definition 4.1. For n ≥ 0, the Hermite polynomials are defined by n d 2 n x2 (4.1) hn (x) = (−1) e e−x dx It is easy to check from the definition that the hn (x) is indeed a polynomial and it is of degree n. The following is easily shown by induction and simple complex analysis arguments. Lemma 4.2. For hn (x) as in (4.1), and for any > 0 and n ≥ 0 I 2 n! e2tx−t (4.2) hn (x) = dt 2πi ∂B (0) tn+1 Proof. First, when n = 0, we would like to show that I 2 e2tx−t 1 (4.3) 1= dt 2πi ∂B (0) t 2
but this is clear because e2tx−t = 1 + f (t, x) where f (t, x) is holomorphic and has a zero at t = 0 for all x, so I 1 f (t, x) dt = 0 2πi ∂B (0) t and then the identity in (4.3) follows. Now, assuming that n I 2 d n! e2tx−t −x2 n x2 (−1) e e = dt dx 2πi ∂B (0) tn+1
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this implies that n I 2 d e2tx−t −x2 n −x2 n! e = (−1) e dt dx 2πi ∂B (0) tn+1 and differentiating, we have that n+1 I 2 d e2tx−t −x2 n+1 −x2 n! e = (−1) 2xe dt dx 2πi ∂B (0) tn+1 I 2 (2t)e2tx−t n −x2 n! + (−1) e dt 2πi ∂B (0) tn+1 I 2 (2x − 2t)e2tx−t n+1 −x2 n! dt = (−1) e 2πi ∂B (0) tn+1 I ∂ 2tx−t2 e n+1 −x2 n! ∂t dt = (−1) e 2πi ∂B (0) tn+1 I 2 e2tx−t n+1 −x2 (n + 1)! = (−1) e dt n+2 2πi ∂B (0) t This completes the induction, proving the lemma.
Additionally we have the following recurrence relations, which follow from (4.1) and (4.2) by induction. Lemma 4.3. For n ≥ 0 h0n (x) = 2nhn−1 (x) hn+1 (x) = 2xhn (x) − 2nhn−1 (x) With these relations, we now prove weighted orthogonality of the Hermite polynomials. Lemma 4.4. The Hermite polynomials are orthogonal in L2 (R, dµ) 2 where dµ is the measure given by dµ(x) = e−x dx. Moreover, for m, n ≥ 0 Z √ (4.4) hm (x)hn (x) dµ = n!2n πδnm R
Proof. Without loss of generality, let m ≥ n Z hm (x)hn (x) dµ n Z d 2 n = (−1) hm (x) e−x dx dx R R
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 13 n+1
= (−1)
Z hm−1 (x)
2m R
d dx
n−1
2
e−x dx
Z
2
hm−n (x)e−x dx R Z m−n d 2 m−n n = (−1) 2 m(m − 1) · · · (m − n + 1) e−x dx dx R n
= 2 m(m − 1) · · · (m − n + 1)
This clearly has the correct value for m = n and the integral evaluates to zero for m > n. Given this orthogonality, we are now ready to define the Hermite functions, which we will see are a complete orthonormal basis for L2 (Rd , dx), as well as being dense in S(Rd ) and S 0 (Rd ), the Schwartz functions and tempered distributions on Rd respectively. Definition 4.5. In Rd , for a multi-index n ∈ Zd0 , we define the n-th Hermite function by (4.5)
φn (x) = p
1
−|x|2 /2
√ e n!2n π
d Y
hnk (xk )
k=1
We have just shown that the Hermite functions are orthonormal in 2 L (Rd , dx). Furthermore, it is easy to check that e−|x| /2 times any polynomial in x is Schwartz, so φn ∈ S(Rd ). Using Lemma 4.3 we now will derive relations for differentiating and multiplying Hermite functions by a polynomial which will be useful later. 2
Lemma 4.6. For a multi-index n ∈ Zd0 and 1 ≤ k ≤ d we have 1 ∂ (4.6) φn+ek (x) = p xk − φn (x) ∂xk 2(nk + 1) 1 ∂ (4.7) φn−ek (x) = p xk + φn (x) ∂xk 2(nk ) Proof. From Lemma 4.3 we have ∂φn 1 −|x|2 /2 (x) = p (−xk hn (x) + 2nk hn−ek (x)) √ e n ∂xk n!2 π 1 2 =p e−|x| /2 (−2xk hn (x) + xk hn (x) + 2nk hn−ek (x)) √ n!2n π p = − 2(nk + 1)φn+ek (x) + xk φn (x) This shows the first relation, and the second follows similarly.
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Motivated by this calculation, we define the creation and annihilation operators as follows. Definition 4.7. For k ∈ {1, . . . , d}, we define the k-th annihilation operator 1 ∂ C k = √ xk + ∂xk 2 and its hermitian conjugate, which we will call the k-th creation operator 1 ∂ † C k = √ xk − ∂xk 2 √ √ Notice that Ck φn = nk φn−ek and Ck† φn = nk + 1φn+ek . It will be useful to rearrange the relations in Lemma 4.6 to obtain Lemma 4.8. For a multi-index n ∈ Zd0 with nk > 0 for 1 ≤ k ≤ d we have r r ∂φn nk + 1 nk (4.8) φn+ek (x) + φn−ek (x) (x) = − ∂xk 2 2 r r nk + 1 nk xk φn (x) = (4.9) φn+ek (x) + φn−ek (x) 2 2 If we take as a convention φn (x) = 0 if nk < 0 for any k, then the above formulas hold for nk = 0 as well. 4.2. Harmonic Oscillator. We immediately see from Lemma 4.6 that the Hermite functions are eigenfunctions of the harmonic oscillator. 2
∂ + x2k Lemma 4.9. Defining the k-th harmonic oscillator by Hk = − ∂x k and on Rd , the total harmonic oscillator by H = H1 + · · · + Hd = ∆ + |x|2 , we have that for a multi-index n ∈ Zd+ 3
(4.10)
Hk φn (x) = (2nk + 1)φn (x)
(4.11)
Hφn (x) = (2|n| + d)φn (x)
The Hermite functions are also eigenvalues of the Fourier transform. Lemma 4.10. For a multi-index n ∈ Zd0 we have (4.12)
Fφn (ξ) = (2π)d/2 (−i)n φn (ξ)
Proof. Applying the Fourier transform to equation (4.6) we have that ∂ Fφn+ek (ξ) = p i − ixk F(φn )(ξ) 2(nk + 1) ∂xk 1
3Here,
by |n| we mean |n| = n1 + n2 + · · · + dn .
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 15
∂ =p xk − Fφn (ξ) ∂xk 2(nk + 1) −i
Additionally, because φ0 (x) =
2 1 e−|x| /2 , π 1/4
we have that
Fφ0 (ξ) = (2π)d/2 φ0 (ξ) Thus, using equation (4.6) repeatedly, we obtain the desired result. 4.3. Density of Hermite Functions. The final results of this section concern density of the Hermite functions in various spaces, L2 (Rd ), S(Rd ), and S 0 (Rd ). For the next results, we follow Chapter 4 in Melrose [4] very closely. It is an easy computation to show that the creation and annihilation operators as in Definition 4.7 obey the following commutation relations for k, j ∈ {1, 2, . . . , d} (4.13)
[Ck , Cj ] = 0
(4.14)
[Ck† , Cj† ] = 0
(4.15)
[Ck , Cj† ] = δk,j
Furthermore, given these relations, we can rewrite the harmonic oscillator as (4.16)
H =d+2
d X
Ck† Ck
k=1
Given this, we have additional commutation relations following from (4.13) through (4.15) as well as (4.16) (4.17)
[Ck , H] = 2Ck
(4.18)
[Ck† , H] = −2Ck†
Lemma 4.11. If λ ∈ C is an eigenvalue of the harmonic oscillator, i.e. Hu = λu for a distribution u ∈ S 0 (Rd ) then λ ∈ {d, d + 2, d + 4, . . . }. Proof. For λ ∈ C, H − λ is an elliptic second order isotropic operator, so elliptic regularity implies that if there is a nonzero distribution u ∈ S 0 (Rd ) with (H − λ)u = 0 then in fact u is Schwartz, i.e. u ∈ S(Rd ). However, H is (formally) self-adjoint, so we have that hHu, uiL2 (Rd ) = hu, HuiL2 (Rd )
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and because kukL2 (Rd ) 6= 0, this implies that λ = λ, so λ ∈ R. Now, from (4.16), we have that λkukL2 (Rd ) = hHu, uiL2 (Rd ) = dkukL2 (Rd ) + 2
d X
kCk ukL2 (Rd )
k=1
implying that λ ≥ d. Furthermore, by (4.16), λ = n if and only if kCk uk = 0 for all k, and we thus have that u satisfies the system of PDE’s ∂u = −xk u ∂xk for k ∈ {1, 2, . . . , d}. However, holding the other xj for j 6= k, fixed, this implies that 2 x u(x) = A(x1 , . . . , xˆk , . . . xd ) exp − k 2 and this clearly implies that (4.19)
|x|2 u(x) = A0 exp − 2
Now, using this calculation, combined with the commutators above, we can finish the proof. Suppose that u ∈ S(Rd ) is an eigenfunction of H with eigenvalue λ ∈ R, λ ≥ d. We may assume that λ > d, as if λ = d, we have shown that u is a gaussian, as in (4.19), which is just a multiple of φ0 . Now, notice that HCk u = (Ck H − [Ck , H])u = λCk u − 2Ck u = (λ − 2)Ck u so if it is nonzero, Ck u is an eigenfunction with eigenvalue λ − 2. For a multi-index α ∈ Nd0 , letting C α = (C1 )α1 · · · (Cd )αd , let m ∈ N0 be the smallest m such that for |α| ≥ m C αu = 0 There must be such an m because we have proven that there are no eigenvalues less than d, and C α u has eigenvalue (λ − |α|) by repeating the above commutator argument. Now, for such an m, for |α| = m − 1, Ck C α u = 0 for all k, so by the above calculation, C α u is a (nonzero) multiple of the gaussian, and in particular, λ − m − 1 = d, so λ = d + m + 1, completing the proof. Now, we will show that we have found all of the eigenfunctions Lemma 4.12. For k ∈ N0 the map Fk : {Degree k homogeneous polynomials on Rd } → Ek
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 17
where Ek is the k’th eigenspace with eigenvalue d + 2k, given by p 7→ p(C † )e−|x|
2 /2
is well defined and an isomorphism. 2
Proof. The expression p(C † )u0 (where u0 = e−|x| /2 ) is well defined because the Ck† all commute. Induction along with the commutator argument from above shows that H(p(C † )u0 ) = (d + 2k)p(C † )u0 so the map is also well defined. Notice that by the same commutator argument, for multi-indexes |α| = |β| we have ( 0 α 6= β C α (C † )β u0 = |α| 2 α!u0 α = β P Thus, if p = |β|=k cβ xβ ∈ ker Fk , then 0 = C α p(C † )u0 = cα 2|α| α!u0 , so p = 0, showing that Fk is injective. Now, if Fk is not surjective, there will be a nonzero v ∈ (im Fk )⊥ ⊂ Ek (with respect to the L2 norm). In particular, v is orthogonal to (C † )α u0 for all α with |α| = k so hC α v, u0 iL2 (Rd ) = hv, (C † )α u0 iL2 (Rd ) = 0 However, by the commutator argument, C α v must be an eigenfunction of H with eigenvalue d. As in the previous proof, it must thus be a multiple of u0 , and thus is zero. We claim that this implies that v = 0. To show this, we prove the following claim by induction on k. If v ∈ Ek and C α v = 0 for all |α| = k, then v = 0. It is trivially true for k = 0, and to show the induction step, notice that for each j ∈ {1, . . . , d} and |α| = k − 1 we have assumed that C α Cj v = 0. However, by induction (which applies because Cj v ∈ Ek−1 ) we have that Cj v = 0 for all j, which implies that v = Au0 , implying in fact v = 0, because Au0 6∈ Ek for k > 0 and A 6= 0. This is a contradiction, showing that Fk is an isomorphism. This lemma, in particular, shows that Ek = span{φα }|α|=k . Now, we will show L2 density of the Hermite functions. In Melrose’s notes [4], he uses the fact that a large negative power of the harmonic oscillator is a compact self adjoint operator and thus applying the relevant version of the spectral theorem gives that the eigenfunctions are dense in L2 (Rd ). We will instead follow Reed-Simon, and use the exponential generating function of the Hermite polynomials from [5], Chapter IX, Problem 7. We first establish the following generating function property of Hermite polynomials on R.
18
OTIS CHODOSH
Lemma 4.13. Denoting the n-th Hermite polynomial by hn (y), we have the following identity for all x, y ∈ R ∞ X xn 2 e−x +2xy = hn (y) n! n=0 Proof. Recall that we have shown in (4.2) that I 2 n! e2tx−t hn (x) = dt 2πi ∂B (0) tn+1 Thus, we have that 2 ∞ ∞ I X xn 1 X e2ty−t n hn (y) = x dt n! 2πi n=0 ∂B (0) tn+1 n=0 I ∞ X 1 xn 2 = e2ty−t dt n+1 2πi ∂B (0) t n=0 I 2ty−t2 e 1 dt = 2πi ∂B (0) t − x = e−x
2 +2xy
Moving the sum inside the integral is justified by dominated convergence as long as is large enough so that |x| < |t|, and this is precisely when the residue of the resulting integrand is contained inside B (0). Lemma 4.14. The Hermite functions {φn }n∈Zd0 form a complete orthonormal basis for L2 (Rd ) Proof. We will first prove this for d = 1. Suppose that there is f ∈ L2 (R) such that for all n ∈ N0 hf, φn iL2 (R) = 0 If we can show that under these assumptions f = 0, then it is a standard argument to show that the φn are dense in L2 (R). Notice that our assumption implies that ∞ q X √ yn 0= n!2n πhf, φn iL2 (R) n! n=0 Z ∞ q X √ yn = n!2n π f (x)φn (x) dx n! R n=0 ∞ Z X yn 2 = f (x)hn (x) e−x /2 dx n! n=0 R
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 19
Z
2
2
f (x)e−y +2xy e−x /2 dx R Z 2 y2 =e f (x)e−(x−2y) /2 dx
=
R
Thus, for any z ∈ R we have that Z 2 F (z) := f (x)e−(x−z) /2 dx = 0 R
However, it is clear that F (z) is entire in z and thus the above equality can be extended to z ∈ C. Taking z = it Z 2 t2 /2 0 = F (it) = e f (x)e−x /2 eitx dx R 2
In particular, this implies that the Fourier transform of f (x)e−x /2 is 2 zero a.e., and thus f (x)e−x /2 = 0 a.e., so f (x) = 0 a.e., as desired. Thus, we have that the Hermite functions are dense in L2 (R). Now, for general d, notice that if we define4 ˜ · · · ⊗L ˜ 2 (R) S := L2 (R)⊗ | {z } d times
then S ⊂ L2 (Rd ). In fact, S is a dense subset of L2 (Rd ), which one can easily see by recalling that finite sums of characteristic functions of rectangles are dense in L2 (Rd ) and these are all in S. However, from our above results about density of Hermite functions in L2 (R), we see that d-dimensional Hermite functions are dense in S and thus L2 (Rd ) as desired. Now, following a combination of the proofs in [4] and [5] we have the following two theorems Lemma 4.15. The Hermite functions are dense in S(Rd ). More specifically, for ϕ ∈ S(Rd ) letting X uN = hϕ, φα iL2 (Rd ) φα |α|≤N
4By
˜ we mean finite sums of products of d functions in L2 (R) where the k-th ⊗ one is a function of xk only. This is the “non-completed tensor product” which is in contrast to the usual Hilbert space (completed) tensor product, i.e. L2 (R) ⊗ L2 (R) is the closure of terms of the form f (x)g(y) in L2 (R2 ). In fact, L2 (R) ⊗ L2 (R) = L2 (R2 ), as described above.
20
OTIS CHODOSH
we have that uN → u in the topology of S(Rd ). Additionally the cα := hϕ, φα iL2 (Rd ) are rapidly decreasing, i.e. for all N sup |cα |(1 + |α|)N < ∞ α∈Nd0
Furthermore, for any cα with this decay, the sum X cα φα (x) |α|≤N
converges in S(Rd ). Proof. For any ϕ ∈ S(Rd ) and M ∈ N0 , we have that H M ϕ ∈ L2 (Rd ), (M ) so if we let cα = hH M ϕ, φα iL2 (Rd ) , as above, we have that X L2 M ) c(M α φα −−→ H ϕ |α|≤M
In particular, this implies that (M ) cα ≤ C M for some CM . Thus, we have that CM ≥ hH M ϕ, φα i = hϕ, H M φα i = |hϕ, φα i| (d + 2|α|)M so |cα | ≤ CM (d + 2|α|)−M which establishes the second claim. Now, we would like to show that for cγ = hϕ, φγ i X uN = cγ φ γ |γ|≤N
converges to ϕ. It is enough to show that uN is a Cauchy sequence S in the topology of S(Rd ), because by completeness of S(Rd ), uN − →u for some u ∈ S(Rd ), but in particular this implies L2 convergence, so u = ϕ a.e. and thus they are equal as smooth functions. To show that uN is Cauchy, recall that S(Rd ) is topologized by the seminorms kϕkα,β := sup |xα Dβ ϕ(x)| x∈Rd
In our case, however, it will be more convenient to consider the seminorms kϕkα,β,2 = kxα Dβ f kL2 (Rd )
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 21
As such, we would like to show show that the two families are equivalent. To see this, notice that if we define g(x) :=
d Y
(1 + x2k )−1
k=1
then g(x) ∈ L2 (Rd ). Thus, using the H¨older inequality, we have that X kϕkα,β,2 ≤ kgkL2 (Rd ) kϕkα+2γ,β |γ|≤1
Conversely, because xα Dβ ϕ(x) Z x1 ∂ xα Dβ ϕ (t, x2 , . . . , xd ) dt = ∂x1 Z−∞ x1 = α1 xα−e1 Dβ ϕ(t, x2 , . . . , xd ) + xα Dβ+e1 ϕ(t, x2 , . . . , xd ) dt −∞ Z x1 X g(x) = x2γ (α1 xα−e1 Dβ ϕ(t, x2 , . . . , xd ) −∞
|γ|≤1
+ xα Dβ+e1 ϕ(t, x2 , . . . , xd )) dt we have that kϕ(x)kα,β ≤ kgkL2 (Rd )
X
α1 kϕkα−e1 +2γ,β,2 + kϕkα+2γ,β+e1 ,2
|γ|≤1
showing that the two collections of norms are equivalent. Now, we will show that uN is Cauchy in the topology given by the seminorms k · kα,β,2 , which by the above argument will show that uN → ϕ in S(Rd ) as desired. Notice that using Lemma 4.8 and induction it is easy to show that X xα Dβ φγ = rη (α, β, γ)φη η∈Nd0 |η−γ|≤|α|+β|
where |rη (α, β, γ)| ≤ Cα,β (1 + |γ|)(|α|+|β|)/2 )
22
OTIS CHODOSH
Thus, setting rη (α, β, γ) = 0 for |η − γ| > |α| + |β|, we have that for M 0 such that 2r < (1 − s)d, for this s, an N large enough such that s(d − 2N ) < −d − 2|r|. This completes the proof. We now prove the following Lemma concerning multiplying a symbol matrix by a function which is polynomially bounded with the difference operator lowering the decay by one order. Lemma 5.4. If f (m, n) is a symbol matrix of order r and g(m, n) is any function g : Zd × Zd → C satisfying | (4γ g) (m, n)| ≤ Cγ (1 + |m| + |n|)s−|γ| then f (m, n)g(m, n) is a symbol matrix of order r + s.
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 27
Proof. It is easy to see that from the conditions on f (m, n) and g(m, n) we have 0 |[f g](m, n)| ≤ CN,0 (1 + |n − m|)−N (1 + |n| + |m|)r+s
so the only thing that is left to check is the difference operator 4ek [f g] (m, n) = 4ek [f ] (m, n) g(m+ek , n+ek )+f (m, n) 4ek [g] (m, n) Applying this inductively finishes the proof.
In the proofs below we will define operators from symbol matrices, and thus will need that Lemma 5.5. If f (m, n) is a symbol matrix of order r, then it defines a continuous operator C ∞ (Td ) → C ∞ (Td ). Proof. It is clear that our desired result is equivalent to proving that the operator X F : {xk }k∈Zd 7→ f (n, k)xk k∈Zd
is a continuous linear operator on the space of rapidly decreasing sequences s(Zd ) = {{xk }k∈Zd : |(1 + |k|)N xk | ≤ CN ∀N ∈ N0 } where the topology on s(Zd ) is given by the seminorms kxk kN = sup |(1 + k)N xk | k
Because a symbol matrix of negative order is also a symbol matrix of zero order, we may assume that r ≥ 0. Notice that multiplication by (1 + |k|)s is a continuous linear bijection s(Zd ) → s(Zd ). Denote this operator by Ls . Choosing any N , notice that kLs ◦ F ◦ Ll ({xk })kN
X
= (1 + |n|)s f (n, k)(1 + |k|)l xk
d
k∈Z N X N +s l = sup(1 + |n|) f (n, k)(1 + |k|) xk d n k∈Z X ≤ C sup(1 + n)N +s (1 + |n| + |k|)r (1 + |k|)l |xk | n
k∈Zd
≤ Ckxk k0 sup(1 + |n|)N +s n
X k∈Zd
(1 + |n| + |k|)r (1 + |k|)l
28
OTIS CHODOSH
X
≤ Ckxk k0 sup(1 + |n|)N +s n
(1 + |n|)r (1 + |k|)r (1 + |k|)l
k∈Zd
X
= Ckxk k0 sup(1 + |n|)N +s+r n
(1 + |k|)l+r
k∈Zd
In particular, taking l < −d − r, and s < −N − r, we have that kLs ◦ F ◦ Ll ({xk })kN ≤ Cl,s kxk k0 Thus, for these s, l we have that kF ({xk })kN = k(L−s (Ls ◦ F ◦ Ll ))(L−l ({xk })kN ≤ kL−s kL (s(Zd ) kLs ◦ F ◦ Ls (L−l ({xk })kN ≤ Cl,s kL−s kL (s(Zd )) kL−l kL (s(Zd )) kxk k0 We will also need the following lemma which says that an order 0 symbol matrix considered as an operator on functions, as in the previous lemma, extends to a bounded operator on L2 (Rd ) Lemma 5.6. Considering an order 0 symbol matrix f (m, n) as an operator A : C ∞ (Td ) → C ∞ (Td ) (as in the previous lemma), A extends to a bounded operator A : L2 (Td ) → L2 (Td ). Proof. This follows readily from Schur’s test (for example, see Theorem 5.2 in [1]) but for completeness we will give the proof. Our proof is essentiall the same as the proof in [1], except in far less generality. For g ∈ L2 (Td ), we have that gˆ(k) := hg, ek iL2 (Td ) is in `2 (Zd ). Thus !2 X X f (n, k)ˆ g (k) n∈Zd
k∈Zd
!2 ≤
=
X
X
n∈Zd
k∈Zd
X
Xp
n∈Zd
k∈Zd
X
X
n∈Zd
k∈Zd
|f (n, k)||ˆ g (k)| !2 p |f (n, k)| |f (n, k)||ˆ g (k)| !
≤
|f (n, k)|
! X
|f (n, k)||ˆ g (k)|2
k∈Zd
! ≤ CN2
X
X
n∈Zd
k∈Zd
(1 + |n − k|)−N
! X k∈Zd
(1 + |n − k|)−N |ˆ g (k)|2
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 29
! ≤ CN2
X
(1 + |k|)−N
k∈Zd
X
(1 + |n − k|)−N |ˆ g (k)|2
n,k∈Zd
!2 = CN2
X
! X
(1 + |k|)−1
k∈Zd
|ˆ g (k)|2
k∈Zd
(A)
5.2. Torus. First, for A ∈ Ψr (Td ), we calculate Km,n , which will serve to motivate our classification theorem below. We denote the Schwartz kernel of A by K(x, y) ∈ D(Td × Td ). We will assume for now that K(x, y) has compact support in a coordinate patch on Td ×Td . To then deal with general A, all that we need to do is take a (finite) partition of unity. Because K(x, y) has compact support in a coordinate patch on Td × Td , we can regard it as a distribution on Rd ×Rd with compact support5. We know that Z K(x, y) = ei(x−y)·ξ a(x, ξ) d¯ξ Rd
in the sense of distributions. Changing variables Z K(x, x − y) = eiy·ξ a(x, ξ) d¯ξ Rd
By Fourier inversion, we have that Z a(x, ξ) = e−iy·ξ K(x, x − y) dy d R Z ei(y−x)·ξ K(x, y) dy =− Rd
This shows that a(x, ξ) is compactly supported in x. Now, we calculate (A) Km,n as follows (A) Km,n = hAen , em i 5It
might not be very clear what we mean by this. For a general manifold, by taking a partition of unity we could assume that A has a kernel supported in a compact coordinate patch. Then we know that by conjugating by the coordinate chart we can just view A as a regular pseudo-differential operator on Rd with compactly supported kernel. However, in the standard coordinates on the torus, a coordinate chart is basically the identity (because we regard the Torus as a quotient of Euclidean space by the integer lattice, Td = Rd /(2πZ)d ) which is why we can consider K(x, y) as just being a distribution on Rd × Rd , without composing with any coordinate charts.
30
OTIS CHODOSH
Z
K(x, y)ei(n·y−m·x) dxdy
= ZT
d ×Td
K(x, y)ei(n·y−m·x) dxdy
= Rd ×Rd
Z
K(x, x − y)ei(n·(x−y)−m·x) dxdy
=− Rd ×Rd
Z
K(x, x − y)ei((n−m)·x−n·y) dxdy
=− Rd ×Rd
Z =− Z =
Rd ×Rd
Fy−1 K(x, x − y)Fy−1 ei(n−m)·x−n·y) dxdy
a(x, ξ)δ(ξ − n)ei(n−m)·x dxd¯ξ
Rd ×Rd
= Fx a(n − m, n) Now, because a(x, ξ) has compact support in x, its Fourier transform in (A) x will be Schwartz, and thus Km,n will be rapidly decaying in |n − m|. (A) Here, we also see that |Km,n | . (1 + |n|)r from the symbol bounds on a. From this calculation it is natural to ask if these two properties are equivalent to A being a pseudo-differential operator of order r on Td , and as the next theorem will show, this is indeed the case. Theorem 5.7. For A a continuous linear operator A : C ∞ (Td ) → (A) C ∞ (Td ), A ∈ Ψr (Td ), if and only if Km,n is a symbol matrix of order r (as in Definition 5.2). Proof. To begin, suppose that A is a pseudo-differential operator of (A) order r, i.e. A ∈ Ψr (Td ). We will show that Km,n is a symbol matrix of order r. We could use the above calculation of the matrix coefficients to establish this, but we will give a second proof instead, because most of the methods used will readily carry over to the isotropic case which we will consider next. First, to deal with the off diagonal decay notice that d Z X 2 (A) |n − m| Km,n = ei(x−y)·ξ a(x, ξ) k=1
Rd ×(Td )2
∂2 ∂2 ∂2 − − ei(y·n−x·m) dyd¯ξdx 2 ∂xk ∂yk ∂x2k ∂yk2 Integrating by parts, the derivatives will fall on the first terms ∂2 ∂2 ∂2 2 − − ei(x−y)·ξ a(x, ξ) ∂xk ∂yk ∂x2k ∂yk2
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 31
= −2ξk2 + ξk2 + ξk2 ei(x−y)·ξ a(x, ξ) − 3(iξk )ei(x−y)·ξ
∂a ∂ 2a (x, ξ) − ei(x−y)·ξ 2 (x, ξ) ∂xk ∂xk
The first term cancels so we can define ∂a ∂ 2a a ˜(x, ξ) = −3(iξk ) (x, ξ) − 2 (x, ξ) ∂xk ∂xk This is still a symbol of order r, defining a new toroidal pseudodifferential operator A˜ ∈ Ψr (Td ). Thus, repeating this argument for (A) higher powers of |n − m|, we have that the original Km,n is bounded by (1 + |m − n|)−N times the matrix for a different pesudo-differential operator of order r. Thus, as long as we can show that the matrix for a pseudo-differential operator of order r is a symbol matrix when N = 0 in Definition 5.2, we will establish this direction of the equivalence. Thus, below we will take N = 0 and establish the rest of the symbol matrix bounds. ∂2 Considering the Laplacian on Td (which we denote by ∆ := − ∂x 2 − 2
1
∂ 2 · · · − ∂x 2 ) we have that (1 + ∆)en = (1 + |n| )en , and that 1 + ∆ is n elliptic with principal symbol 1 + |ξ1 |2 + · · · + |ξd |2 . We make use of the following development in [7] concerning complex powers of elliptic operators.
Theorem 5.8 (Shubin II.10.1, II.11.2). Let B be an elliptic differential operator of order j on a closed d-dimensional manifold M and b(x, ξ), the principal symbol of B. Assume that b(x, ξ) does not take values in a closed angle Λ of the complex plane for ξ 6= 0. Then for appropriately chosen contour Γ, and 1} ⊂ Rd x = tω for ω ∈ S d−1 and t > 1 In polar coordinates, our assumption on f can be thus written as ∂ αf f (x) ∈ ts−|α| L2 (R+ × S d−1 , td−1 dtdω) ∂xα It is clear that xj ∈ tC ∞ (S d−1 ), so we see that d
X ∂xj ∂f ∂f = ∈ ts L2 (R+ × S d−1 , td−1 dtdω) ∂ωk ∂ω ∂x k j j=1 because differentiating in xj lowers the power of t by one, but a power ∂x of t is also regained from the ∂ωkj , and multiplying by a function in C ∞ (S d−1 ) cannot hurt the L2 (S d−1 ) bounds. Similarly d
X ∂xj ∂f ∂f = ∈ ts−1 L2 (R+ × S d−1 , td−1 dtdω) ∂t ∂t ∂x j j=1 Thus, we have that for any differential operator on S d−1 , P , and any l ∈ N0 the assumption on f gives ∂l (P f (t, ω)) ∈ ts−l L2 (R+ × S d−1 , td−1 dtdω) l ∂t We can rewrite this as being integrable on R+ with values in L2 (S d−1 ) ∂l (P f (t, ω)) ∈ ts−l−(d−1)/2 L2 (R+ ; L2 (S d−1 )) ∂tl This does not depend on the exact form of P , or even its order, so we can take P to be very large order and elliptic, and thus for any k, we can consider original f as a L2 function on R+ taking values in H k (S d−1 ) ∂l (f (t, ω)) ∈ ts−l−(d−1)/2 L2 (R+ ; H k (S d−1 )) ∂tl Thus, for any differential operator on S d−1 , Q of order q we have ∂l (Qf (t, ω)) ∈ ts−l−(d−1)/2 L2 (R+ ; H k−q (S d−1 )) ∂tl
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 37
For large enough k, by Sobolev embedding H k−q (S d−1 ) ⊂ L∞ (S d−1 ), so we have that l ∂ sup l (Qf (t, ω)) ∈ ts−l−(d−1)/2 L2 (R+ ) d−1 ∂t w∈S
Writing l ∂ ∂ p sup t l (Qf (t, ω)) = ts−l+p−1−(d−1)/2 g(t) ∂t w∈S d−1 ∂t for g(t) ∈ L2 (R+ ). Because L∞ membership for smooth functions does not depend on its values on any compact set, we may assume that f and thus g are zero for t < 2. notice that Z t l+1 l ∂ 0 0p ∂ p ∂ (Qf (t , ω)) dt t t l (Qf (t, ω)) = ∂t ∂tl+1 0 ∂t so p ∂l t sup (Qf (t, ω)) ∂tl t∈R+ ,w∈S d−1 Z ∞ l ∂ ∂ 0 p dt (Qf (t , ω)) ≤ t ∂t ∂tl 0 Z ∞ = ts−l−(d−1)/2+p−1 |g(t)| dt 1
Z ≤
∞ 2s−2l−d−1+2p
t 1
1/2 Z dt
∞
1/2 |g(t)| dt
1
This is finite for 2s − 2l − d − 1 + 2p < −1, which is equivalent to s − l − d/2 + p < 0. Reverting to regular coordinates, this proves the lemma. By this Lemma combined with (5.9), we see that for r0 > r (5.10)
∂γ 0 a(x, ξ) ∈ (1 + |ξ|2 )(r −|γ|)/2 L∞ (Rdξ ) γ ∂ξ
uniformly in x. Finally, we need to consider derivatives in x of a(x, ξ) to ensure they do not increase the order. Calculating, we see that if we were to replace a(x, ξ) with ∂xl a(x, ξ), defining a distribution Z ∂ ei(x−y)·ξ a(x, ξ) d¯ξ L(x, y) = ∂xl Rd
38
OTIS CHODOSH
we have that Z hL(x, y)en , em i = (Rd )3
∂ [a(x, ξ)] ei(x·(ξ−m)+y·(n−ξ)) dyd¯ξdx ∂xl
Z
a(x, ξ)(ξl − ml )ei(x·(ξ−m)+y·(n−ξ)) dyd¯ξdx
=−i Rd ×(Td )2
Z
a(x, ξ)ei(x·(ξ−m)+y·(n−ξ)) dyd¯ξdx
=iml Rd ×(Td )2
Z −
a(x, ξ)(iξl )ei(x·(ξ−m)+y·(n−ξ)) dyd¯ξdx
Rd ×(Td )2
=iml hAen , em i − hA(∂xl en ), em i =i(ml − nl )hAen , em i From this we see that differentiating in x gives a symbol matrix of the same order, so we could have repeated the argument with x derivatives, showing that derivatives in x do not change the order. Thus, we have 0 that A ∈ Ψr (Td ) for r0 > r. This is quite close to what we would like to prove, but we want to be able to take r0 = r above. To show this, we will use the idea of the proof of the L2 boundedness argument along with the formula for getting at the symbol via the action on exponentials we showed in (5.3). Letting B γ = Aγ (1 + ∆)−(r−|γ|)/2 , we have shown that B γ ∈ Ψ (Td ) for all > 0. The symbol of B γ is bγ (x, ξ) = (1 + |ξ|2 )−(r−|γ|)/2
∂γ a(x, ξ) ∂ξ γ
We would like to show that B γ ∈ Ψ0 (Td ), which will show that a(x, ξ) is a symbol of order r, so A ∈ Ψr (Td ). To do so, let C be a fixed constant and consider C − (B γ )∗ B γ ∈ Ψ (Td ) for all > 0. Because (B γ )∗ B γ has a symbol matrix of order 0 by assumption, we know that it is bounded on L2 (Td ). Thus, we see that it is a positive operator for large enough C: choosing C > k(B γ )∗ B γ kL (L2 (Td )) , we have that for f ∈ L2 (Td ) (5.11)
h(C − (B γ )∗ B γ )f, f iL2 (Td ) ≥ 0
Now, we will make use of the following theorem from [7]. Theorem 5.11. (Shubin III.18) For X an open set in Rn and properly supported A ∈ Ψs (X), ψ(x) ∈ C ∞ (X), with nonzero gradient, ψx0 (x) 6=
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 39
0 for x ∈ X. Then for any f ∈ C ∞ (X) and N ≥ 0 and λ ≥ 1 we have X Dα (f (z)eiλρx (z) ) e−iλψ A(f eiλψ ) = |z=x a(α) (x, λψx0 (x)) z α! |α| λ0 the second term is smaller in absolute value than 2 . Thus, we have that Z χx0 (x)(C − |bγ (x, λψx0 (x)|2 ) dx ≥ −1 Td
This implies that
inf (C − |bγ (x, λψx0 (x))|2 ) ≥ −2
x∈B (x0 )
40
OTIS CHODOSH
Thus
2 This immediately implies that for ξ with |ξ| > λ0 r 2 γ |b (x, ξ)| ≤ C + |bγ (x, λψx0 (x))|2 ≤ C +
From this, we see that B γ ∈ Ψ0 (Td ), implying that A ∈ Ψr (Td ). 5.3. Isotropic Calculus. Now, we will state and prove a similar statement about isotropic pseudo-differential operators on R. Here, we will use use the Hermite functions as a basis for L2 (R) as described in (A) Section 4. As before, we define Km,n = hAφn , φm iL2 (R) (this time for m, n ∈ N0 ). With some modifications, the method of proof employed Theorem 5.7 will allow us to prove the following Theorem 5.12. For A a continuous linear operator A : S(R) → S(R), (A) A ∈ Ψriso (R), if and only if Km,n is a symbol matrix of order r/2 (as in Definition 5.2)6. Proof. Let A ∈ Ψriso (R). Multiplying by powers of the harmonic oscillator and using L2 (R) boundedness arguments, the estimates (A) |Km,n | . (1 + |m| + |n|)r/2
follow exactly as in the toroidal case above. Now, we establish off diagonal decay. First, because φn is an eigenfunction of the Fourier transform, as in Lemma 4.10, we can rewrite (A) Km,n as Z √ n (5.12) 2π(−i) eix·ξ a(x, ξ)φn (ξ)φm (x) dxd¯ξ Rd ×Rd
This allows us to make the following calculation, where Hx = x2 − 2 and Hξ = ξ 2 − ∂∂2 ξ
∂2 ∂x2
(A) 2(m − n)Km,n (A) = 2(m + 1 − n − 1)Km,n 6Note
(A)
that the order of Km,n is r/2, as opposed to r in Theorem 5.7. This should not come as a surprise, because n-th eigenvalue of the Laplacian on T is of order |n|2 , while the harmonic oscillator on R (which is a fundamental example of an isotropic operator) has its n-th eigenvalue on the order of |n|. Both of these are second order differential operators, so we see here an indication that an order r isotropic pseudo-differential operator should (and it will turn out to be true) correspond to r/2 symbol matrixes.
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 41
=
√
n
Z
2π(−i)
eix·ξ a(x, ξ)(Hx − Hξ ) [φn (ξ)φm (x)] dxd¯ξ
Rd ×Rd
Z √ n = 2π(−i)
(Hx − Hξ ) eix·ξ a(x, ξ) φn (ξ)φm (x) dxd¯ξ
Rd ×Rd
Now, we will show that the difference of harmonic oscillator applied to eixξ a(x, ξ) results in eixξ a ˜(x, ξ) where a ˜(x, ξ) is a new symbol, still of order r. (Hx − Hξ ) eixξ a(x, ξ) ∂2 ∂2 2 2 ixξ x −ξ − 2 + 2 =e eixξ a(x, ξ) ∂ x ∂ ξ = x2 − ξ 2 + ξ 2 − x2 eixξ a(x, ξ) ∂ ∂2 ∂2 ∂ ixξ + ix − 2 + 2 a(x, ξ) +e −iξ ∂x ∂ξ ∂ x ∂ ξ ∂ ∂2 ∂2 ∂ ixξ + ix − 2 + 2 a(x, ξ) =e −iξ ∂x ∂ξ ∂ x ∂ ξ = eixξ a ˜(x, ξ) (A)
Applying this repeatedly, we see that for any N ≥ 0, multiplying Km,n ˜ (A) by (m − n)N is the same as considering Km,n for A˜ ∈ Ψriso (R), and thus by the L2 boundedness argument (5.13)
|Km,n | . CN (1 + |n − m|)−N (1 + |m| + |n|)r/2
For the difference operator, notice that (A) =hAφn+1 , φm+1 i − hAφn , φm i 4 Km,n 1 =p hAC † φn , C † φm i − hAφn , φm i (m + 1)(n + 1) 1 =p hCAC † φn , φm i − hAφn , φm i (m + 1)(n + 1) 1 =p h(AC + [C, A])C † φn , φm i − hAφn , φm i (m + 1)(n + 1) ! n+1 = p − 1 hAφn , φm i (m + 1)(n + 1) 1 +p h([C, A])C † φn , φm i (m + 1)(n + 1)
42
OTIS CHODOSH
r =
! n+1 − 1 hAφn , φm i m+1
1 +p h([C, A])C † φn , φm i (m + 1)(n + 1) √ √ n+1− m+1 √ hAφn , φm i = m+1 1 +p h([C, A])C † φn , φm i (m + 1)(n + 1) n−m √ √ = √ hAφn , φm i m + 1( n + 1 + m + 1) 1 +p h([C, A])C † φn , φm i (m + 1)(n + 1) Because p √ √ ( m + 1 + n + 1)2 = m + n + 2 + 2 (m + 1)(n + 1) ≥ (1 + |m| + |n|) and (m + 1)(n + 1) = (1 + m + n + mn) & (1 + |m| + |n|)2 We can combine these inequalities with (5.13) (we can repeat the above argument for [C, A]C † , which is a degree r+1 isotropic operator) we see that the difference operator will lower the degree in |m| + |n| by 1 and still preserve the off diagonal boundedness. Repeating this argument, (A) we see that Km,n is a symbol matrix of order r/2. (A) To prove the converse, assume that Km,n is a symbol matrix of order r/2 of a continuous linear operator A : S(R) → S(R). Letting K(x, y) be the Schwartz kernel of A, we define a distribution Z a(x, ξ) = − ei(y−x)·ξ K(x, y) d¯y R
This implies that (A) Km,n
=
√
n
Z
2π(−i)
eix·ξ a(x, ξ)φn (ξ)φm (x) dxd¯ξ
R×R ∂a However, if we were to replace a(x, ξ) with ∂x (x, ξ) we see from the following calculation that this results in a a symbol matrix of order (r − 1)/2.
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 43
Z √ n 2π(−i)
∂a eix·ξ (x, ξ)φn (ξ)φm (x)dξdx ∂x R×R Z √ ∂ ix·ξ n+1 = 2π(−i) e a(x, ξ) φn (ξ)φm (x)dξdx ξ+i ∂x R×R Z √ ∂ n+1 ix·ξ = 2π(−i) e a(x, ξ) ξ − i φn (ξ)φm (x)dξdx ∂x R×R r Z √ n+1 eix·ξ a(x, ξ) φn+1 (ξ)φm (x) = 2π(−i)n+1 2 R×R r r n m+1 + φn−1 (ξ)φm (x) + i φn (ξ)φm+1 (x) 2 2 r m −i φn (ξ)φm−1 (x) dξdx 2 r r r r n + 1 (A) n (A) m + 1 (A) m (A) = Km,n+1 − Km,n−1 + Km+1,n − K 2 2 2 2 m−1,n r r r n + 1 (A) n + 1 (A) n + 1 (A) = Km,n+1 − Km−1,n + Km−1,n 2 2 2 r r r m (A) n (A) n (A) Km−1,n + Km+1,n − K − 2 2 2 m,n−1 r r m + 1 (A) n (A) + Km+1,n − K 2 2 m+1,n r r r ! n+1 n+1 m (A) (A) = 4 Km−1,n + − Km−1,n 2 2 2 r r r ! n m+1 n (A) (A) + 4 Km,n−1 + − Km+1,n 2 2 2
As claimed above, by Lemma 5.4, this is a symbol matrix of order (r − 1)/2. Thus, we define for α, β ∈ N0 , Aα,β to be the operator with Schwartz kernel Z ∂ α+β a α,β K (x, y) = ei(x−y)·ξ α β (x, ξ) d¯ξ ∂x ∂ξ R From our above results, we have Aα,β : S(R) → S(R) and its matrix is a symbol matrix of order (r − α − β)/2. Now, for s ∈ R, we define Bsα,β = (1 + H)s Aα,β . By Lemma 5.3, if r − s − α − β < −1, then Bsα,β is Hilbert Schmidt, so by Theorem 5.9 there is Ksα,β (x, y) ∈ L2 (R × R)
44
OTIS CHODOSH
such that for f ∈ S(R) Bsα,β f
Z =
Ksα,β (x, y)f (y) dy
R
This implies that for r(x, ξ) the symbol of (1 + H)−s Z α,β −s α,β A f = (1 + H) Ks (y, z)f (z) dz R Z = ei(x−y)ξ r(x, ξ)Ksα,β (y, z)f (z) dzdyd¯ξ Taking the Fourier transform ∂ α+β a (x, ξ) ∈ (1 + |x|2 + |ξ|2 )s L2 (R × R) ∂xα ∂ξ β for r − s − α − β < −1 because the Fourier transform of Ksα,β is square integrable, and the principal symbol of (1 + H)−s is (1 + x2 + ξ 2 )−s and the lower order terms will only improve the integrability. By Sobolev embedding this implies that a(x, ξ) ∈ C ∞ (R). Then by Lemma 5.10 for r0 with r − r0 − α − β < 0 it also implies ∂ α+β a 0 (x, ξ) ∈ (1 + |x|2 + |ξ|2 )r L∞ (R × R) α β ∂x ∂ξ 0
0
Thus, we see that A ∈ Ψriso (R) for r0 > r, and Bsα,β ∈ Ψriso+2s−α−β (R) for r0 > r. Thus, to complete the proof, all that we must show is that A ∈ Ψriso (R). Notice that this implies that for r + 2s = α + β 0 (R) for any 0 > 0. However, by Lemma 5.6, we see that for Bsα,β ∈ Ψiso r + 2s = α + β, Bsα,β is bounded on L2 (R). Thus, using the following lemma, we can thus conclude that the symbol of Bsα,β is bounded, which them implies that A ∈ Ψriso (R) because that would give the necessary bounds on each derivative. 0 Lemma 5.13. For Q ∈ Ψiso (R) for some 0 < 1/2, if Q is bounded on 2 L (R), then the symbol of Q, σL (Q) = q(x, ξ) is bounded, i.e.
sup |q(x, ξ)| < ∞ (x,ξ)∈R2
Proof. Because Q is bounded on L2 (R) there is some C > 0 such that √ kQf kL2 (R) ≤ Ckf kL2 (R) Choose > 0, z0 ∈ R and ϕ(x) ∈ C0∞ (R) a bump function with ϕ(x) ∈ [0, 1] and ϕ ≡ 1 for |x| < /2 and ϕ ≡ 0 for |x| > . Let ϕz0 (x) = ϕ(x − z0 ). For v ∈ [−1, 1] we thus have that 2
kQ(eiy v ϕz0 (y))k2L2 (R) ≤ Ckϕz0 k2L2 (R) = Ckϕk2L2 (R)
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 45
Thus, we have that (5.14)
D E 2 2 Iv (z0 ) := e−ix v Q∗ Q(eiy v ϕ(y)), ϕ
L2 (R)
≤ Ckϕk2L2 (R)
Letting p(x, ξ) = σL (Q∗ Q) Z 2 2 Iv (z0 ) = ei(x−y)ξ p(x, ξ)ei(y −x )v ϕz0 (y)ϕz0 (x) dyd¯ξdx Z 2 2 = ei(x−y)ξ p(x + z0 , ξ)ei(y −x )v ei(y−x)2vz0 ϕ(y)ϕ(x) dyd¯ξdx Z 2 2 = ei(x−y)ξ p(x + z0 , ξ + 2vz0 )ei(y −x )v ϕ(y)ϕ(x) dyd¯ξdx Z 2 = ei(x−y)η p(x + z0 , 2vx + 2vz0 + η)ei(y−x) v ϕ(y)ϕ(x) dyd¯ξdx In the last line, we made the substitution ξ = 2vx + η. Now, expanding around η = 0 p(x + z0 , 2vx + 2vz0 + η) Z = p(x + z0 , 2vx + 2vz0 ) +
1
p2 (x + z0 , 2vx + 2vz0 + tη)η dt 0
Combining this with the above, the first term simplifies via Fourier inversion giving Z p(x + z0 , 2vx + 2vz0 )ϕ(x)2 dx Now, we will show that as |z0 | → ∞ the second term goes to zero. Let χ ∈ Cc∞ be a bump function χ(η) ∈ [0, 1], χ(η) ≡ 1 for |η| < 1 and χ(η) ≡ 0 for |η| > 2. Using this we can estimate the second term where η is small Z Z t=1 ei(x−y)η p2 (x + z0 , 2vx + 2vz0 + tη) t=0 i(y−x)2 v ηχ(η)e ϕ(y)ϕ(x) dtdyd¯ηdx Z Z 1 ≤ |p2 (x + z0 , 2vx + 2vz0 + tη)||η||χ(η)||ϕ(y)||ϕ(x)| dtdyd¯ηdx 0 Z Z 1 ≤C (1 + |x + z0 | + |2vx + 2vz0 + tη|)0 −1 0
|η||χ(η)||ϕ(y)||ϕ(x)| dtdyd¯ηdx ≤ C 0 (1 − + |z0 |)0 −1
46
OTIS CHODOSH
In the last line, we used that on the support of ϕ(x), |x| ≤ , so |x + z0 | ≥ |z0 | − |x| ≥ |z0 | − which gives the above inequality. Similarly, when η is bounded away from zero, we can integrate by parts, obtaining the bounds Z Z t=1 1 ∂y3 (ei(x−y)η )p2 (x + z0 , 2vx + 2vz0 + tη) 3 t=0 (−iη) 2 η(1 − χ(η))ei(y−x) v ϕ(y)ϕ(x) dtdyd¯ηdx Z Z t=1 1 = (ei(x−y)η )p2 (x + z0 , 2vx + 2vz0 + tη) 3 t=0 η 3 i(y−x)2 v η(1 − χ(η))∂y (e ϕ(y))ϕ(x) dtdyd¯ηdx Z Z t=1 1 3 i(y−x)2 v 00 0 −1 ≤ C (1 − + |z0 |) |∂ (e ϕ(y))||ϕ(x)| dtdyd¯ηdx 2 y t=0 |η| ≤ C 000 (1 − + |z0 |)0 −1 Combining all of this into (5.14) we thus have Z (5.15) p(x + z0 , 2vx + 2vz0 )ϕ(x)2 dx + R(z0 ) ≤ Ckϕk2L2 (R) where R(z0 ) is the sum of the above two terms, and by the above bounds, we know that R(z0 ) → 0 as |z0 | → ∞. Because all derivatives of p go to zero, p is uniformly continuous, and thus for δ > 0, we can take in the definition of ϕ small enough so that Z [ sup p(x + z0 , 2vx + 2vz0 ) − p(x + z0 , 2vx + 2vz0 )]ϕ(x)2 dx < δ x∈B (0)
This combined with (5.15) gives bounds of the form |p(z0 , 2vz0 )| ≤ P for some P ≥ 0 and for large |z0 | (and thus for all z0 because p is certainly bounded inside of a compact set). From the above proof, it is clear that P does not depend on v ∈ [−1, 1], so we know that sup |p(x, ξ)| ≤ P |x|≥2|ξ|
by taking appropriate v and z0 in these bounds. For q the symbol of Q, this shows that √ sup |q(x, ξ)| ≤ P |x|≥2|ξ|
Now, to extend this to all of R × R, taking the Fourier transform of ˆ as discussed in Section 3, applying the the operator Q, denoted Q
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 47
ˆ (which is still bounded on L2 because the Fourier above argument to Q transform is an isometry L2 (R) → L2 (R)) we have that √ sup |q(ξ, x)| ≤ P 2 |x|≥2|ξ|
This shows that q is bounded, proving the Lemma.
This shows that A ∈ Ψriso (R), finishing the proof of the Theorem. This theorem does not hold as stated in Rd for d ≥ 2. For example, consider C1† , the creation operator in the first coordinate. Recall that √ C1† φn = n1 + 1φn+e1 Thus
√ (C1† ) Km,n = hC1† φn , φm i = n1 + 1δn+e1 ,m This has the appropriate decay; away from the diagonal it is zero, and for n + e1 = m it is order 1/2 in |n|, as in the d = 1 case. However, taking the difference operator in the first coordinate gives √ √ (C1† ) 41 Km,n = ( n1 + 3 − n1 + 1)δn+e1 ,m 2 √ =√ δn+e1 ,m n1 + 3 + n1 + 1 This is order −1/2 in n1 , but merely bounded for e.g. n2 , so in general the order in |n| is only 0. Furthermore, taking further difference operators in the first coordinate do not change the order in |n| at all. It is possible to prove an analogous theorem for d ≥ 2. By using arguments similar to the proof of Theorem 5.12, one could prove the following Theorem 5.14. For A a continuous linear operator S(R) → S(R), (A) A ∈ Ψriso (Rd ) if and only if Km,n has the following property for all multi-indexes α, β ∈ Nd0 (5.16)
(A) |α,β Km,n | ≤ (1 + |m| + |n|)(r−|α|−|β|)/2
where we define xk Knm by r r r r nk + 1 nk mk + 1 mk Kn+ek ,m − Kn−ek ,m + Kn,m+ek − Kn,m−ek 2 2 2 2 and ξk Knm by r r r r mk + 1 mk nk + 1 nk Kn,m+ek + Kn,m−ek − Kn+ek ,m − Kn−ek ,m 2 2 2 2 and let α,β = (x1 )α1 · · · (xd )αd (ξ1 )β1 · · · (ξd )βd .
48
OTIS CHODOSH
The α,β operator is really the matrix analogue of applying ∂xα ∂ξβ to the symbol of A, as can be seen via calculations similar to those on page 43. The conditions of (5.16) are somewhat unsatisfying, as they are considerably more complex than the simple difference operator results in the other cases. For example, it can be shown that infinite matrixes obeying (5.16) are rapidly decreasing off of the diagonal, which is not obvious from the condition. It is likely that the issues relating to the difference operator in higher dimensions is a consequence of the high multiplicity of the harmonic oscillator eigenspaces. It seems possible that a better result could be obtained through a “rearranging” of the eigenspaces. That is, the difference operator compares the operator’s action on φn and φn+ek , elements of the |n| + d and the |n| + d + 1 eigenspaces, but there is no reason that these are the proper elements from these two eigenspaces to compare. However, we have been unable to find a satisfactory manner in which to compare the two eigenspaces in order to obtain a simpler condition on the matrix. 6. Acknowledgements I am very grateful to my advisor, Andr´as Vasy, for teaching me about pseudo-differential operators, suggesting and then greatly assisting me with this work, and for his endless encouragement and support over the last few years. I could not have wished for a better advisor. I would like to thank the entire Stanford Mathematics Department for everything I have learned over these past four years, as well as financial support in the summer of 2009 in the form of a VPUE department grant. I am indebted to Michael Eichmair, for his support and teaching, particularly for his astute recommendation that I work on my honors thesis with Professor Vasy. I would also like to thank all of the friends I have made at Stanford, in particular, Yakov Shlapentokh-Rothman for countless hours spent learning mathematics, as well as Alison Law for her support, understanding and encouragement. Finally, I would like to convey my incredible gratitude to my parents and sister, Abigail, James and Ursula. References [1] P. R. Halmos and V. S. Sunder, Bounded integral operators on L2 spaces, A Series of modern surveys in mathematics ; 96, Springer-Verlag, Berlin, 1978. ¨ rmander, The Analysis of Linear Partial Differential Operators I, II, [2] L. Ho III, Springer, 2007.
MATRIX REPRESENTATIONS OF PSEUDO-DIFFERENTIAL OPERATORS 49
[3] M. S. Joshi, Introduction to pseudo-differential operators, http://arxiv.org/ abs/math/9906155v1, (1999). [4] R. Melrose, Unpublished lecture notes: Introduction to microlocal analysis, http://math.mit.edu/~rbm/18.157-F09/18.157-F09.html, (2009). [5] M. Reed and B. Simon, Methods of modern mathematical physics, Academic Press, New York, 1980. [6] M. Ruzhansky and V. Turunen, On the toroidal quantization of periodic pseudo-differential operators, Numerical Functional Analysis and Optimization, 30 (2009), pp. 1098–1124. [7] M. A. Shubin, Pseudodifferential operators and spectral theory, SpringerVerlag, Berlin, 1987. Department of Mathematics, Stanford University, Stanford, CA 94305-2125 USA E-mail address:
[email protected]
