Fourier Analysis Second Semester 2009/10 Homework Assignment 4 (Due on 16th March 2010. Please staple multiple sheets together.)

Only the questions marked with an asterisk (*) will count towards the assessment for this course. *1. In lectures we have proved the Heisenberg uncertainty principle, which says for ψ ∈ S(R)

such that ∥ψ∥_{L^{2}(R)} = 1 we have that

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

Use this to prove that

(∫_{−∞}^{∞}(x − x_{0})^{2}|ψ(x)|^{2} dx) (∫_{−∞}^{∞}(ξ − ξ_{0})^{2}|ψ̂(ξ)|^{2} dξ) ≥ \frac{1}{16π^{2}}

for each x_{0}, ξ_{0} ∈ R.

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 Consider the initial value problem of finding u: R × [0, ∞) → C such that

\{ \begin{matrix}∂_{t}u(x, t) − ∂_{x}^{2}u(x, t) = 0 & for x ∈ R and t > 0, \\ u(x, 0) = f(x) & for x ∈ R, \end{matrix}

given data f ∈ S(R).

(a) Use the Fourier transform to derive a formula for a solution to be

u(x, t) = (H_{t} ∗ f)(x)  (⋆)

where H_{t} is the heat kernel and is defined as

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

and ∗ denotes convolution in R.

(b) Show directly that the Heat kernel satisfies

∂_{t}H_{t}(x) − ∂_{x}^{2}H_{t}(x) = 0

for x ∈ R and t > 0.

(c) Show that {H_{t}}_{t>0} is a family of good kernels and so ∥u(·, t) − f∥_{L^{2}(R)} → 0 as t → 0.

(d) 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, for u is given by formula (⋆). (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. For each t ∈ R, consider the operators A_{t} and A_{t}^{∗} defined on f ∈ S(R) by

A_{t}f(x) = f′(x) + txf(x) and A_{t}^{∗}f(x) = −f′(x) + txf(x).

When t = 1 the operators A_{t} and A_{t}^{∗} 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) (A_{t}^{∗}f, g) = (f, A_{t}g), (b) (A_{t}^{∗}A_{t}f, g) ≥ 0,

(c) A_{t}^{∗}A_{t} = L_{t} − tI, where L_{t}(f)(x) = −f′′(x) + t^{2}x^{2}f(x) and I is the identity operator. Use these facts to prove (L_{t}(f), f) ≥ t(f, f) for all f ∈ S(R) and use it to provide an alternative proof of the Heisenberg uncertainty principle.

6. Suppose g: R × [0, 1] → C is such that g(·, t) ∈ M(R) and ∂_{2}g(·, t) ∈ M(R) for each t ∈ [0, 1] and there exists a function Φ ∈ M(R) such that

|∂_{2}g(x, t) − ∂_{2}g(x, s)| ≤ |t − s|Φ(x)

for all s, t, x ∈ R. Here ∂_{2} denotes differentiation with respect to the second variable. Prove that G(·) = ∫_{−∞}^{∞} g(x, ·)dx is a differentiable function and

G′(t) = ∫_{−∞}^{∞} ∂_{2}g(x, t)dx.