Download Asymptotic distribution of eigenvalues of Laplace operator

Asymptotic distribution of eigenvalues of Laplace
operator
Martin Plávala
23.8.2013
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Topics
We will talk about:
the number of eigenvalues of Laplace operator smaller than some λ
as a function of λ
asymptotic behaviour of this function for λ → ∞
the use of all of this in statistical physics
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Motivation
About 100 years ago, Hermann Weyl has
proven his theorem about asymptotic
behaviour of the number of eigenvalues of
Laplace operator. Modern proof of his
theorem can be found in Reed and Simon’s
book.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Motivation
About 100 years ago, Hermann Weyl has
proven his theorem about asymptotic
behaviour of the number of eigenvalues of
Laplace operator. Modern proof of his
theorem can be found in Reed and Simon’s
book.
Still, this theorem is used in statistical physics without any proof, nor
proper formulation. We just say that entropy is the same in all
containers, depending only on their volume.
Let’s discover how and why this works!
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Number of states
Number of states of self-adjoint operator A smaller than some λ is
number of eigenfunctions that have eigenvalues smaller than this λ. We
denote it NA (Ω, λ), where Ω is the domain of the functions in the domain
of the operator A.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Number of states
Number of states of self-adjoint operator A smaller than some λ is
number of eigenfunctions that have eigenvalues smaller than this λ. We
denote it NA (Ω, λ), where Ω is the domain of the functions in the domain
of the operator A.
Correct for operators with only point spectrum!
Definition
Let A be self-adjoint operator, defined on some dense subset of Hilbert
space and bounded from below. Number of states of this operator is
equal to:
NA (Ω, λ) = dim ImP(−∞,λ)
where P(−∞,λ) is projection-valued measure.
We just have to count the number of point of spectra smaller than λ and
if there is essential spectrum, than number of states is infinity.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Density of states and statistical physics
Density of states is just the value of derivation of the number of states in
some point. We use it to obtain entropy of ideal gas. The important
point is that entropy is function of:
d
ND (Ω, λ)
ln
dλ
where ND (Ω, λ) stands for number of states of (minus) Laplace operator
with Dirichlet boundary condition on the bounder of Ω. We mark this
operator −4Ω
D . In 3 dimensions it looks like this:
−4Ω
D =−
∂2
∂2
∂2
−
−
∂x 2
∂y 2
∂z 2
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Density of states and statistical physics
Density of states is just the value of derivation of the number of states in
some point. We use it to obtain entropy of ideal gas. The important
point is that entropy is function of:
d
ND (Ω, λ)
ln
dλ
where ND (Ω, λ) stands for number of states of (minus) Laplace operator
with Dirichlet boundary condition on the bounder of Ω. We mark this
operator −4Ω
D . In 3 dimensions it looks like this:
−4Ω
D =−
∂2
∂2
∂2
−
−
∂x 2
∂y 2
∂z 2
But that is Hamilton’s operator for a particle trapped inside a box.
Shape and volume of the box is defined by Ω, since Ω is the box.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Formulation of the problem in statistical mechanics
Let’s consider scalar particle trapped inside a box. We want to find all
possible states of this particle by solving Schroedringers equation. The
potential in Hamilton’s operator looks like this:
0 (x, y , z) ∈ Ω
V (x, y , z) =
∞ (x, y , z) ∈
/Ω
Since we require the wave function to vanish outside Ω, this is the same
as solving eigenvector-eigenvalue problem for operator −4Ω
D . We will
ignore some constants in Hamilton’s operator. The solution for Ω shaped
like a cube is known.
We want to find number of states (and density of states) to get entropy.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Formulation of the problem in mathematics
Theorem (Weyl)
Let Ω be bounded, open and contented (Jordan measurable) subset of
R3 . Let ND (Ω, λ) denote number of states of operator −4Ω
D , who’s
eigenvalues are smaller than λ and let V denote Jordan measure of Ω.
Then it holds that:
V
ND (Ω, λ)
=
lim
3
λ→∞
6π 2
λ2
If we prove this theorem, we can estimate the entropy of a particle of
ideal gas.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Important parts of the proof
The proof is based on combining few ideas:
Known estimates for the cube
Inequality of operators
Min-max principle
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Known estimates for the cube
We denote by −4Ω
N Laplace’s operator defined on subset of functions
who’s normal derivative vanishes on the border of Ω.
