Periodic Schrödinger Operators
Let Γ be a lattice of static ions that generate an electric
potential V (x) that is periodic with respect to Γ.
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Then the Hamiltonian for a single electron moving in
this lattice is
1
∆ + V (x)
H = − 2m
This Hamiltonian commutes with all of the translation
operators
(Tγ φ (x) = φ(x + γ )
γ∈Γ
PS 1
Simultaneous eigenfunctions for these operators obey
Hφα = eα φα
Tγ φα = λα,γγ φα
∀γγ ∈ Γ
Tγ is unitary ⇒
λα,γγ = 1 ⇒ λα,γγ = eiβα,γγ
Tγ Tγ 0 φα = Tγ +γγ 0 φα ⇒
⇒ λα,γγ λα,γγ 0 φα = λα,γγ +γγ 0 φα
⇒ βα,γγ + βα,γγ 0 = βα,γγ +γγ 0 mod 2π
∀γγ , γ 0 ∈ Γ
Write
Γ=
n1γ 1 + · · · ndγ d n1 , · · · , nd ∈ ZZ
For each α, all βα,γγ , γ ∈ Γ are determined, mod 2π, by
βα,γγ i , 1 ≤ i ≤ d. Given any d numbers β1 , · · · , βd the
system of linear equations (with unknowns k1 , · · · , kd )
that is
d
P
γ i · k = βi
1≤i≤d
γi,j kj = βi
1≤i≤d
j=1
PS 2
(where γi,j is the j th component of γ i ) has a unique
solution. So, for each α, there exists a kα ∈ IRd such
that kα · γ i = βα,γγ i for all 1 ≤ i ≤ d and hence
βα,γγ = kα · γ mod 2π
∀γγ ∈ Γ
Notice that, for each α, kα is not uniquely determined.
Indeed
βα,γγ = kα · γ mod 2π, βα,γγ = k0α · γ mod 2π ∀γγ ∈ Γ
⇐⇒ (kα − k0α ) · γ ∈ 2πZZ
∀γγ ∈ Γ
⇐⇒ kα − k0α ∈ Γ#
where the dual lattice, Γ# , of Γ is
d #
Γ = b ∈ IR b · γ ∈ 2πZZ for all γ ∈ Γ
Relabel, replacing the index α by the corresponding
value of k ∈ IRd /Γ# and another index n. Under the
new labeling the eigenvalue/eigenvector equations are
Hφn,k = en (k, V )φn,k
Tγ φn,k = eik·γγ φn,k
i.e. φn,k (x + γ ) = eik·γγ φn,k (x)
∀γγ ∈ Γ
∀γγ ∈ Γ
PS 3
or equivalently, with φn,k (x) = eik·x ψn,k (x),
2
1
2m
i∇
∇ − k ψn,k + V ψn,k = en (k, V )ψn,k
ψn,k (x + γ ) = ψn,k (x)
Denote by INk the set of values of n that appear in pairs
α = (k, n) and define
Hk = span
H̃k = span
φn,k
ψn,k
n ∈ INk
n ∈ INk
Then, formally, and in particular ignoring that k runs
over an uncountable set,
2
d
L (IR ) = span
φn,k
k ∈ IRd /Γ# , n ∈ INk
= ⊕k∈IRd /Γ# Hk
unitary
∼
=
⊕k∈IRd /Γ# H̃k
The restriction of the Schrödinger operator H to H̃k is
2
1
∇ − k + V applied to functions that are periodic
2m i∇
with respect to Γ.
PS 4
So what have we gained? At least formally, we now
know that to find the spectrum of
H=
2
d
1
2m
2
i∇
∇ + V (x)
acting on L IR , it suffices to find, for each k ∈
IRd /Γ# , the spectrum of
Hk =
2
d
1
2m
i∇
∇−k
2
+ V (x)
acting on L IR /Γ . Unlike H, Hk has compact resolvent. So, the spectrum of Hk necessarily consists of
a sequence of eigenvalues en (k) converging to ∞. The
functions en (k) are continuous in k and periodic with
respect to Γ# and the spectrum of H is precisely
d
#
en (k) n ∈ IN, k ∈ IR /Γ
PS 5
Rigorousification
⊕k∈IRd /Γ# H̃k = ⊕k∈IRd /Γ# span
is implemented using
d
d
#
S IR /Γ × IR /Γ
n
d
d ∞
= ψ ∈ C IR × IR ψn,k
n ∈ INk
ψ(k, x + γ ) = ψ(k, x) ∀γγ ∈ Γ
e
ib·x
ψ(k + b, x) = ψ(k, x) ∀b ∈ Γ
#
with inner product
hψ, φiΓ =
Z
1
|Γ# |
dk
IRd /Γ#
Z
o
dx ψ(k, x) φ(k, x)
IRd /Γ
and completion
2
d
#
d
L IR /Γ × IR /Γ
Also definen
S IRd = f ∈C ∞ IRd Q
d
2n
sup (1 + x )
x
Its completion is L2 (IRd ).
f
(x)
<∞
ij
ij
∂
j=1 ∂xj
∀n, i1 , · · · id ∈ IN
o
PS 6
Set
(uψ)(x) =
(ũf )(k, x) =
1
|Γ# |
X
Z
dd k eik·x ψ(k, x)
IRd /Γ#
e−ik·(x+γγ ) f (x + γ )
γ ∈Γ
∞
(IRd /Γ) and set
Let V ∈ CIR
2
d
Dh = S IR
h = i∇
∇ + V (x)
2
d
d
#
κ = i∇
∇x − k + V (x) Dκ = S IR /Γ × IR /Γ
Proposition S.3, S.5 There is a unitary map U such
that U extends u, U ∗ = U −1 extends ũ and
d
#
d
S IR /Γ × IR /Γ
dense
2
d
#
d
L IR /Γ × IR /Γ
u
ũ
U
U∗
S IR
d
dense
d
2
L IR
Furthermore, h and κ have unique self–adjoint extensions, H and K, and
ũhu = κ
U ∗ HU = K
PS 7
∞
Now fix any k ∈ IRd and V ∈ CIR
(IRd /Γ) and set
2
d
∞
hk = i∇
∇x − k + V (x)
Dhk = C IR /Γ
Lemma S.7,S.8
a) The operator hk has a unique self–adjoint extension,
d
2
Hk , in L IR /Γ .
b) If =λ 6= 0 or λ < − supx |V (x)|, then λ is not in the
spectrum of Hk . If λ is not in the spectrum of Hk , the
−1
resolvent Hk − λ1l
is compact.
c) Let R > 0 and λ < − supx |V (x)|. There is a constant
C 0 such that
−1 −1 − Hk0 − λ1l ≤ C 0 |k − k0 |
Hk − λ1l
for all k, k0 ∈ IRd with |k|, |k0 | ≤ R.
d) Let c ∈ Γ# and define Uc to be the multiplication
operator eic·x on L2 (IRd /Γ). Then Ub is unitary and
U∗c Hk Uc = Hk+c
PS 8
Idea of Proof:
Hk is a bounded perturbation of
2
d
2
i∇
∇x − k , acting on L IR /Γ . The latter is diago-
nalized by the Fourier transform. It’s spectrum is
2
#
(b − k) b ∈ Γ
Proposition S.9 The spectrum of Hk consists of a sequence of eigenvalues
e1 (k) ≤ e2 (k) ≤ e3 (k) ≤ · · ·
with, for each n, en (k) continuous in k and periodic
with respect to Γ# and limn→∞ en (k) = ∞. The limit
is uniform in k.
Theorem S.10 The spectrum of H is
d
#
en (k) k ∈ IR /Γ , n ∈ IN
PS 9