Fourier Analysis Second Semester 2008/9 Homework Assignment 5 (Due on 20th March 2009)

Only the questions marked with an asterisk (*) will count towards the assessment for this course. Most of these exercises are taken from Stein and Shakarchi.

*1. The aim of this question is to prove the Heisenberg uncertainty principle: Suppose ψ is a

Schwartz function such that

∫_{−∞}^{∞} |ψ(x)|^{2} dx = 1.

Then

(∫_{−∞}^{∞} x^{2}|ψ(x)|^{2} dx) (∫_{−∞}^{∞} ξ^{2}|ψ̂(ξ)|^{2} dξ) ≥ \frac{1}{16π^{2}} .  (†)

(a) Show that, under the hypothesis on ψ,

1 = − ∫_{−∞}^{∞} (xψ′(x)\overline{ψ(x)} + x\overline{ψ′(x)}ψ(x)) dx.  (‡)

(b) Use (‡) to conclude

1 ≤ 2 (∫_{−∞}^{∞} x^{2}|ψ(x)|^{2} dx)^{\frac{1}{2}} (∫_{−∞}^{∞} |ψ′(x)|^{2} dx)^{\frac{1}{2}}

(c) Use the above and what you know about the Fourier transform to complete the proof of

(†). *2. Define the Landau kernels L_{n}: R → C by

L_{n}(x) = \{ \begin{matrix}\frac{(1−x^{2})^{n}}{c_{n}} , & if |x| ≤ 1, \\ 0, & if |x| > 1. \end{matrix}

where c_{n} is chosen so that ∫_{R} L_{n} = 1.

(a) Prove that {L_{n}}_{n∈N} is a family of good kernels as n → ∞.

(b) Prove that, for a bounded uniformly continuous function f, the convolution (f ∗L_{n})(x) → f(x) uniformly in x.

(c) Show that given a (uniformly) continuous function f: R → C which is zero outside [−1/2, 1/2] and ε > 0, we can find a polynomial P such that |f(x) − P (x)| < ε for all x ∈ [−1/2, 1/2].

*3 Recall the heat kernel H_{t} is defined as

H_{t}(x) = \frac{1}{(4πt)^{\frac{1}{2}}} e^{−x^{2}/(4t)}.

Show that the function

v(x, t) = \frac{x}{t}H_{t}(x)

satisfies the heat equation for t > 0 and lim_{t→0} v(x, t) = 0 for all x ≠ 0, but v is not continuous at the origin.

4. For f ∈ S(R), suppose f(x) is non-negative for all x ∈ R, positive for |x| ≤ 1 and zero for |x| ≥ 2. Show that u(x, t) > 0 for all x ∈ R and t > 0. (This means that the heat equation propagates heat at infinite speed: here our initial heat distribution f was only non-zero in a neighbourhood of the origin, but for any positive t that heat is felt everywhere on R.)

5. Consider the operators A and A^{∗} defined on f ∈ S(R) by

Af(x) = f′(x) + xf(x) and A^{∗}f(x) = −f′(x) + xf(x).

The operators A and A^{∗} are sometimes called the annihilation and creation operators, respectively. Prove that for all f, g ∈ S(R) and

(f, g) = ∫_{R} f(x)\overline{g(x)} dx

we have

(a) (Af, g) = (f, A^{∗}g),

(b) (A^{∗}Af, g) ≥ 0,

(c) A^{∗}A = L − I, where L(f)(x) = −f′′(x) + x^{2}f(x) and I is the identity operator. Use these facts to prove L(f) ≥ f for all f ∈ S(R) and show this is equivalent to the Heisenberg uncertainty principle of question 1.