Hermite polynomials and Schwartz space

[Anden]

Homework Statement

I'm supposed to show that the Hermite Polynomials are in Schwartz space

$h_n = \frac{1}{\sqrt{n!}}(A^{\dagger})^n h_0$

where

$A^{\dagger} = \frac{1}{\sqrt{2}}(-\frac{d}{dx} + x)$

and

$h_0 = \pi^{-1/4}e^{-x^2/2}$

Homework Equations

Seminorm: $\|\phi\|_{\alpha,\beta} = \sup_{x\in\mathbb{R}^n}|x^{\alpha}\partial^{\beta}\phi(x)|$

($\alpha$ and $\beta$ are multi-indices)

Schwartz space is defined as being the set
$S(\mathbb{R}^n) = \{\phi\in C^{\infty}(\mathbb{R}^n)|\|\phi\|_{\alpha,\beta}< \infty \ \forall\alpha,\beta\in\mathbb{N}^n\}$

The Attempt at a Solution

My idea was to show that $A^{\dagger}: S(\mathbb{R})\rightarrow S(\mathbb{R})$ is a linear map from Schwartz space to itself. Then it would be enough to show that $h_0$ is in Schwartz space for the problem to be solved

Multiplication with $x$ and $d/dx$ are linear maps from Schwartz space to itself since according to the definition of the seminorm $\|\cdot\|_{\alpha,\beta},\ \alpha,\beta\in\mathbb{Z}_{>0}$ $\|x\phi(x)\|\leq \|\phi(x)\|_{\alpha+1,\beta} + \beta\|\phi(x)\|_{\alpha,\beta-1}$ and $\|\frac{d}{dx}\phi(x)\| = \|\phi(x)\|_{\alpha,\beta+1}$.
This means that the first part is finished.

Is this approach valid or have I completely misunderstood the concept? Also, how do I show that $e^{-x^2/2}\in S(\mathbb{R})$?

 

[mighty2000]

Your approach seems to be on the right track. To show that A^{\dagger} is a linear map from Schwartz space to itself, you can use the definition of linearity: for any two functions f and g in Schwartz space and any scalar c, A^{\dagger}(cf+g) = cA^{\dagger}(f) + A^{\dagger}(g).

To show that h_0 is in Schwartz space, you can use the definition of the seminorm and the properties of the exponential function. Specifically, you can use the fact that for any multi-index \alpha, \partial^{\alpha}e^{-x^2/2} is a polynomial in x, and therefore its absolute value is bounded by a constant for all x. This would show that h_0 satisfies the condition for all \alpha and \beta, and therefore is in Schwartz space.

I hope this helps. Let me know if you have any further questions.