# Asymptotic distribution of eigenvalues of Laplace tatry/texty13/plavala- آ 2013-08-27آ number

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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.

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

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.

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

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:

ln (

d

dλ ND(Ω, λ)

) 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 −4ΩD . In 3 dimensions it looks like this:

−4ΩD = − ∂2

∂x2 − ∂

2

∂y2 − ∂

2

∂z2

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

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:

ln (

d

dλ ND(Ω, λ)

) 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 −4ΩD . In 3 dimensions it looks like this:

−4ΩD = − ∂2

∂x2 − ∂

2

∂y2 − ∂

2

∂z2

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:

V (x , y , z) =

{ 0 (x , y , z) ∈ Ω ∞ (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 −4ΩD . 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 −4ΩD , who’s eigenvalues are smaller than λ and let V denote Jordan measure of Ω. Then it holds that:

lim λ→∞

ND(Ω, λ)

λ 3 2

= V

6π2

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 −4ΩN 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:∣∣∣∣∣ND(�, λ)− Vλ

3 2

6π2

∣∣∣∣∣ ≤ C (1 + V 23λ)∣∣∣∣∣NN(�, λ)− Vλ 3 2

6π2

∣∣∣∣∣ ≤ C (1 + V 23λ) 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 ∞∑ n=1

anψn = ∞∑ n=1

anλnψn

which is true if and only if ϕ is in operator domain D(A). ϕ ∈ D(A) if and only if:

(Aϕ,Aϕ) = ∞∑ n=1

|anλn|2

Quadratic forms of operators −4ΩD and −4ΩN

Definition

Let Ω be open region in Rn. Dirichlet Laplace operator on Ω, −4ΩD , is the only self-adjoint operator defined on L2(Ω, dmx), who’s quadratic form is the closure of the form:

q(f , g) =

∫ (∇f )∗ · ∇gdmx

with form domain C∞0 (Ω).

Definition

Let Ω be open region in Rn. Neumann Laplace operator on Ω, −4ΩN , is the only self-adjoint operator defined on L2(Ω, dmx), who’s quadratic form is:

q(f , g) =

∫ (∇f )∗∇gdmx

with form domain H1(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 ϕ1...ϕn−1

{ inf

χ∈Q(A), ‖χ‖=1, χ∈[ϕ1...ϕn−1]⊥ (χ,Aχ)

} 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), ‖χ‖=1, χ∈[ϕ1...ϕn−1]⊥ (χ,Hχ)

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