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Rapid Hermite coefficient decay for Schwartz functions

Statement

Assume Countable Choice. Let $S (R^{2}; C)$ be the space of smooth complex-valued functions $f$ for which $\sup_{x \in R^{2}} ∣ x^{α} \partial^{β} f (x) ∣ < \infty$ for every pair of multi-indices $α, β \in N_{0}^{2}$ . Give this space the locally convex topology with basic neighborhoods ${f : p_{α_{j} β_{j}} (f - f_{0}) < ε_{j}, 1 \leq j \leq k}$ , where $k < \infty$ , $ε_{j} > 0$ , and $p_{α β} (u) = \sup_{x} ∣ x^{α} \partial^{β} u (x) ∣$ . Let $(h_{α})_{α \in N_{0}^{2}}$ be the product Hermite basis of $L^{2} (R^{2})$ obtained from the multidimensional oscillator at $ℏ = 1$ , $m_{1} = m_{2} = 1 / 2$ , and $ω_{1} = ω_{2} = 2$ , and write $∣ α ∣ = α_{1} + α_{2}$ . For $H = - \partial_{x_{1}}^{2} - \partial_{x_{2}}^{2} + ∣ x ∣^{2}$ on $S (R^{2}; C)$ , every integer $r \geq 0$ and every $f \in S (R^{2}; C)$ satisfy

$\sum_{α \in N_{0}^{2}} (1 + ∣ α ∣)^{2 r} ∣ ⟨ f, h_{α} ⟩_{L^{2}} ∣^{2} \leq ∥ H^{r} f ∥_{L^{2}}^{2} < \infty .$

Consequently the map $f \mapsto (⟨ f, h_{α} ⟩)_{α}$ is continuous from Schwartz space to each weighted square-summable sequence space with norm $(\sum_{α} (1 + ∣ α ∣)^{2 r} ∣ c_{α} ∣^{2})^{1 / 2}$ .

Statement references

Teschl, Mathematical Methods in Quantum Mechanics, §8.3

Laugesen, Spectral Theory of Partial Differential Equations, Chapter 3

Facts & Assumptions

Given: Countable Choice, the two-dimensional Schwartz space, the common product Hermite basis in the Statement, and the differential operator $H$ acting on Schwartz functions.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function.

[F1]

Anisotropic Hermite oscillator and quadratic form basis: at $N = 2$ and $a_{1} = a_{2} = c = 1$ , the product Hermite functions form a complete orthonormal basis and have oscillator eigenvalues $2 ∣ α ∣ + 2$ .

[F2]

Multidimensional oscillator completeness: at $d = 2$ , $ℏ = 1$ , $m_{j} = 1 / 2$ , and $ω_{j} = 2$ , the unscaled product basis is $h_{α} = g_{α_{1}} (x_{1}) g_{α_{2}} (x_{2})$ , with eigenvalue $2 ∣ α ∣ + 2$ .

[F3]

Hermite recurrence: $g_{n} = H_{n} g_{0} / \sqrt{2^{n} n!}$ , where $H_{n}$ is a degree- $n$ polynomial and $g_{0} (y) = π^{- 1 / 4} e^{- y^{2} / 2}$ .

[F4]

Fourier expansion in a Hilbert space: a complete orthonormal family in a Hilbert space gives the norm-convergent coefficient expansion and Parseval's identity.

Proof

technique · direct

Given: [A1], [F1]–[F3], $f \in S (R^{2}; C)$ , and an integer $r \geq 0$ .

1.1A1F1F2algebra

With these parameters, $ℏ^{2} / (2 m_{j}) = 1$ , $m_{j} ω_{j}^{2} / 2 = 1$ , and $ℓ_{j} = \sqrt{ℏ / (m_{j} ω_{j})} = 1$ . The parameter substitution in the inspected proof of [F1] uses precisely this scaling; [F2] specifies the resulting unscaled product basis $h_{α} = g_{α_{1}} (x_{1}) g_{α_{2}} (x_{2})$ and its eigenvalue $λ_{α} = \sum_{j = 1}^{2} (2 α_{j} + 1) = 2 ∣ α ∣ + 2$ .

2.1F3step 1.1algebra

By [F3] each Hermite factor is a polynomial times $e^{- x_{j}^{2} / 2}$ , so $h_{α}$ and all its derivatives decay rapidly enough for integration by parts against any Schwartz function. The operator $H^{r}$ is a finite sum of polynomial-coefficient derivatives and maps Schwartz space continuously to itself, hence $H^{r} f \in L^{2}$ . Repeated integration by parts in $x_{1}$ and $x_{2}$ has no boundary terms; it gives $⟨ H^{r} f, h_{α} ⟩ = ⟨ f, H^{r} h_{α} ⟩ = λ_{α}^{r} ⟨ f, h_{α} ⟩$ .

3.1F1F2F4step 2.1algebra

Parseval for the complete orthonormal basis [F1]–[F2] is supplied by [F4] and gives $∥ H^{r} f ∥_{2}^{2} = \sum_{α} ∣ ⟨ H^{r} f, h_{α} ⟩ ∣^{2} = \sum_{α} (2 ∣ α ∣ + 2)^{2 r} ∣ ⟨ f, h_{α} ⟩ ∣^{2}$ . Since $1 + ∣ α ∣ \leq 2 ∣ α ∣ + 2$ , the asserted weighted bound follows.

4.1step 3.1algebra∎

Each $H^{r}$ is a finite differential expression with polynomial coefficients, so its Schwartz seminorms bound the pointwise polynomially weighted derivatives in that expression. In two dimensions, $∥ u ∥_{2} \leq C \sup_{x} (1 + ∣ x ∣)^{2} ∣ u (x) ∣$ because $(1 + ∣ x ∣)^{- 4}$ is integrable. Thus $f \mapsto ∥ H^{r} f ∥_{2}$ is a continuous seminorm on Schwartz space, and the weighted coefficient map is continuous.

Rapid Hermite coefficient decay for Schwartz functions

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources