C. Cueni - Department Mathematik

Mathematisches Institut
LMU München
WS 2016-17
Prof. H. Siedentop
Dr. R. Helling
Dr. J.-C. Cuenin
Mathematical Quantum Mechanics
Problem Sheet 13
Hand-in deadline: 03/03/2017 before noon in the designated MQM box (1st
floor, next to the library).
Ex. 1: Determine whether the following potentials in R3 are dilation-analytic,
1{|x| ≤ 1},
1
,
1 + |x|2
1+
x21
1
,
+ x42 + x63
2
e−|x| ,
sin(|x|)
.
|x|
More precisely, determine for which α > 0 these potentials belong to Fα .
Ex. 2: Consider H0 = −d2 /dx2 on L2 (R) and Ra rank one perturbation
H = H0 + |ψihψ|. Here ψ ∈ Cc∞ (R) is such that ψ 6= 0 and ∈ R \ {0}.
1. Derive the resolvent formula
(H − k 2 )−1 = (H0 − k 2 )−1 − |(H0 − k 2 )−1 ψih(H0 − k 2 )−1 ψ|
1 + hψ, (H0 − k 2 )−1 ψi
for k ∈ C.
2. Prove that for any f, g ∈ Cc∞ (R), the functions k 7→ hf, (H0 − k 2 )−1 gi
and k 7→ hf, (H − k 2 )−1 gi are meromorphic, and find the eigenvalues
and resonances of H in the weak-coupling limit → 0.

Download Report