The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Potentials with compact support
Pedro Fernando Morales
Department of Mathematics
Baylor University
pedro morales@baylor.edu
July, 9th 2010
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Outline
1
2
The problem
The Idea
The Statement
The Assumptions
Zeta Function
Secular Equation
Characteristic Function
Asymptotics for large y
3
4
Analytic Continuation
General Setting
Analytic Continuation
Casimir Energy and Force
Determinant and Energy
Force
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Idea
Pistons with Compact Supported Potentials
Consider a piston
with Dirichlet boundary conditions and middle plate represented by
V (x)
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Idea
Zeta Function
• State the problem as an eigenvalue equation
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Idea
Zeta Function
• State the problem as an eigenvalue equation
• Write down the spectral zeta function
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Idea
Zeta Function
• State the problem as an eigenvalue equation
• Write down the spectral zeta function
• Find the Casimir energy and functional determinant
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Idea
Zeta Function
• State the problem as an eigenvalue equation
• Write down the spectral zeta function
• Find the Casimir energy and functional determinant
• Find analytic continuation
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Idea
Zeta Function
• State the problem as an eigenvalue equation
• Write down the spectral zeta function
• Find the Casimir energy and functional determinant
• Find analytic continuation
• Evaluate ζ 0 (0) and ζ − 12
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Statement
Eigenvalue Problem
We want to solve the eigenvalue equation
∂2
− 2 + V (x) φ(x) = λ2 φ(x)
∂x
in [0, L] with Dirichlet boundary conditions.
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Statement
Eigenvalue Problem
We want to solve the eigenvalue equation
∂2
− 2 + V (x) φ(x) = λ2 φ(x)
∂x
in [0, L] with Dirichlet boundary conditions. Let V (x) be supported
in [a, a + w ] such that w > 0 and
a+w
Z
V (x)dx = σ
a
is fixed.
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Assumptions
Assumptions on φ
• Continuous
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Assumptions
Assumptions on φ
• Continuous
• Incommensurable lengths (a/L ∈
/ Q)
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
The Assumptions
Assumptions on φ
• Continuous
• Incommensurable lengths (a/L ∈
/ Q)
• Normalization (φ(a) = φ(a + w ) = 1)
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Secular Equation
Secular Condition
Integrating the equation around the support gives
λ
2
a+w
Z +
a+w
Z +
φ(x)dx = −
a−
Pedro Fernando Morales
Potentials with Compact Support
a−
∂ 2 φ(x)
dx +
∂x 2
a+w
Z +
V (x)φ(x)dx
a−
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Secular Equation
Letting → 0 and using the solutions of the differential equation,
lead to
λ cot(λa) + λ cot(λ(L − w − a)) + σ − λ2 w
a+w
Z x
Z
0
2
φ (x)
V (u) − λ du dx
=
a
Pedro Fernando Morales
Potentials with Compact Support
a
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Characteristic Function
Characteristic Function
In order to remove singularities at the origin, define
1
σ
F (λ) = − sin(λ(L − w )) − 2 sin(λa) sin(λ(L − w − a))
λ
λ
+w sin(λa) sin(λ(L − w − a))
 x

a+w
Z
Z
sin(λa) sin(λ(L − w − a))
φ0 (x) 
V (u) − λ2 du  dx
+
λ2
a
Pedro Fernando Morales
Potentials with Compact Support
a
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Characteristic Function
Asymptotics for large y
In order to write the zeta function as a contour integral and
deform it to the imaginary axis, we set λ = iy , and when y → ∞,
∂ 2 φ(x)
∂ 2 φ(x)
+
V
(x)φ(x)
+
y
φ(x)
∼
+ y φ(x)
∂x 2
∂x 2
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Characteristic Function
so that
a+w
Z
φ(x)dx ∼ cV
a
where cV is a constant depending on V (x) and independent of a
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Characteristic Function
Thus,
dV
1
+ 2
e y (L−w )
F (iy ) ∼ w −
2y
4y
where
dV = σ + c V
depends only on the potential V (x) and is independent of a.
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Characteristic Function
Therefore, we have the asymptotics of the logarithm
ln F (iy ) ∼ y (L − w ) + ln w −
∞
X
k=1
Pedro Fernando Morales
Potentials with Compact Support
b k2 c X
k − j −k j−k j
−k
y
2 w dV
j
j=0
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Characteristic Function
Therefore, we have the asymptotics of the logarithm
ln F (iy ) ∼ y (L − w ) + ln w −
∞
X
k=1
b k2 c X
k − j −k j−k j
−k
y
2 w dV
j
j=0
Remark
In the limit when w = 0 the asymptotic expansion contains a
logarithmic term in y recovering the semitransparent result
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
General Setting
[0, L] × N setting
Considering the problem in [0, L] × N , where N is a
(D − 1)-dimensional compact Riemannian manifold possibly with
boundary, we have that
ζ(s) =
X
λ2 + η`2
λ,`
−s
for <(s) >
D
2
where η` is the spectrum in N
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Analytic Continuation
Therefore
ζ(s) =
∞
X sin πs Z
`
π
dy
y 2 − η`2
−s
η`
)
N
X
dk
ln F (iy ) − y (L − w ) −
yk
k=0
(
Γ s−1
1
1
−
(L − w ) √ 2 ζN s −
2Γ(s)
2
π
)
N
X
Γ s + k2
k
D −N −2
ζN s +
+
kdk
for <(s) >
k
2
2
Γ 1+ 2
d
×
dy
(
k=1
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Analytic Continuation
where
d0 = dV
b k2 c X
k − j −k j−k j
2 w dV
and dk =
j
j=0
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Determinant and Energy
Functional Determinant and Casimir Energy
Letting N = D − 1 we can calculate ζ 0 (0) which gives the
functional determinant
exp ζ 0 (0)
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Determinant and Energy
Functional Determinant and Casimir Energy
Letting N = D − 1 we can calculate ζ 0 (0) which gives the
functional determinant
exp ζ 0 (0)
and setting N = D leads to the Casimir energy
1
ζ −
2
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Force
Casimir Force
In order to calculate the Casimir Force, we can compute
1
1 ∂
ζ −
−
2 ∂a
2
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Force
Attractive or repulsive?
• For some specific potentials, the force is attractive to the
closest wall (delta, rectangular)
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Force
Attractive or repulsive?
• For some specific potentials, the force is attractive to the
closest wall (delta, rectangular)
• Attractive for all potentials? (It seems to be the case)
Pedro Fernando Morales
Potentials with Compact Support
Math Department
The problem
Zeta Function
Analytic Continuation
Casimir Energy and Force
Force
Thank You!
Pedro Fernando Morales
Potentials with Compact Support
Math Department