Let denote Ω shaped like a cube, that satisfy the requirements of
Weyl’s theorem. It holds that:
3
2
V λ 2 3λ
≤
C
1
+
V
ND (, λ) −
6π 2 3
2
V λ 2 3λ
≤
C
1
+
V
NN (, λ) −
6π 2 where V is the volume of the cube and C is some constant.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Operators and quadratic forms
Let A be dense defined, semi-bounded and self-adjoint operator on some
infinite dimensional Hilbert space H, with eigenvectors ψn and
eigenvalues λn . If we pass to the spectral representation, image of some
vector ϕ ∈ H is:
Aϕ = A
∞
X
an ψn =
n=1
∞
X
an λn ψn
n=1
which is true if and only if ϕ is in operator domain D(A). ϕ ∈ D(A) if
and only if:
∞
X
2
(Aϕ, Aϕ) =
|an λn | < ∞
n=1
Although some vector ϑ ∈ H is in the domain of quadratic quadratic
form assigned to A if and only if:
(ϕ, Aϕ) =
∞
X
2
|λn | |bn | < ∞
n=1
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Ω
Quadratic forms of operators −4Ω
D and −4N
Definition
Let Ω be open region in Rn . Dirichlet Laplace operator on Ω, −4Ω
D , is
the only self-adjoint operator defined on L2 (Ω, d m x), who’s quadratic
form is the closure of the form:
Z
q(f , g ) = (∇f )∗ · ∇gd m x
with form domain C0∞ (Ω).
Definition
Let Ω be open region in Rn . Neumann Laplace operator on Ω, −4Ω
N , is
the only self-adjoint operator defined on L2 (Ω, d m x), who’s quadratic
form is:
Z
q(f , g ) = (∇f )∗ ∇gd m x
with form domain H 1 (Ω).
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Inequality of operators
Definition
Let operators A,B have form domains Q(A), Q(B) and let both be
self-adjoint and positive. We say that:
0≤A≤B
if and only if
Q(A) ⊃ Q(B)
∀ψ ∈ Q(B) : (ψ, Aψ) ≤ (ψ, Bψ)
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Min-max principle
Theorem
Let A denote bounded from below and self-adjoint operator. Let
ϕ1 . . . ϕn−1 be any n − 1 tuple of vectors. We define:
µn = sup
inf
(χ, Aχ)
ϕ1 ...ϕn−1
χ∈Q(A), kχk=1, χ∈[ϕ1 ...ϕn−1 ]⊥
The if we arrange the element of spectra of A from the smallest up,
counting even their multiplicity, then for a fix n ∈ N one of the following
holds:
(a) µn is the nth eigenvalue of A.
(b) there is at most n − 1 eigenvalues of A smaller than µn and it holds
that:
µn = µn+1 = µn+2 = . . .
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Demonstration of min-max principle
Let’s demonstrate min-max principle on a well known quantum system:
particle bound to a finite line.
The formula for µn :
µn =
sup
ϕ1 ...ϕn−1
inf
χ∈Q(H), kχk=1, χ∈[ϕ1 ...ϕn−1 ]⊥
(χ, Hχ)
According to the theorem µn is the nth eigenvalue of Hamilton’s
operator. Let’s proceed from inside to outside. We have take all of the
unit vectors χ, which are orthogonal on some vectors ϕ1 . . . ϕn−1 and
find the infimum of set containing elements (χ, Hχ). If we for example
set n = 3, ϕ1 . . . ϕn−1 to be ψ12 , ψ42 , then the minimum is the first
energy level E1 and the χ for which the minimum happens is ψ1 .
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Demonstration of min-max principle
Let’s demonstrate min-max principle on a well known quantum system:
particle bound to a finite line.
The formula for µn :
µn =
sup
ϕ1 ...ϕn−1
inf
χ∈Q(H), kχk=1, χ∈[ϕ1 ...ϕn−1 ]⊥
(χ, Hχ)
Now we have to do this for all tuples of vectors ϕ1 . . . ϕn−1 and find the
supremum of this set. For the case of n = 3 we took ψ12 , ψ42 . But if we
change one of them for ψ1 then clearly mimicking the steps we did before
the number we would get would be E2 , the second energy level. The
”correct” choice of vectors ϕ1 . . . ϕn−1 is to take ψ1 , ψ2 , which would
give us E3 , the third eigenvalue of Hamilton’s operator.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Merging min-max principle and inequality of operators
Theorem
If:
0≤A≤B
as defined before, then:
NA (Ω, λ) ≥ NB (Ω, λ)
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Merging min-max principle and inequality of operators
Theorem
If:
0≤A≤B
as defined before, then:
NA (Ω, λ) ≥ NB (Ω, λ)
Proof.
All we have to prove is that k th element of spectrum of operator A is
smaller or equal the k th element of spectrum of operator B. Since one of
the requirements of definition of the inequality of operators is that:
∀ψ ∈ Q(B) : (ψ, Aψ) ≤ (ψ, Bψ)
and because adding supremum and infimum preserves the inequality
above, we get the expressions of elements of spectrum of both operators
as in min-max principle.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Operator inequality theorem
Theorem
Let Ω1 and Ω2 be disjoint open subsets of open set Ω. Let
int
Ω1 ∪ Ω 2
= Ω and let Ω \ (Ω1 ∪ Ω2 ) have measure of 0. Then:
Ω1 ∪Ω2
0 ≤ −4Ω
D ≤ −4D
1 ∪Ω2
0 ≤ −4Ω
≤ −4Ω
N
N
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
One more inequality theorem
Theorem
For any Ω it holds that:
Ω
0 ≤ −4Ω
N ≤ −4D
Theorem
Let Ω ⊂ Ω0 , then it holds that:
Ω
0
0 ≤ −4Ω
D ≤ −4D
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
The proof
Let Ω be a contented set. We will call inner cubes all cubes of the same
size that fit inside Ω without crossing the bounder of Ω and we will
denote them Ω−
n . We will call outer cubes all cubes of the same size that
cover all of Ω and denote them Ω+
n . We get inequalities in given order:
Ω+
0 ≤ −4Dn ≤ −4Ω
D
Ω+
Ω+
0 ≤ −4Nn ≤ −4Dn
A+ (n)
C+
A+ (n)
C+
Ω+
0 ≤ ⊕α=1 − 4Nn,α ≤ −4Nn
which sums up to:
0 ≤ ⊕α=1 − 4Nn,α ≤ −4Ω
D
That means:
A+ (n)
ND (Ω, λ) ≤
X
+
NN (Cn,α
, λ)
α=1
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
The proof
Similarly we get:
A− (n)
C−
n,α
0 ≤ −4Ω
D ≤ ⊕α=1 − 4D
which leads us to:
A− (n)
X
−
ND (Cn,α
, λ) ≤ ND (Ω, λ)
α=1
Now we have to put together both inequalities of numbers of states:
A− (n)
X
A+ (n)
−
ND (Cn,α
, λ)
≤ ND (Ω, λ) ≤
α=1
X
+
NN (Cn,α
, λ)
α=1
3
2
which after dividing by λ and passing to limits as λ → ∞ and n → ∞
we get:
3
3
V
V
≤ lim inf ND (Ω, λ)λ− 2 ≤ lim sup ND (Ω, λ)λ− 2 ≤
2
λ→∞
6π 2
6π
λ→∞
which proves Weyl’s theorem.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
The use in statistical physics
We have proven that:
ND (Ω, λ) =
Martin Plávala
3
V 3
λ 2 + o(λ 2 )
6π 2
Asymptotic distribution of eigenvalues of Laplace operator
The use in statistical physics
We have proven that:
ND (Ω, λ) =
3
V 3
λ 2 + o(λ 2 )
6π 2
Lets suppose that:
ND (Ω, λ) =
1
V 3
λ 2 + C1 λ + C2 λ 2
2
6π
as in the case of cube.This, when used in formula for entropy, gives us:
V 1
4π 2 C1 − 1
4π 2 C2 −1
d
2
2 +
ND (Ω, λ) = ln
λ
+
ln
1
+
λ
λ
ln
dλ
4π 2
V
V
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
The use in statistical physics
We have proven that:
ND (Ω, λ) =
3
V 3
λ 2 + o(λ 2 )
6π 2
Lets suppose that:
ND (Ω, λ) =
1
V 3
λ 2 + C1 λ + C2 λ 2
2
6π
as in the case of cube.This, when used in formula for entropy, gives us:
V 1
4π 2 C1 − 1
4π 2 C2 −1
d
2
2 +
ND (Ω, λ) = ln
λ
+
ln
1
+
λ
λ
ln
dλ
4π 2
V
V
If we consider one electron in box shaped like a cube with the volume of
1m3 at temperature 300K , then:
λ ≈ 1018 m−2
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Numerical simulation
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator
Thank you for your attention.
Martin Plávala
Asymptotic distribution of eigenvalues of Laplace operator