The Schwartz Space: Tools for Quantum Mechanics and Infinite Dimensional Analysis

In 1950–1951, Laurent Schwartz published a two volumes work Théorie des Distributions [1,2], where he provided a convenient formalism for the theory of distributions. The purpose of this paper is to present a self-contained account of the main ideas, results, techniques, and proofs that underlie the approach to distribution theory that is central to aspects of quantum mechanics and infinite dimensional analysis. This approach develops the structure of the space of Schwartz test functions by utilizing the operator

This operator arose in quantum mechanics as the Hamiltonian for a harmonic oscillator and, in that context as well as in white noise analysis, the operator N = T − 1 is called the number operator. The physical context provides additional useful mathematical tools such as creation and annihilation operators, which we examine in detail.

In this paper we include under one roof all the essential necessary notions of this approach to the test function space $\mathcal{S}({\mathbf{R}}^n)$. The relevant notions concerning topological vector spaces are presented so that the reader need not wade through the many voluminous available works on this subject. We also describe in brief the origins of the relevant notions in quantum mechanics.

We present

• the essential notions and results concerning topological vector spaces;
• a detailed analysis of the creation operator C, the annihilation operator A, the number operator N and the harmonic oscillator Hamiltonian T;
• a detailed account of the Schwartz space $\mathcal{S}({\mathbf{R}}^d)$, and its topology, as a decreasing intersection of subspaces ${\mathcal{S}}_p({\mathbf{R}}^d)$, for p ∈ {0, 1, 2, …}:

  $$\mathcal{S}({\mathbf{R}}^d)={\displaystyle \underset{p\ge 0}{\cap }{\mathcal{S}}_p({\mathbf{R}}^d)\subset \cdots }\subset {\mathcal{S}}_2({\mathbf{R}}^d)\subset {\mathcal{S}}_1({\mathbf{R}}^d)\subset L^2({\mathbf{R}}^d)$$
• an exact characterization of the functions in the space ${\mathcal{S}}_p(\mathbf{R})$;
• summary of notions from spectral theory and quantum mechanics;

Our exposition of the properties of T and of $\mathcal{S}(\mathbf{R})$ follows Simon’s paper [3], but we provide more detail and our notational conventions are along the lines now standard in infinite-dimensional distribution theory.

The classic work on spaces of smooth functions and their duals is that of Schwartz [1,2]. Our purpose is to present a concise and coherent account of the essential ideas and results of the theory. Of the results that we discuss, many can be found in other works such as [1–6], which is not meant to be a comprehensive list. We have presented portions of this material previously in [7], but also provide it here for convenience. The approach we take has a direct counterpart in the theory of distributions over infinite dimensional spaces [8,9].

In this section we summarize the basic notions, notation, and results that we discuss in more detail in later sections. Here, and later in this paper, we work mainly with the case of functions of one variable and then describe the generalization to the multi-dimensional case.

We use the letter W to denote the set of all non-negative integers:

The Schwartz space is a topological vector space, i.e., it is a vector space equipped with a Hausdorff topology with respect to which the vector space operations (addition, and multiplication by scalar) are continuous. In this section we shall go through a few of the basic notions and results for topological vector spaces.

Let V be a real vector space. A vector topology τ on V is a topology such that addition V × V → V : (x, y) ⟼ x + y and scalar multiplication R × V → V : (t, x) ⟼ tx are continuous. If V is a complex vector space we require that C × V → V : (α, x) ⟼ αx be continuous.

It is useful to observe that when V is equipped with a vector topology, the translation maps are continuous, for every x ∈ V, and are hence also homeomorphisms since $t_x^{-1}=t_{-x}$.

A topological vector space is a vector space equipped with a Hausdorff vector topology. A local base of a vector topology τ is a family of open sets {U_α}_α∈I containing 0 such that if W is any open set containing 0 then W contains some U_α. If U is any open set and x any point in U then U − x is an open neighborhood of 0 and hence contains some U_α, and so U itself contains a neighborhood x + U_α of x:

Doing this for each point x of U, we see that each open set is the union of translates of the local base sets U_α.

