Asymptotic distribution of eigenvalues of Laplaceoperator

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 λas 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 hasproven his theorem about asymptoticbehaviour of the number of eigenvalues ofLaplace operator. Modern proof of histheorem can be found in Reed and Simon’sbook.

Still, this theorem is used in statistical physics without any proof, norproper formulation. We just say that entropy is the same in allcontainers, 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 hasproven his theorem about asymptoticbehaviour of the number of eigenvalues ofLaplace operator. Modern proof of histheorem can be found in Reed and Simon’sbook.

Still, this theorem is used in statistical physics without any proof, norproper formulation. We just say that entropy is the same in allcontainers, 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 isnumber of eigenfunctions that have eigenvalues smaller than this λ. Wedenote it NA(Ω, λ), where Ω is the domain of the functions in the domainof the operator A.

Correct for operators with only point spectrum!

DefinitionLet A be self-adjoint operator, defined on some dense subset of Hilbertspace and bounded from below. Number of states of this operator isequal 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 andif 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 isnumber of eigenfunctions that have eigenvalues smaller than this λ. Wedenote it NA(Ω, λ), where Ω is the domain of the functions in the domainof the operator A.

Correct for operators with only point spectrum!

DefinitionLet A be self-adjoint operator, defined on some dense subset of Hilbertspace and bounded from below. Number of states of this operator isequal 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 andif 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 insome point. We use it to obtain entropy of ideal gas. The importantpoint is that entropy is function of:

ln(

d

dλND(Ω, λ)

)where ND(Ω, λ) stands for number of states of (minus) Laplace operatorwith Dirichlet boundary condition on the bounder of Ω. We mark thisoperator −4Ω

D . 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 insome point. We use it to obtain entropy of ideal gas. The importantpoint is that entropy is function of:

ln(

d

dλND(Ω, λ)

)where ND(Ω, λ) stands for number of states of (minus) Laplace operatorwith Dirichlet boundary condition on the bounder of Ω. We mark thisoperator −4Ω

D . 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 allpossible states of this particle by solving Schroedringers equation. Thepotential 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 sameas solving eigenvector-eigenvalue problem for operator −4Ω

D . We willignore some constants in Hamilton’s operator. The solution for Ω shapedlike 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 ofR3. Let ND(Ω, λ) denote number of states of operator −4Ω

D , who’seigenvalues are smaller than λ and let V denote Jordan measure of Ω.Then it holds that:

limλ→∞

ND(Ω, λ)

λ32

=V

6π2

If we prove this theorem, we can estimate the entropy of a particle ofideal 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 ofWeyl’s theorem. It holds that:∣∣∣∣∣ND(, λ)− Vλ

32

6π2

∣∣∣∣∣ ≤ C(

1 + V23λ)

∣∣∣∣∣NN(, λ)− Vλ32

6π2

∣∣∣∣∣ ≤ C(

1 + V23λ)

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 someinfinite dimensional Hilbert space H, with eigenvectors ψn andeigenvalues λn. If we pass to the spectral representation, image of somevector 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) ifand only if:

(Aϕ,Aϕ) =∞∑n=1

|anλn|2 <∞

Although some vector ϑ ∈ H is in the domain of quadratic quadraticform assigned to A if and only if:

(ϕ,Aϕ) =∞∑n=1

|λn| |bn|2 <∞

Martin Plávala Asymptotic distribution of eigenvalues of Laplace operator

Quadratic forms of operators −4ΩD and −4Ω

N

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 quadraticform 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 quadraticform 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 beself-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

TheoremLet A denote bounded from below and self-adjoint operator. Letϕ1 . . . ϕn−1 be any n − 1 tuple of vectors. We define:

µn = supϕ1...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 followingholds:

(a) µn is the nth eigenvalue of A.

(b) there is at most n − 1 eigenvalues of A smaller than µn and it holdsthat:

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 = supϕ1...ϕn−1

inf

χ∈Q(H), ‖χ‖=1, χ∈[ϕ1...ϕn−1]⊥(χ,Hχ)

According to the theorem µn is the nth eigenvalue of Hamilton’soperator. Let’s proceed from inside to outside. We have take all of theunit vectors χ, which are orthogonal on some vectors ϕ1 . . . ϕn−1 andfind the infimum of set containing elements (χ,Hχ). If we for exampleset n = 3, ϕ1 . . . ϕn−1 to be ψ12, ψ42, then the minimum is the firstenergy level E1 and the χ for which the minimum happens is ψ1.

Now we have to do this for all tuples of vectors ϕ1 . . . ϕn−1 and find thesupremum of this set. For the case of n = 3 we took ψ12, ψ42. But if wechange one of them for ψ1 then clearly mimicking the steps we did beforethe 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 wouldgive us E3, the third eigenvalue of Hamilton’s operator.

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 = supϕ1...ϕn−1

inf

χ∈Q(H), ‖χ‖=1, χ∈[ϕ1...ϕn−1]⊥(χ,Hχ)

According to the theorem µn is the nth eigenvalue of Hamilton’soperator. Let’s proceed from inside to outside. We have take all of theunit vectors χ, which are orthogonal on some vectors ϕ1 . . . ϕn−1 andfind the infimum of set containing elements (χ,Hχ). If we for exampleset n = 3, ϕ1 . . . ϕn−1 to be ψ12, ψ42, then the minimum is the firstenergy level E1 and the χ for which the minimum happens is ψ1.

Now we have to do this for all tuples of vectors ϕ1 . . . ϕn−1 and find thesupremum of this set. For the case of n = 3 we took ψ12, ψ42. But if wechange one of them for ψ1 then clearly mimicking the steps we did beforethe 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 wouldgive 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

TheoremIf:

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 issmaller or equal the k th element of spectrum of operator B. Since one ofthe 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 inequalityabove, we get the expressions of elements of spectrum of both operatorsas in min-max principle.

Martin Plávala Asymptotic distribution of eigenvalues of Laplace operator

Merging min-max principle and inequality of operators

TheoremIf:

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 issmaller or equal the k th element of spectrum of operator B. Since one ofthe 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 inequalityabove, we get the expressions of elements of spectrum of both operatorsas in min-max principle.

Martin Plávala Asymptotic distribution of eigenvalues of Laplace operator

Operator inequality theorem

TheoremLet Ω1 and Ω2 be disjoint open subsets of open set Ω. Let(Ω1 ∪ Ω2

)int= Ω and let Ω \ (Ω1 ∪ Ω2) have measure of 0. Then:

0 ≤ −4ΩD ≤ −4

Ω1∪Ω2D

0 ≤ −4Ω1∪Ω2N ≤ −4Ω

N

Martin Plávala Asymptotic distribution of eigenvalues of Laplace operator

One more inequality theorem

TheoremFor any Ω it holds that:

0 ≤ −4ΩN ≤ −4Ω

D

TheoremLet Ω ⊂ Ω0, then it holds that:

0 ≤ −4Ω0D ≤ −4

ΩD

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 samesize that fit inside Ω without crossing the bounder of Ω and we willdenote them Ω−n . We will call outer cubes all cubes of the same size thatcover all of Ω and denote them Ω+

n . We get inequalities in given order:

0 ≤ −4Ω+n

D ≤ −4ΩD

0 ≤ −4Ω+n

N ≤ −4Ω+

n

D

0 ≤ ⊕A+(n)α=1 −4

C+n,α

N ≤ −4Ω+n

N

which sums up to:

0 ≤ ⊕A+(n)α=1 −4

C+n,α

N ≤ −4ΩD

That means:

ND(Ω, λ) ≤A+(n)∑α=1

NN(C+n,α, λ)

Martin Plávala Asymptotic distribution of eigenvalues of Laplace operator

The proof

Similarly we get:

0 ≤ −4ΩD ≤ ⊕

A−(n)α=1 −4

C−n,αD

which leads us to:

A−(n)∑α=1

ND(C−n,α, λ) ≤ ND(Ω, λ)

Now we have to put together both inequalities of numbers of states:

A−(n)∑α=1

ND(C−n,α, λ) ≤ ND(Ω, λ) ≤A+(n)∑α=1

NN(C+n,α, λ)

which after dividing by λ32 and passing to limits as λ→∞ and n→∞

we get:

V

6π2≤ lim inf

λ→∞ND(Ω, λ)λ−

32 ≤ lim sup

λ→∞ND(Ω, λ)λ−

32 ≤ V

6π2

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) =V

6π2λ32 + o(λ

32 )

Lets suppose that:

ND(Ω, λ) =V

6π2λ32 + C1λ+ C2λ

12

as in the case of cube.This, when used in formula for entropy, gives us:

ln(

d

dλND(Ω, λ)

)= ln

(V

4π2λ12

)+ ln

(1 +

4π2C1V

λ−12 +

4π2C2V

λ−1)

If we consider one electron in box shaped like a cube with the volume of1m3 at temperature 300K , then:

λ ≈ 1018m−2

Martin Plávala Asymptotic distribution of eigenvalues of Laplace operator

The use in statistical physics

We have proven that:

ND(Ω, λ) =V

6π2λ32 + o(λ

32 )

Lets suppose that:

ND(Ω, λ) =V

6π2λ32 + C1λ+ C2λ

12

as in the case of cube.This, when used in formula for entropy, gives us:

ln(

d

dλND(Ω, λ)

)= ln

(V

4π2λ12

)+ ln

(1 +

4π2C1V

λ−12 +

4π2C2V

λ−1)

If we consider one electron in box shaped like a cube with the volume of1m3 at temperature 300K , then:

λ ≈ 1018m−2

Martin Plávala Asymptotic distribution of eigenvalues of Laplace operator

The use in statistical physics

We have proven that:

ND(Ω, λ) =V

6π2λ32 + o(λ

32 )

Lets suppose that:

ND(Ω, λ) =V

6π2λ32 + C1λ+ C2λ

12

as in the case of cube.This, when used in formula for entropy, gives us:

ln(

d

dλND(Ω, λ)

)= ln

(V

4π2λ12

)+ ln

(1 +

4π2C1V

λ−12 +

4π2C2V

λ−1)

If we consider one electron in box shaped like a cube with the volume of1m3 at temperature 300K , then:

λ ≈ 1018m−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