Our objective in this section is to show that the Schwartz space is complete, in the sense that every Cauchy sequence converges. Recall that $\mathcal{S}(\mathbf{R})$ is the set of all C^∞ functions f on R for which for all a, b ∈ W = {0, 1, 2, …}. The functions p_a,b are semi-norms, with ║ · ║_0,0, being just the sup-norm. Thus the family of semi-norms given above specify a Hausdorff vector topology on $\mathcal{S}(\mathbf{R})$. We will call this the Schwartz topology on $\mathcal{S}(\mathbf{R})$.

Theorem 4. The topology on$\mathcal{S}(\mathbf{R})$ generated by the family of semi-norms ║·║_a,b for all a, b ∈ {0, 1, 2, …}, is complete.

Proof. Let (f_n)_n≥1 be a Cauchy sequence on $\mathcal{S}(\mathbf{R})$. Then this sequence is Cauchy in each of the semi-norms ║·║_a,b, and so each sequence of functions x^aD^bf_n(x) is uniformly convergent. Let

Let f = g₀. Using a Taylor theorem argument it follows that g_b is D^bf. For instance, for b = 1, observe first that and so, letting n → ∞, we have which implies that f′(x) exists and equals g₁(x).

In this way, we have x^aD^bf_n(x) → x^aD^bf(x) pointwise. Note that our Cauchy hypothesis implies that the sequence of functions x^aD^bf_n(x) is Cauchy in sup-norm, and so the convergence is uniform. In particular, the sup-norm of x^aD^bf(x) is finite, since it is the limit of a uniformly convergent sequence of bounded functions. Thus $f\in \mathcal{S}(\mathbf{R})$.

Finally, we have to check that f_n converges to f in the topology of $\mathcal{S}(\mathbf{R})$. We have noted above that x^aD^bf_n(x) → x^aD^bf(x) uniformly. Thus f_n → f relative to the semi-norm ║·║_a,b. Since this holds for every a, b ∈ {0, 1, 2, 3, …}, we have f_n → f in the topology of $\mathcal{S}(\mathbf{R})$. □

Now let’s take a quick look at the Schwartz space $\mathcal{S}({\mathbf{R}}^d)$. First some notation. A multi-index a is an element of {0, 1, 2, …}^d, i.e., it is a mapping

If a is a multi-index, we write |a| to mean the sum a₁+⋯+a_d, x^a to mean the product $x_1^{a_1}\dots x_d^{a_d}$, and D^a to mean the differential operator $x_{x_1}^{a_1}\dots x_{x_d}^{a_d}$. The space $\mathcal{S}({\mathbf{R}}^d)$ consists of all C^∞ functions f on Rd such that each function x^aD^bf(x) is bounded. On $\mathcal{S}({\mathbf{R}}^d)$ we have the semi-norms for each pair of multi-indices a and b. The Schwartz topology on $\mathcal{S}({\mathbf{R}}^d)$ is the smallest topology making each semi-norm ║·║_a,b continuous. This makes $\mathcal{S}({\mathbf{R}}^d)$ a topological vector space.

The argument for the proof of the preceding theorem goes through with minor alterations and shows that:

Theorem 5. The topology on$\mathcal{S}({\mathbf{R}}^d)$ generated by the family of semi-norms ║·║_a,b for all a, b ∈ {0, 1, 2, …}^d, is complete.

We shall summarize the definition and basic properties of Hermite polynomials (our approach is essentially that of Hermite’s original [11]). We repeat for convenience of reference much of the presentation in Section 2.1 of [7].

A central role is played by the Gaussian kernel

Properties of translates of p are obtained from

Expanding the right side in a Taylor series we have where the Taylor coefficients, denoted H_n(x), are

This is the n–th Hermite polynomial and is indeed an n–th degree polynomial in which xⁿ has coefficient 1, facts which may be checked by induction.

Observe the following

Going over to the Taylor series and comparing the appropriate Taylor coefficients (differentiation with respect to y and z can be carried out under the integral) we have

Thus an orthonormal set of functions is given by

Because these are orthogonal polynomials, the n–th one being exactly of degree n, their span contains all polynomials. It can be shown that the span is in fact dense in L²(p(x)dx). Thus the polynomials above constitute an orthonormal basis of L²(p(x)dx).

Next, consider the derivative of H_n:

So

The operator is called the creation operator in L²(R;p(x)dx).

Officially, we can take the creation operator to have domain consisting of all functions f which can be expanded in L²(p(x)dx) as ∑_n≥0 a_nh_n, with each a_n a complex number, and satisfying the condition ∑_n≥0(n + 1)|a_n|² < ∞; the action of the operator on f yields the function ${\displaystyle {\sum }_{n\ge 0}\sqrt{n+1}{a}_n{h}_{n+1}}$. This makes the creation operator unitarily equivalent to a multiplication operator (in the sense discussed later in subsection A.5) and hence a closed operator (see A.1 for definition). For the type of smooth functions f we will mostly work with, the effect of the operator on f will in fact be given by application of $\left(-\frac{d}{dx}+x\right)$ to f.

Next, from the fundamental generating relation Equation (41) we have :

Using Equation (41) again on the left we have

Letting y = 0 allows us to equate the n = 0 terms, and then, successively, the higher order terms. From this we see that where H₋₁ = 0. Thus:

The operator is the annihilation operator in L²(R;p(x)dx). As with the creation operator, we may define it in a more specific way, as a closed operator on a specified domain.

In the preceding section we studied Hermite polynomials in the setting of the Gaussian space L²(R;p(x)dx). Let us translate the concepts and results back to the usual space L²(R; dx).

To this end, consider the isomorphism:

Then the orthonormal basis polynomials h_n go over to the functions ϕ_n given by

The family {ϕ_n}_n≥0 forms an orthonormal basis for L²(R, dx).

We now determine the annihilation and creation operators on L²(R, dx). If f ∈ L²(R, dx) is differentiable and has derivative f′ also in L²(R, dx), we have:

So, on L²(R, dx), the annihilator operator is which will satisfy where ϕ₋₁ = 0. For the moment, we proceed by taking the domain of A to be the Schwartz space $\mathcal{S}(\mathbf{R})$.

Next,

Thus the creation operator is

The reason we have written A* is that, as is readily checked, we have the adjoint relation with the inner-product being the usual one on L²(R, dx). Again, for the moment, we take the domain of C to be the Schwartz space $\mathcal{S}(\mathbf{R})$ (though, technically, in that case we should not write C as A^∗, since the latter, if viewed as the L²–adjoint operator, has a larger domain).

For this we have

Observe also that which imply:

Next observe that and so CA is called the number operator N:

As noted above in Equation (59), the number operator N has the eigenfunctions ϕ_n:

Integration by parts (see Lemma 10) shows that for every $f,g\in \mathcal{S}(\mathbf{R})$, and so also

It follows that the operator N satisfies for every $f,g\in \mathcal{S}(\mathbf{R})$.

Now consider the case of R^d. For each j ∈ {1, …, d}, there are creation, annihilation, and number operators:

These map $\mathcal{S}({\mathbf{R}}^d)$ into itself and, as is readily verified, satisfy the commutation relations

Now let us be more specific about the precise definition of the creation and annihilation operators. The basis {ϕ_n}_n≥0 for L²(R) yields an orthonormal basis ${\left\{{\varphi }_m\right\}}_m{}_{\in W^d}$ of L²(R^d) given by ${\varphi }_m={\varphi }_{m_1}(x_1)\cdots {\varphi }_{m_d}(x_d)$. For convenience we say ϕ_m=0 if some m_j <0. Given its effect on the orthonormal basis ${\left\{{\varphi }_m\right\}}_m{}_{\in W^d}$ the operator C_k has the form: where ${m^{\prime }}_i=m_i$ for all i ∈ {1, …, d} except when i = k, in which case ${m^{\prime }}_k=m_k+1$. The domain of C_k is the set $\mathcal{D}(C_k)$ given by

The operator C_k is then officially defined by specifying its action on a typical element of its domain: where m′ is as before. The operator C_k is essentially the composite of a multiplication operator and a bounded linear map taking ϕ_m → ϕ_m′ where m′ is as defined above. (See subsection A.5 for precise formulation of a multiplication operator.) Noting this, it can be readily checked that C_k is a closed operator using the following argument: Let T be a bounded linear operator and M_h a multiplication operator (any closed operator will do); we show that the composite M_hT is a closed operator. Suppose x_n → x. Since T is a bounded linear operator, Tx_n → Tx. Now suppose also that M_h(Tx_n) → y. Since M_h is closed, it follows then that $Tx\in \mathcal{D}(M_h)$ and y = M_hTx.

The operators A_k and N_k are defined analogously.

Proposition 6. Let $ℒ$₀ be the vector subspace of L²(R^d) spanned by the basis vectors${\left\{{\varphi }_m\right\}}_m{}_{\in W^d}$. Then for k ∈ {1, 2, …, d}, C_k|$ℒ$₀ and A_k|$ℒ$₀ have closures given by C_k and A_k, respectively (see subsection A.4 for the notion of closure).

Proof. We need to show that the graph of C_k, denoted Gr(C_k), is equal to the closure of the graph of C_k|$ℒ$₀, i.e., to $\overline{Gr(C_k|{ℒ}_0)}$ (see to subsection A.1 for the notion of graph). It is clear that $Gr(C_k|{ℒ}_0)\subseteq Gr(C_k)$. Using this and the fact that Ck is a closed operator, we have

Going in the other direction, take (f, C_kf) ∈ Gr(C_k). Now ${\displaystyle {\sum }_{m\in W^d}{a}_m{\varphi }_m}$ where a_m = ⟨f, ϕ_m⟩. Let f_N be given by

Observe that f_N ∈ $ℒ$₀. Moreover in L²(R^d). Thus $(f,C_{k}f)\in \overline{Gr(C_k|L_0)}$ and so we have $Gr(C_k)\subseteq \overline{Gr(C_k|L_0)}$.

The proof for A_k follows similarly. □

Linking this new definition for C_k with our earlier formulas Equation (63) we have:

Proposition 7. If$f\in \mathcal{S}({\mathbf{R}}^d)$ then

Proof. Let $g=\frac{\partial f}{\partial x_k}+\frac{x_k}{2}f$. Since $f\in \mathcal{S}({\mathbf{R}}^d)$, we have g ∈ L²(R^d). So we can write g as $g={\displaystyle {\sum }_{j\in W^d}{a}_j{\varphi }_j}$ where a_j=⟨g, ϕ_j⟩. Let us examine these a_j’s more closely. Observe where ${j^{″}}_i=j_i$ for all i ∈ {1, …, d} except when i = k, in which case ${j^{″}}_k=j_k-1$.

Bringing this information back to our expression for g we see that

The second equality is obtained by letting m = j″ and noting that ϕ_j″ = 0 when ${j^{″}}_k$ is −1. The proof follows similarly for A_k. □

Our aim here is to obtain a complete characterization of the functions in ${\mathcal{S}}_p(\mathbf{R})$. We will prove that ${\mathcal{S}}_p(\mathbf{R})$ consists of all square-integrable functions f for which all derivatives f^(k) exist for k ∈ {1, 2, …, p} and for all a, b ∈ {0, 1, …, p − 1} with a + b ≤ p − 1.

A significant tool we will use is the Fourier transform: