text: and crossover transitions for representative values of the quark mass and chemical potential. This critical quark mass is found to be independent of µ, implying that − this ﬂavor-symmetric model cannot produce a critical point in the T µ plane. 0.10 μ=0GeV 0.12 μ=0GeV 0.12 μ=0GeV μ=0.3GeV μ=0.3GeV μ=0.3GeV 0.08 0.10 0.10 0.06 0.08 0.08 0.06 0.06 0.04 0.04 0.04 0.02 0.02 0.02 0.000.12 0.14 0.16 0.18 0.20 0.000.12 0.14 0.16 0.18 0.20 0.22 0.000.12 0.14 0.16 0.18 0.20 0.22 T(GeV) T(GeV) T(GeV) (a) (b) (c) Figure 3: Dependence of σ on T , for zero and non-zero values of µ and (a) m = l m = 10 MeV, (b) m = m = 35 MeV, and (c) m = m = 45 MeV. In all cases, s l s l s chemical potential has no eﬀect on the order of the phase transition. 3.2 2+1-ﬂavors Finding no critical point in the ﬂavor-symmetric case, we extend the model to 2+1 ﬂavors. The shooting is extended to ﬁnd values of σ and σ that yield regular l s solutions for both chiral ﬁelds. Figure 4 shows a representative case for m = 40 l MeV, m = 70 MeV at µ = 0 MeV and µ = 300 MeV. As in the ﬂavor-symmetric s case, the order of the phase transition is not aﬀected by the chemical potential. The chiral phase structure at a particluar µ is summarized by ﬁnding the second- order line in the Columbia Plot. Figure 5 shows representative Columbia Plots for µ = 0 and µ = 300 MeV. We ﬁnd the tri-critical point, marking the end of the ﬁrst-order region, at m = 200 MeV. At greater m values, second-order transitions s s occur at m = 0 and rapid crossover transitions occur for ﬁnite values. The physical l point m = 5 MeV, m = 95 MeV is within the ﬁrst-order region, contradicting l s expectations from lattice results at µ = 0. The location of the second-order line may be adjusted by the choice of input parameters, but this work is focused on the dependence of the phase transition order on chemical potential. At small m there are deviations in second-order line the on the order of 1 MeV s as µ is varied. However, these small changes are likely numerical artifacts rather than evidence of a qualitative change. Thus, the phase transition order is unaﬀected 7 )3VeG(σ )3VeG(σ )3VeG(σ ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 0.14 0.14 σl σl 0.12 σs 0.12 σs 0.10 0.10 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0.00 0.00 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.10 0.12 0.14 0.16 0.18 0.20 0.22 T (GeV) T (GeV) (a) (b) Figure 4: Dependence of σ and σ on T with m = 40 MeV, m = 70 MeV and (a) l s l s µ = 0 MeV and (b) µ = 300 MeV. by µ, indicating that the critical surface depicted in Figure 2 has zero curvature in − this model, and no critical point in the T µ plane is possible. 3.3 Chiral potential Further analysis reveals how the solutions to (13) relate to the ﬂavor-symmetric scalar potential (9). In particular, the eﬀects of spontaneous symmetry breaking is ≡ → evident from the vacuum states of potential. We examine V (χf), where χf χ(u 1) is the near-horizon value of the chiral ﬁeld. Figure 6 shows the temperature- independent potential for a representative case with m = m = 10 MeV and µ = 10 l s MeV. The near-horizon solution is projected onto the potential, showing ﬁrst-order → chiral symmetry restoration as T T . The trivial solution χ(u) = 0 is also plotted, c but below T = T it is energetically disfavored. Because m is small, the eﬀects of c q explicit symmetry breaking are small, and at χf = 0, chiral symmetry is restored. The potential is asymmetric in χf, in contrast to the 2-ﬂavor model which also permits negative solutions for χ, σ. We perform a similar analysis for the 2+1-ﬂavor potential (9). With three ﬂavors, we now project the temperature dependence of χf and χf onto the surface given by l s V (χ , χ ). Figure 7 shows a representative case with m = 30 MeV, m = 100 MeV l s l s and µ = 0.1 GeV. The path along the surface is parameterized by temperature, ranging from 0.1 GeV to 0.23 GeV. The transition is crossover and the chiral ﬁelds do not vanish, a result of explicit chiral symmetry breaking due to the nonzero quark → masses. As in the two-ﬂavor case, when T T the eﬀects of spontaneous symmetry c 8 )3VeG( σ )3VeG( σ ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: σl

text: σl

text: σs

text: σs

text: T

text: (GeV)

text: T

text: (GeV)

text: (a)

text: (b)

text: Figure

text: Dependence

label: caption

text: of

text: σ

text: and

text: σ

text: on

text: T

text: with

text: m

text: MeV,

text: m

text: MeV

text: and

text: (a)

text: l

text: s

text: l

text: s

text: µ

text: MeV

text: and

text: (b)

text: µ

text: MeV.

text: by

text: µ,

text: indicating

text: that

text: the

text: critical

text: surface

text: depicted

text: in

text: Figure

text: has

text: zero

text: curvature

text: in

text: this

text: model,

text: and

text: no

text: critical

text: point

text: in

text: the

text: T

text: µ

text: plane

text: is

text: possible.

text: Chiral

text: potential

text: Further

text: analysis

text: reveals

text: how

text: the

text: solutions

text: to

text: relate

text: to

text: the

text: ﬂavor-symmetric

text: scalar

text: potential

text: In

text: particular,

text: the

text: eﬀects

text: of

text: spontaneous

text: symmetry

text: breaking

text: is

text: evident

text: from

text: the

text: vacuum

text: states

text: of

text: potential.

text: We

text: examine

text: V

text: (χf),

text: where

text: χf

text: χ(u

text: is

text: the

text: near-horizon

text: value

text: of

text: the

text: chiral

text: ﬁeld.

text: Figure

text: shows

text: the

text: temperature-

text: independent

text: potential

text: for

text: a

text: representative

text: case

text: with

text: m

text: m

text: MeV

text: and

text: µ

text: l

text: s

text: MeV.

text: The

text: near-horizon

text: solution

text: is

text: projected

text: onto

text: the

text: potential,

text: showing

text: ﬁrst-order

text: chiral

text: symmetry

text: restoration

text: as

text: T

text: T

text: The

text: trivial

text: solution

text: χ(u)

text: is

text: also

text: plotted,

text: c

text: but

text: below

text: T

text: T

text: it

text: is

text: energetically

text: disfavored.

text: Because

text: m

text: is

text: small,

text: the

text: eﬀects

text: of

text: c

text: q

text: explicit

text: symmetry

text: breaking

text: are

text: small,

text: and

text: at

text: χf

text: chiral

text: symmetry

text: is

text: restored.

text: The

text: potential

text: is

text: asymmetric

text: in

text: χf,

text: in

text: contrast

text: to

text: the

text: model

text: which

text: also

text: permits

text: negative

text: solutions

text: for

text: χ,

text: σ.

text: We

text: perform

text: a

text: similar

text: analysis

text: for

text: the

text: potential

text: With

text: three

text: ﬂavors,

text: we

text: now

text: project

text: the

text: temperature

text: dependence

text: of

text: χf

text: and

text: χf

text: onto

text: the

text: surface

text: given

text: by

text: l

text: s

text: V

text: (χ

text: χ

text: Figure

text: shows

text: a

text: representative

text: case

text: with

text: m

text: MeV,

text: m

text: MeV

text: l

text: s

text: l

text: s

text: and

text: µ

text: GeV.

text: The

text: path

text: along

text: the

text: surface

text: is

text: parameterized

text: by

text: temperature,

text: ranging

text: from

text: GeV

text: to

text: GeV.

text: The

text: transition

text: is

text: crossover

text: and

text: the

text: chiral

text: ﬁelds

text: do

text: not

text: vanish,

text: a

text: result

text: of

text: explicit

text: chiral

text: symmetry

text: breaking

text: due

text: to

text: the

text: nonzero

text: quark

text: masses.

text: As

text: in

text: the

text: two-ﬂavor

text: case,

text: when

text: T

text: T

text: the

text: eﬀects

text: of

text: spontaneous

text: symmetry

text: c

text: σ

text: σ

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text: (cid:90) − − t u(1) = σ 1g dτ S(t, τ)p (τ) (332) 0 (cid:90) ex 0 p(1) = 0 (333) u(2) = −σ−1g(cid:18)(cid:90) t dτ S(t, τ)p (τ(cid:90)) (cid:90) (cid:2) (cid:3) (cid:19) (334) 0 ex 0 t t t (cid:48) ∈ M p(2) = ρ c2g dτ S (t, τ) + dτ dτ S (t, τ) δA(1) ∂2uδ((t, x) ∂ ). (335) 0 0 0 2,2 t 0 0 0 τ ∼ ∼ ∼ We observe that the second contribution in p(2) scales as u3 g3 if p 1. Since p is an acoustic ex ex O quantity as well, it must be of the order of acoustic quantities as well, i.e., of order p = (ε). We ex can choose the linearization parameter ε from the derivation of the acoustic wave equation from Euler’s equation in curved space-time to satisfy, O ≈ − ε = (g) 10 3. We see that we have the scaling O u(2) = (g2 = (cid:15)) O p(2) = (g2 = (cid:15)), | | M as we need for consistency because max u = (cid:15) by deﬁnition of the perturbation bundle . We can stop ∈ N in linear order perturbation theory and just notice that we have by mathematical induction for all k additional contributions, − O (cid:15)u gk := u(k+2) u(k+1) = ((cid:15)gk) (336) k − O (cid:15)p gk := p(k+2) p(k+1) = ((cid:15)gk), (337) k O i.e., u = (1) = p as we would expect from a perturbation theory. In particular, we have for the k k ﬁxed points of the iteration scheme u and p expressed as a telescope sum over u and p , (cid:88) (cid:0) (cid:1) (cid:88) k k ∞ ∞ − u = (cid:88) (cid:0)u(k+1) u(k)(cid:1) = (cid:15)(cid:88) u gk (338) k k=0 k=1 ∞ ∞ − p = p(k+1) p(k) = (cid:15) p gk, (339) k k=0 k=1 ≈ noting that u(1) = u(2) and u(0) = 0 = p(0) = p(1). Using that g2 (cid:15), we can set in linear order perturbation theory in (cid:15), p(2) = p and u(2) = u. (340) 101

label: equation

text: t

text: σ

text: dτ

text: S(t,

text: τ)p

text: (τ)

text: ex

text: t

text: dτ

text: S(t,

text: τ)p

text: ex

text: t

text: t

text: t

text: M

text: ρ

text: dτ

text: S

text: (t,

text: τ)

text: dτ

text: dτ

text: S

text: (t,

text: τ)

text: x)

text: t

text: τ

text: We

text: observe

text: that

text: the

text: second

text: contribution

text: in

text: scales

text: as

text: if

text: p

text: Since

text: p

text: is

text: an

text: acoustic

text: ex

text: ex

text: O

text: quantity

text: as

text: well,

text: it

text: must

text: be

text: of

text: the

text: order

text: of

text: acoustic

text: quantities

text: as

text: well,

text: i.e.,

text: of

text: order

text: p

text: (ε).

text: We

text: ex

text: can

text: choose

text: the

text: linearization

text: parameter

text: ε

text: from

text: the

text: derivation

text: of

text: the

text: acoustic

text: wave

text: equation

text: from

text: Euler’s

text: equation

text: in

text: curved

text: space-time

text: to

text: satisfy,

text: O

text: ε

text: (g)

text: We

text: see

text: that

text: we

text: have

text: the

text: scaling

text: O

text: O

text: M

text: as

text: we

text: need

text: for

text: consistency

text: because

text: max

text: u

text: by

text: deﬁnition

text: of

text: the

text: perturbation

text: bundle

text: We

text: can

text: stop

text: N

text: in

text: linear

text: order

text: perturbation

text: theory

text: and

text: just

text: notice

text: that

text: we

text: have

text: by

text: mathematical

text: induction

text: for

text: all

text: k

text: additional

text: contributions,

text: O

text: gk

text: k

text: O

text: gk

text: k

text: O

text: i.e.,

text: u

text: p

text: as

text: we

text: would

text: expect

text: from

text: a

text: perturbation

text: theory.

text: In

text: particular,

text: we

text: have

text: for

text: the

text: k

text: k

text: ﬁxed

text: points

text: of

text: the

text: iteration

text: scheme

text: u

text: and

text: p

text: expressed

text: as

text: a

text: telescope

text: sum

text: over

text: u

text: and

text: p

text: k

text: k

text: u

text: u

text: gk

text: k

text: p

text: p(k)

text: p

text: gk,

text: k

text: noting

text: that

text: and

text: Using

text: that

text: we

text: can

text: set

text: in

text: linear

text: order

text: perturbation

text: theory

text: in

text: p

text: and

text: u.


text: Comparison of the coeﬃcients of ∂kw for k = 0, 1, 2 yields the identiﬁcations for the still unknown t functions f = f(t) and q = q(t). For the latter function, we ﬁnd an equation which still involves the conformal factor f, (cid:48) (cid:48) (cid:48)(cid:48) (cid:48) (cid:48) (cid:48)(cid:48) q = (f )∂ log Σ + (f )2 + f = (f )∂ log Σ + (f )2 + f (346) t t The determining ordinary diﬀerential equation for f is obtained by comparing coeﬃcients of ∂kw for t k = 1 and is given by (cid:112) √ (cid:48) − ⇔ (cid:48) − ↔ − 2f = ∂ log Σ f = ∂ log Σ f(t) = log Σ(t). (347) t t As initial condition to ﬁrst order elementarily integrable ordinary diﬀerential equation for f, we must take f(0) = 0 to mirror the fact that Σ(0) = 1/D2(0) = 1/12 = 1 by deﬁnition of a damping function. ≤ We observe that the right-hand-side is of the diﬀerential equation is well-deﬁned since 0 < D 1 and ≥ ≥ ±∞ √ Σ = 1/D2 1 for all t 0. Thus, the argument of the logarithm does not diverge to . Solving the ordinary diﬀerential equation for f results in the equation right to the equivalence arrow, i.e., f = log Σ. We are now in the position to ﬁnd an explicit expression of the function q in terms of the time-lapse function Σ, √ √ − q(t) = ∂2 log Σ (∂ log Σ)2. (348) t t The whole derivation works exclusively because we have [∆∂ C] = 0, where C[w] = exp(f(t))[w] is a g0 linear multiplication operator acting on w. The commutativity of the Laplacian ∆∂ and C is ensured g0 by the time-dependence of f and the time-independence of ∆∂ plus the dependence of ∆∂ on the ﬁber { } g0 { } g0 coordinates yi ≤ ≤ restricted from ∂Ω to Γ and the independence of C on yi ≤ ≤ . In other words, C 1 i n 0 i { } 1 i n is a constant w.r.t. diﬀerentiation w.r.t. the coordinates yi ≤ ≤ . It remains to relate the source term Ψ 1 i n in the diﬀerential equation for u to the source term Ψ in the diﬀerential equation for w. This is attained − − f by acting with C 1 = exp( f) on the diﬀerential equation for u from the left. Since the Ansatz for u with the previous identiﬁcations recasts the right hand side of the diﬀerential equation for u in exp(f) times − the generalized Klein-Gordon operator acting on w, the C 1 cancels the redundant contribution from the conformal factor f and leaves us with, − − − ≡ ∂2w (p(∆ )w q(t)) = Ψ exp( f) Ψ . (349) t g0 f Naturally, the approach has its limitations, namely we have to require a well-behaved damping. The − ≥ − ≥ ≥ well-behavedness is to be understood s.t. p(∆ ) q(t) 0, i.e., p( γ ) q(t) for all t 0 where γ g0 − 1 1 denotes the smallest eigenvalue of the Dirichlet Laplacian ∆∂ in the sense of the Lichnernowicz theorem. g0 − γ is strictly positive. Let us deﬁne the operators D2(t) = p(∆∂ ) q(t). We now recast the second order 1 g0 partial diﬀerential equation in a ﬁrst order system of ordinary operator equations by deﬁning W = ∂ w t 103 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Comparison

text: of

text: the

text: coeﬃcients

text: of

text: ∂kw

text: for

text: k

text: yields

text: the

text: identiﬁcations

text: for

text: the

text: still

text: unknown

text: t

text: functions

text: f

text: f(t)

text: and

text: q

text: q(t).

text: For

text: the

text: latter

text: function,

text: we

text: ﬁnd

text: an

text: equation

text: which

text: still

text: involves

text: the

text: conformal

text: factor

text: f,

text: q

text: (f

text: log

text: Σ

text: (f

text: f

text: (f

text: log

text: Σ

text: (f

text: f

text: t

text: t

text: The

text: determining

text: ordinary

text: diﬀerential

text: equation

text: for

text: f

text: is

text: obtained

text: by

text: comparing

text: coeﬃcients

text: of

text: ∂kw

text: for

text: t

text: k

text: and

text: is

text: given

text: by

text: log

text: Σ

text: f

text: log

text: Σ

text: f(t)

text: log

text: Σ(t).

text: t

text: t

text: As

text: initial

text: condition

text: to

text: ﬁrst

text: order

text: elementarily

text: integrable

text: ordinary

text: diﬀerential

text: equation

text: for

text: f,

text: we

text: must

text: take

text: to

text: mirror

text: the

text: fact

text: that

text: by

text: deﬁnition

text: of

text: a

text: damping

text: function.

text: We

text: observe

text: that

text: the

text: right-hand-side

text: is

text: of

text: the

text: diﬀerential

text: equation

text: is

text: well-deﬁned

text: since

text: D

text: and

text: Σ

text: for

text: all

text: t

text: Thus,

text: the

text: argument

text: of

text: the

text: logarithm

text: does

text: not

text: diverge

text: to

text: Solving

text: the

text: ordinary

text: diﬀerential

text: equation

text: for

text: f

text: results

text: in

text: the

text: equation

text: right

text: to

text: the

text: equivalence

text: arrow,

text: i.e.,

text: f

text: log

text: Σ.

text: We

text: are

text: now

text: in

text: the

text: position

text: to

text: ﬁnd

text: an

text: explicit

text: expression

text: of

text: the

text: function

text: q

text: in

text: terms

text: of

text: the

text: time-lapse

text: function

text: Σ,

text: q(t)

text: ∂2

text: log

text: Σ

text: log

text: t

text: t

text: The

text: whole

text: derivation

text: works

text: exclusively

text: because

text: we

text: have

text: C]

text: where

text: C[w]

text: exp(f(t))[w]

text: is

text: a

text: linear

text: multiplication

text: operator

text: acting

text: on

text: w.

text: The

text: commutativity

text: of

text: the

text: Laplacian

text: and

text: C

text: is

text: ensured

text: by

text: the

text: time-dependence

text: of

text: f

text: and

text: the

text: time-independence

text: of

text: plus

text: the

text: dependence

text: of

text: on

text: the

text: ﬁber

text: coordinates

text: yi

text: restricted

text: from

text: ∂Ω

text: to

text: Γ

text: and

text: the

text: independence

text: of

text: C

text: on

text: yi

text: In

text: other

text: words,

text: C

text: i

text: n

text: i

text: i

text: n

text: is

text: a

text: constant

text: w.r.t.

text: diﬀerentiation

text: w.r.t.

text: the

text: coordinates

text: yi

text: It

text: remains

text: to

text: relate

text: the

text: source

text: term

text: Ψ

text: i

text: n

text: in

text: the

text: diﬀerential

text: equation

text: for

text: u

text: to

text: the

text: source

text: term

text: Ψ

text: in

text: the

text: diﬀerential

text: equation

text: for

text: w.

text: This

text: is

text: attained

text: f

text: by

text: acting

text: with

text: C

text: exp(

text: f)

text: on

text: the

text: diﬀerential

text: equation

text: for

text: u

text: from

text: the

text: left.

text: Since

text: the

text: Ansatz

text: for

text: u

text: with

text: the

text: previous

text: identiﬁcations

text: recasts

text: the

text: right

text: hand

text: side

text: of

text: the

text: diﬀerential

text: equation

text: for

text: u

text: in

text: exp(f)

text: times

text: the

text: generalized

text: Klein-Gordon

text: operator

text: acting

text: on

text: w,

text: the

text: C

text: cancels

text: the

text: redundant

text: contribution

text: from

text: the

text: conformal

text: factor

text: f

text: and

text: leaves

text: us

text: with,

text: (p(∆

text: )w

text: q(t))

text: Ψ

text: exp(

text: f)

text: Ψ

text: t

text: f

text: Naturally,

text: the

text: approach

text: has

text: its

text: limitations,

text: namely

text: we

text: have

text: to

text: require

text: a

text: well-behaved

text: damping.

text: The

text: well-behavedness

text: is

text: to

text: be

text: understood

text: s.t.

text: p(∆

text: q(t)

text: i.e.,

text: p(

text: γ

text: q(t)

text: for

text: all

text: t

text: where

text: γ

text: denotes

text: the

text: smallest

text: eigenvalue

text: of

text: the

text: Dirichlet

text: Laplacian

text: in

text: the

text: sense

text: of

text: the

text: Lichnernowicz

text: theorem.

text: γ

text: is

text: strictly

text: positive.

text: Let

text: us

text: deﬁne

text: the

text: operators

text: p(∆∂

text: q(t).

text: We

text: now

text: recast

text: the

text: second

text: order

text: partial

text: diﬀerential

text: equation

text: in

text: a

text: ﬁrst

text: order

text: system

text: of

text: ordinary

text: operator

text: equations

text: by

text: deﬁning

text: W

text: w

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: and rewriting the above equation in the form of a ﬁrst order system of ordinary diﬀerential equations which are in fact ordinary op(cid:18)erato(cid:19)r equ(cid:18)ations. The(cid:19)su(cid:18)bstitu(cid:19)tion(cid:18)yields(cid:19) ∂w ≡ ∂ w 0 1 w 0 ≡ = + A(t)w + Ψ(cid:126) . (350) ∂t ∂t W D2(t) 0 W Φ f f We have ﬁnished the necessary preliminary work to apply Duhamel’s principle to the diﬀerential equa- ≥ ∈ tion for w. We let t , t 0 and g H1,2(Γ ). Let us calculate the commutator [D2(t ), D2(t )], where we 1 2 · i 1 2 will suppress the composition sign for operators. (cid:0) (cid:1) − [D2(t ), D2(t )]g = (cid:0)D2(t )D2(t ) D2(t )D2(t )g (cid:1) 1 2 1 2 2 1 − − = (cid:0)p(∆∂ )p(∆∂ ) p(∆∂ )q(t ) (cid:1)q(t )p(∆∂ ) + q(t )q(t ) g − g0 g0 − g0 2 − 1 g0 1 2 p(∆∂ )p(∆∂ ) q(t )p(∆∂ ) p(∆∂ )q(t ) + q(t )q(t ) g g0 g0 − 2 g0 −g0 1 2− 1 = p(∆∂ )p(∆∂ ) p(∆∂ )p(∆∂ ) g (q(t )p(∆∂ ) p(∆∂ )q(t ))g g0 g−0 g0 g0 1 − g0 g0 1 + (q(t )p(∆∂ ) p(∆∂ )q(t ))g + (q(t )q(t ) q(t )q(t ))g − 2 g0 − g0 2 1 2 −1 2 = (q(t )p(∆∂ ) p(∆∂ )q(t ))g + (q(t )p(∆∂ ) p(∆∂ )q(t ))g 1 g0 g0 1 2 g0 g0 2 = 0[g], where O is the zero operator, sending each function it acts on, identically to zero. In the ﬁfth step, we ∈ have used our previous observation that p(∆∂ )q(t) = 0. By arbitrariness of g H1,2(Γ ), we have found g0 i the result [D2(t ), D2(t )] = 0. (351) 1 2 Let us concentrate on the D2(t)’s a family of operators. The function q(τ) can is bounded for all ≥ t 0 because Σ(t = 0) = 1 excludes the nasty cases where Σ(t) is hyper-exponential, i.e., a function like R → R C∞ Σ(t) = exp(exp(α(t))) where α(t) : + is w.l.o.g. . However, the Laplacian ∆∂ is unbounded in (cid:107) (cid:107) 0 g0 the norm . on H1,2(Γ ). However, if we require our functions to be a bit more regular, namely to be Hg10,2 i (cid:107) (cid:107) → (cid:107) (cid:107) in H2,2(Γ ) as well, we assure that ∆∂ : (H2,2(Γ ), . (H2,2(Γ , . ) is bounded. The restriction i g0 i L2g0 i L2g0 of the domain dom(∆∂ ) are necessary and suﬃcient to ensure self-adjointness of ∆∂ w.r.t. the L2 norm g0 ∈ g0 g0 restricted to Γ . Indeed, we have for g H2,2(Γ ), i i (cid:107) (cid:107) ≤ (cid:107) (cid:107) ∞ ∆∂ g g < , (352) g0 L2g0 Hg20,2 (cid:113) by restriction of ∆∂ to H2,2(Γ ). Then we have in the graph norm applied to ∆∂ , g0 i g0 (cid:107) (cid:107) ≡ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) ≤ (cid:107) (cid:107) ∞ . ∆∂ g = g + ∆ g < . gr g0 gr L2g0 g0 L2g0 Hg20,2 With suitable restrictions, we can turn the on all of L2(Γ ) unbounded Dirichlet Laplace-Beltrami operator i → ∆∂ into a bounded operator. Notice that the imbedding H2,2(Γ ) (cid:44) H1,2(Γ ) is a compact linear operator g0 i i by the Rellich imbedding theorem and in H2,2(Γ ) is by construction dense in both H1,2(Γ ) and L2(Γ ) i i i 104 ##LTLine## ##LTLine## ##LTLine##

text: and

text: rewriting

text: the

text: above

text: equation

text: in

text: the

text: form

text: of

text: a

text: ﬁrst

text: order

text: system

text: of

text: ordinary

text: diﬀerential

text: equations

text: which

text: are

text: in

text: fact

text: ordinary

text: ∂w

text: w

text: w

text: A(t)w

text: ∂t

text: ∂t

text: W

text: W

text: Φ

text: f

text: f

text: We

text: have

text: ﬁnished

text: the

text: necessary

text: preliminary

text: work

text: to

text: apply

text: Duhamel’s

text: principle

text: to

text: the

text: diﬀerential

text: equa-

text: tion

text: for

text: w.

text: We

text: let

text: t

text: t

text: and

text: g

text: Let

text: us

text: calculate

text: the

text: commutator

text: where

text: we

text: i

text: will

text: suppress

text: the

text: composition

text: sign

text: for

text: operators.

text: )]g

text: )g

text: )p(∆∂

text: p(∆∂

text: )q(t

text: )p(∆∂

text: q(t

text: )q(t

text: g

text: p(∆∂

text: )p(∆∂

text: q(t

text: )p(∆∂

text: p(∆∂

text: )q(t

text: q(t

text: )q(t

text: g

text: 2−

text: p(∆∂

text: )p(∆∂

text: p(∆∂

text: )p(∆∂

text: g

text: (q(t

text: )p(∆∂

text: p(∆∂

text: )q(t

text: ))g

text: (q(t

text: )p(∆∂

text: p(∆∂

text: )q(t

text: ))g

text: (q(t

text: )q(t

text: q(t

text: )q(t

text: ))g

text: −1

text: (q(t

text: )p(∆∂

text: p(∆∂

text: )q(t

text: ))g

text: (q(t

text: )p(∆∂

text: p(∆∂

text: )q(t

text: ))g

text: where

text: O

text: is

text: the

text: zero

text: operator,

text: sending

text: each

text: function

text: it

text: acts

text: on,

text: identically

text: to

text: zero.

text: In

text: the

text: ﬁfth

text: step,

text: we

text: have

text: used

text: our

text: previous

text: observation

text: that

text: p(∆∂

text: )q(t)

text: By

text: arbitrariness

text: of

text: g

text: we

text: have

text: found

text: i

text: the

text: result

text: Let

text: us

text: concentrate

text: on

text: the

text: a

text: family

text: of

text: operators.

text: The

text: function

text: q(τ)

text: can

text: is

text: bounded

text: for

text: all

text: t

text: because

text: Σ(t

text: excludes

text: the

text: nasty

text: cases

text: where

text: Σ(t)

text: is

text: hyper-exponential,

text: i.e.,

text: a

text: function

text: like

text: R

text: R

text: C∞

text: Σ(t)

text: exp(exp(α(t)))

text: where

text: α(t)

text: is

text: w.l.o.g.

text: However,

text: the

text: Laplacian

text: is

text: unbounded

text: in

text: the

text: norm

text: on

text: However,

text: if

text: we

text: require

text: our

text: functions

text: to

text: be

text: a

text: bit

text: more

text: regular,

text: namely

text: to

text: be

text: i

text: in

text: as

text: well,

text: we

text: assure

text: that

text: is

text: bounded.

text: The

text: restriction

text: i

text: i

text: i

text: of

text: the

text: domain

text: dom(∆∂

text: are

text: necessary

text: and

text: suﬃcient

text: to

text: ensure

text: self-adjointness

text: of

text: w.r.t.

text: the

text: norm

text: restricted

text: to

text: Γ

text: Indeed,

text: we

text: have

text: for

text: g

text: i

text: i

text: g

text: g

text: by

text: restriction

text: of

text: to

text: Then

text: we

text: have

text: in

text: the

text: graph

text: norm

text: applied

text: to

text: i

text: g

text: g

text: g

text: gr

text: gr

text: With

text: suitable

text: restrictions,

text: we

text: can

text: turn

text: the

text: on

text: all

text: of

text: unbounded

text: Dirichlet

text: Laplace-Beltrami

text: operator

text: i

text: into

text: a

text: bounded

text: operator.

text: Notice

text: that

text: the

text: imbedding

text: is

text: a

text: compact

text: linear

text: operator

text: i

text: i

text: by

text: the

text: Rellich

text: imbedding

text: theorem

text: and

text: in

text: is

text: by

text: construction

text: dense

text: in

text: both

text: and

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: such that under the above restrictions ∆∂ is still densely deﬁned with Dom(∆∂ ). Choosing our Laplacian g0 g0 with the suitable restrictions, we have using linearity of ∆∂ and boundedness that ∆∂ is a continuous g0 g0 linear operator and moreover completely continuous, i.e., for a weakly convergent sequence (g ) ∈N in (cid:107) (cid:107) (cid:107) (cid:107) n n (H2,2(Γ ), . ), the sequence (∆∂ g ) ∈N converges in (H2,2(Γ ), . ) in the norm topology. The i L2g0 g0 n n i L2g0 compactness follows from the complete continuity of the linear operator ∆∂ the reﬂexivity of H2,2, equipped g0 with the L2 -norm, as Banach spaces and. We seek to relate the compactness of ∆∂ under our restrictions g0 g0 to the compactness of D2(t). By boundedness of q for all t and as a function of t, we have that the operators − (cid:107) (cid:107) → (cid:107) (cid:107) D2(t) = p(∆∂ ) q(t) : (H2,2(Γ ), . ) (H2,2(Γ ), . ) are linear, bounded, continuous and compact g0 ≥ i L2g0 i L2g0 operators for all t 0. This allows us to deﬁne suitable algebras of operators containing the family (D2(t)) ≥ . Indeed, the family of operators (D2(t)) ≥ generates a Banach-algebra, and, by self-adjointness t 0 ∗ ∗ { } t 0 ∗ ∗ of each D2(t), a C algebra C ( D2(t) ≥ ), which is minimal in the sense that if C denotes all C -algebras ∗ t 0 C containing the family (D2(t)) ≥ , we have (cid:92) t 0 ≡ ∗ ∗ W C ((D2(t)) ≥ ) = C . (353) t 0 ∗∈ ∗ C C ∗ ∗ By boundedness of the ∆ ’s subject to the suitable restrictions, the C -algebra W is even a W -algebra, g0 i.e., a von-Neumann algebra. Since it is a von-Neumann algebra over the real (and thus also complex) (cid:107) (cid:107) Hilbert space H2,2(Γ ), endowed with with the L2 norm restricted to Γ , . , we can apply von- i g0 i L2g0 Neumann’s theorem on the decomposition of von-Neumann algebras. Observing that [D2(t ), D2(t )] = 0 ≥ ∗ 1 2 ∈ for all t , t 0 and by minimality of W, the W -algebra W is even commutative,i.e., for all A, B W, 1 2 ∗ we have [A, B] = 0. As a commutative W -algebra, W has a trivial decomposition in terms of commutant sub-algebras. In the decomposition, we have (cid:77) ∞ adk [W] W W = W = , (354) adk [W] ad [W] k=0 W W by commutativity of W. We have deﬁne iteratively adk+1[W] = ad [adk [W]] with ad0 [W] = W and { ∈ ∈ } W W W W ad [W] = A W : [A, B] = 0 for all B W denotes the centralisator of W w.r.t. W and denote by W adk+1[W] the (k + 1)-th centralisator of W w.r.t. W or, equivalently, the centralisator of adk [W] w.r.t. W W W. Thus in the Magnus expansion, the generator G(τ) only consists the ﬁrst contribution G (τ) resulting ∗ 1 from the ﬁrst factor of the decomposition of the W -algebras W, i.e., in practical calculus-like notation again, we have the exact expres(cid:90)sion, (cid:90) (cid:18) (cid:19) (cid:18) (cid:19) (cid:82) τ τ (cid:48) 0 1 0 τ G(τ) = G (τ) = dτ A(τ) = dτ (cid:48) = (cid:48) (cid:48) (355) (1) D2(τ ) 0 τ dτ D2(τ ) 0 0 0 0 → − It remains to evaluate exp(G(τ)) and replace in the solution formula given above τ t τ because we use Duhamel’s principle to solve an inhomogeneous partial diﬀerential equation. We have using com- ∗ ∈ N mutativity of the W algebra, the fo(cid:32)llowin(cid:0)g(cid:82) two equatio(cid:1)ns for k (cid:33) 0 (cid:0)(cid:82) (cid:1) (cid:48) (cid:48) τk τ dτ D2(τ ) k 0 G2k(τ) = 0 (cid:48) (cid:48) (356) 0 τk τ dτ D2(τ ) k 0 105 ##LTLine## ##LTLine##

text: such

text: that

text: under

text: the

text: above

text: restrictions

text: is

text: still

text: densely

text: deﬁned

text: with

text: Dom(∆∂

text: Choosing

text: our

text: Laplacian

text: with

text: the

text: suitable

text: restrictions,

text: we

text: have

text: using

text: linearity

text: of

text: and

text: boundedness

text: that

text: is

text: a

text: continuous

text: linear

text: operator

text: and

text: moreover

text: completely

text: continuous,

text: i.e.,

text: for

text: a

text: weakly

text: convergent

text: sequence

text: (g

text: ∈N

text: in

text: n

text: n

text: the

text: sequence

text: g

text: ∈N

text: converges

text: in

text: in

text: the

text: norm

text: topology.

text: The

text: i

text: n

text: n

text: i

text: compactness

text: follows

text: from

text: the

text: complete

text: continuity

text: of

text: the

text: linear

text: operator

text: the

text: reﬂexivity

text: of

text: equipped

text: with

text: the

text: -norm,

text: as

text: Banach

text: spaces

text: and.

text: We

text: seek

text: to

text: relate

text: the

text: compactness

text: of

text: under

text: our

text: restrictions

text: to

text: the

text: compactness

text: of

text: By

text: boundedness

text: of

text: q

text: for

text: all

text: t

text: and

text: as

text: a

text: function

text: of

text: t,

text: we

text: have

text: that

text: the

text: operators

text: p(∆∂

text: q(t)

text: are

text: linear,

text: bounded,

text: continuous

text: and

text: compact

text: i

text: i

text: operators

text: for

text: all

text: t

text: This

text: allows

text: us

text: to

text: deﬁne

text: suitable

text: algebras

text: of

text: operators

text: containing

text: the

text: family

text: Indeed,

text: the

text: family

text: of

text: operators

text: generates

text: a

text: Banach-algebra,

text: and,

text: by

text: self-adjointness

text: t

text: t

text: of

text: each

text: a

text: C

text: algebra

text: C

text: which

text: is

text: minimal

text: in

text: the

text: sense

text: that

text: if

text: C

text: denotes

text: all

text: C

text: -algebras

text: t

text: C

text: containing

text: the

text: family

text: we

text: have

text: t

text: W

text: C

text: C

text: t

text: C

text: C

text: By

text: boundedness

text: of

text: the

text: ’s

text: subject

text: to

text: the

text: suitable

text: restrictions,

text: the

text: C

text: -algebra

text: W

text: is

text: even

text: a

text: W

text: -algebra,

text: i.e.,

text: a

text: von-Neumann

text: algebra.

text: Since

text: it

text: is

text: a

text: von-Neumann

text: algebra

text: over

text: the

text: real

text: (and

text: thus

text: also

text: complex)

text: Hilbert

text: space

text: endowed

text: with

text: with

text: the

text: norm

text: restricted

text: to

text: Γ

text: we

text: can

text: apply

text: von-

text: i

text: i

text: Neumann’s

text: theorem

text: on

text: the

text: decomposition

text: of

text: von-Neumann

text: algebras.

text: Observing

text: that

text: for

text: all

text: t

text: t

text: and

text: by

text: minimality

text: of

text: W,

text: the

text: W

text: -algebra

text: W

text: is

text: even

text: commutative,i.e.,

text: for

text: all

text: A,

text: B

text: W,

text: we

text: have

text: [A,

text: B]

text: As

text: a

text: commutative

text: W

text: -algebra,

text: W

text: has

text: a

text: trivial

text: decomposition

text: in

text: terms

text: of

text: commutant

text: sub-algebras.

text: In

text: the

text: decomposition,

text: we

text: have

text: adk

text: [W]

text: W

text: W

text: W

text: adk

text: [W]

text: ad

text: [W]

text: W

text: W

text: by

text: commutativity

text: of

text: W.

text: We

text: have

text: deﬁne

text: iteratively

text: ad

text: [adk

text: [W]]

text: with

text: [W]

text: W

text: and

text: W

text: W

text: W

text: W

text: ad

text: [W]

text: A

text: W

text: [A,

text: B]

text: for

text: all

text: B

text: W

text: denotes

text: the

text: centralisator

text: of

text: W

text: w.r.t.

text: W

text: and

text: denote

text: by

text: W

text: the

text: (k

text: centralisator

text: of

text: W

text: w.r.t.

text: W

text: or,

text: equivalently,

text: the

text: centralisator

text: of

text: adk

text: [W]

text: w.r.t.

text: W

text: W

text: W.

text: Thus

text: in

text: the

text: Magnus

text: expansion,

text: the

text: generator

text: G(τ)

text: only

text: consists

text: the

text: ﬁrst

text: contribution

text: G

text: (τ)

text: resulting

text: from

text: the

text: ﬁrst

text: factor

text: of

text: the

text: decomposition

text: of

text: the

text: W

text: -algebras

text: W,

text: i.e.,

text: in

text: practical

text: calculus-like

text: notation

text: again,

text: we

text: have

text: the

text: exact

text: τ

text: τ

text: τ

text: G(τ)

text: G

text: (τ)

text: dτ

text: A(τ)

text: dτ

text: τ

text: dτ

text: It

text: remains

text: to

text: evaluate

text: exp(G(τ))

text: and

text: replace

text: in

text: the

text: solution

text: formula

text: given

text: above

text: τ

text: t

text: τ

text: because

text: we

text: use

text: Duhamel’s

text: principle

text: to

text: solve

text: an

text: inhomogeneous

text: partial

text: diﬀerential

text: equation.

text: We

text: have

text: using

text: com-

text: N

text: mutativity

text: of

text: the

text: W

text: algebra,

text: the

text: two

text: for

text: k

text: τk

text: τ

text: dτ

text: k

text: τk

text: τ

text: dτ

text: k

text: ##LTLine##

text: ##LTLine##


text: (cid:32) (cid:33) and (cid:0)(cid:82) (cid:1) (cid:0)(cid:82) (cid:1) (cid:48) (cid:48) 0 τk τ dτ D2(τ ) k+1 G2k+1(τ) = (cid:48) (cid:48) 0 (357) τk+1 τ dτ D2(τ ) k 0 0 We can now apply the Taylor series expansion of the exponential functions of compact self-adjoint operators, (cid:88) ∞ 1 exp(G(τ)) = (cid:88) Gk(τ) (cid:88) (358) k! k=0 ∞ ∞ 1 1 = G2k(τ) + G2k+1(τ) (359) (2k)! (2k + 1)! k=0 k=0 ≡ exp + exp . (360) even odd ≥ Noting that D2 is a negative operator for all t 0, the sums over even and odd summation indices k can be expressed in terms of sine and cosine functions with operator values arguments. For the sum over  (cid:16)(cid:113) (cid:17)  even indices, we have (cid:82)  (cid:16)(cid:113) (cid:17)  (cid:82) τ (cid:48) − (cid:48) cos τ dτ ( D2)(τ ) 0 expeven =  (cid:18) (cid:113)(cid:82)0 (cid:19) τ (cid:48) − (cid:48)  (361) 0 cos τ dτ ( D2)(τ )  0   (cid:18) (cid:113) (cid:19)   − · (cid:82)  1 cos τ dζ( D2)(τ ζ) 0 0 = , (362) − 1 0 cos τ dζ( D2)(ζ) 0 (cid:48) ≥ where we have rescaled in the last step in the integration variable τ = τζ and used τ 0 to pull τ out of the square root. In case D2 is time-independent this gives the well-known formulas for the constantly  (cid:113) (cid:48) (cid:18) (cid:113) (cid:19)  damped wave-equation. For the sum over odd indices, we use ag(cid:82)ain τ = ζτ as a transfor(cid:82)mation and ﬁnd    (cid:113) (cid:18) (cid:113) (cid:19)   (cid:82) (cid:82) − −  1 1 0 dζ ( D2)(τζ) sin τ dζ ( D2)(τζ) 0 0 exp = − . odd − 1 − 1 1 dζ ( D2)(τζ) sin τ dζ ( D2)(τζ) 0 0 0 (363)  (cid:18) (cid:113) (cid:19) G (cid:113)≡ (cid:18) (cid:113) (cid:19)  In total, we have evaluate(cid:82)d the Magnus exponential (τ) (cid:82) exp(G(τ)) with the resu(cid:82)lt    (cid:113) (cid:18) (cid:113) (cid:19) (cid:18) (cid:113) (cid:19)   (cid:82) − (cid:82) · − (cid:82) −  1 1 1 cos τ dζ( D2)(τ ζ) dζ ( D2)(τζ) sin τ dζ ( D2)(τζ) G 0 0 0 (τ) = − . − 1 − − · 1 1 1 dζ ( D2)(τζ) sin τ dζ ( D2)(τζ) cos τ dζ( D2)(τ ζ) 0 0 0 (364) 106 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: τk

text: τ

text: dτ

text: τ

text: dτ

text: k

text: We

text: can

text: now

text: apply

text: the

text: Taylor

text: series

text: expansion

text: of

text: the

text: exponential

text: functions

text: of

text: compact

text: self-adjoint

text: operators,

text: exp(G(τ))

text: Gk(τ)

text: k!

text: exp

text: exp

text: even

text: odd

text: Noting

text: that

text: is

text: a

text: negative

text: operator

text: for

text: all

text: t

text: the

text: sums

text: over

text: even

text: and

text: odd

text: summation

text: indices

text: k

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: sine

text: and

text: cosine

text: functions

text: with

text: operator

text: values

text: arguments.

text: For

text: the

text: sum

text: over

text: even

text: indices,

text: we

text: have

text: τ

text: cos

text: τ

text: dτ

text: expeven

text: τ

text: cos

text: τ

text: dτ

text: cos

text: τ

text: dζ(

text: ζ)

text: cos

text: τ

text: dζ(

text: where

text: we

text: have

text: rescaled

text: in

text: the

text: last

text: step

text: in

text: the

text: integration

text: variable

text: τ

text: τζ

text: and

text: used

text: τ

text: to

text: pull

text: τ

text: out

text: of

text: the

text: square

text: root.

text: In

text: case

text: is

text: time-independent

text: this

text: gives

text: the

text: well-known

text: formulas

text: for

text: the

text: constantly

text: damped

text: wave-equation.

text: For

text: the

text: sum

text: over

text: odd

text: indices,

text: we

text: use

text: τ

text: ζτ

text: as

text: a

text: and

text: ﬁnd

text: dζ

text: sin

text: τ

text: dζ

text: exp

text: odd

text: dζ

text: sin

text: τ

text: dζ

text: G

text: In

text: total,

text: we

text: have

text: the

text: Magnus

text: exponential

text: (τ)

text: exp(G(τ))

text: with

text: the

text: cos

text: τ

text: dζ(

text: ζ)

text: dζ

text: sin

text: τ

text: dζ

text: G

text: (τ)

text: dζ

text: sin

text: τ

text: dζ

text: cos

text: τ

text: dζ(

text: ζ)

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Next, we deﬁne the operator O2(t) = ∆ + W. We reformulate the acoustic wave equation by means g0 of setting P = ∂ p as (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) t ∂p ≡ ∂ p 0 1 p 0 ≡ = + B(t)p + h. (369) ∂t ∂t P c2O2(t) 0 P ρ c2∂2u 0 t (cid:54) ∈ R By non-commutativity of ∆ and W, [∆ , W] in general and [O(t ), O(t )] = 0 for general t , t +, (cid:54) g0 ∈ Rg0 1 2 ∗ 1 2 0 we also have [B(t ), B(t )] = 0 for t , t + typically. This means, that the W -algebra generated by ≥ 1 2 1 2 0 O(t), t 0, say W, cannot be decomposed using the Lie lower central series such that the W is nil-potent, ∈ N i.e., such that the series truncates after M factors. By the previous considerations, the endowment of gl(2, W) with the usual Lie-bracket [., .] actually turns gl(2, W) into a Lie algebra. The acoustic wave equation has the form of an operator evolution equation, and O is linear and continuous, as well as bounded ∈ M (cid:107) (cid:107) → M (cid:107) (cid:107) in the H2,2-norm: For f H2,2(Ω ), we have for ∆ : (H2,2( ), . ) (H2,2( ), . ) 0 0 g0 0 0 Hg20,2 0 0 L2g0 (cid:107) (cid:107) ≤ (cid:107) | ⇒ (cid:107) (cid:107) ≤ ∆ f f ∆ → 1. (370) g0 L2g0 Hg20,2 g0 Hg20,2 L2g0 Thus, we can apply the Magnus series and our previous considerations for the abstract problem Magnus problem ensure convergence of the Magnus series. We make the usual Ansatz and notice p(t = 0) = 0 by our speciﬁcation of initial conditions p(t =(cid:90)0) = 0 = ∂ p(t = 0), t t − p = dτ exp(G(t τ))h(τ), (371) 0 ∈ by Duhamel’s principle. By Magnus’ theorem, the Magnus generator G gl(2, W) satisﬁes the following evolution equation, (cid:88) ∞ ∂G − B = d exp 1(B) = k adk [B], (372) ∂t G k! G k=0 where B denotes the k-th Bernoulli number and ad denotes the left adjoint action of the Lie-algebra k G M gl(2, W) on itself. Using the above convergence properties in the (H2,2( ))-Frobenius norm, a Banach 0 0 ≤ ≤ ≤ · · · ≤ ≤ ﬁxed-point argument [58] resulted in the following Picard iteration scheme (0 τ τ − τ n n 1 1 τ = τ) 0 (cid:88) ∞ (cid:90) G = G (373) (k) k=0 τ (cid:48) (cid:48) (cid:18)(cid:90) (cid:19) (cid:90) G (τ) = (cid:88) dτ B(τ ) (cid:88) (cid:89) (374) (1) 0 − G (τ = τ ) = n 1 Bj (cid:80) j τi−1 dτ ad τn−1 dτ [B(τ )]. (375) (n) 0 j! i Gki(τi) n n − ≥ 0 0 j=1 ji=1 ki=n 1;ki 1 i=1 108 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Next,

text: we

text: deﬁne

text: the

text: operator

text: W.

text: We

text: reformulate

text: the

text: acoustic

text: wave

text: equation

text: by

text: means

text: of

text: setting

text: P

text: p

text: as

text: t

text: ∂p

text: p

text: p

text: B(t)p

text: h.

text: ∂t

text: ∂t

text: P

text: P

text: ρ

text: t

text: R

text: By

text: non-commutativity

text: of

text: and

text: W,

text: W]

text: in

text: general

text: and

text: [O(t

text: O(t

text: for

text: general

text: t

text: t

text: we

text: also

text: have

text: [B(t

text: B(t

text: for

text: t

text: t

text: typically.

text: This

text: means,

text: that

text: the

text: W

text: -algebra

text: generated

text: by

text: O(t),

text: t

text: say

text: W,

text: cannot

text: be

text: decomposed

text: using

text: the

text: Lie

text: lower

text: central

text: series

text: such

text: that

text: the

text: W

text: is

text: nil-potent,

text: N

text: i.e.,

text: such

text: that

text: the

text: series

text: truncates

text: after

text: M

text: factors.

text: By

text: the

text: previous

text: considerations,

text: the

text: endowment

text: of

text: W)

text: with

text: the

text: usual

text: Lie-bracket

text: actually

text: turns

text: W)

text: into

text: a

text: Lie

text: algebra.

text: The

text: acoustic

text: wave

text: equation

text: has

text: the

text: form

text: of

text: an

text: operator

text: evolution

text: equation,

text: and

text: O

text: is

text: linear

text: and

text: continuous,

text: as

text: well

text: as

text: bounded

text: M

text: M

text: in

text: the

text: For

text: f

text: we

text: have

text: for

text: f

text: f

text: Thus,

text: we

text: can

text: apply

text: the

text: Magnus

text: series

text: and

text: our

text: previous

text: considerations

text: for

text: the

text: abstract

text: problem

text: Magnus

text: problem

text: ensure

text: convergence

text: of

text: the

text: Magnus

text: series.

text: We

text: make

text: the

text: usual

text: Ansatz

text: and

text: notice

text: p(t

text: by

text: our

text: speciﬁcation

text: of

text: initial

text: conditions

text: p(t

text: p(t

text: t

text: t

text: p

text: dτ

text: exp(G(t

text: τ))h(τ),

text: by

text: Duhamel’s

text: principle.

text: By

text: Magnus’

text: theorem,

text: the

text: Magnus

text: generator

text: G

text: W)

text: satisﬁes

text: the

text: following

text: evolution

text: equation,

text: ∂G

text: B

text: d

text: exp

text: k

text: adk

text: [B],

text: ∂t

text: G

text: k!

text: G

text: where

text: B

text: denotes

text: the

text: k-th

text: Bernoulli

text: number

text: and

text: ad

text: denotes

text: the

text: left

text: adjoint

text: action

text: of

text: the

text: Lie-algebra

text: k

text: G

text: M

text: W)

text: on

text: itself.

text: Using

text: the

text: above

text: convergence

text: properties

text: in

text: the

text: ))-Frobenius

text: norm,

text: a

text: Banach

text: ﬁxed-point

text: argument

text: resulted

text: in

text: the

text: following

text: Picard

text: iteration

text: scheme

text: τ

text: τ

text: τ

text: n

text: n

text: τ

text: τ)

text: G

text: G

text: (k)

text: τ

text: G

text: (τ)

text: dτ

text: B(τ

text: G

text: (τ

text: τ

text: n

text: Bj

text: j

text: τi−1

text: dτ

text: ad

text: τn−1

text: dτ

text: [B(τ

text: (n)

text: j!

text: i

text: Gki(τi)

text: n

text: n

text: ki=n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Let us now investigate the integral in the integral equation more closely. By deﬁnition of O2(t) and (cid:107) (cid:107) (cid:107) (cid:107) O using the previo(cid:115)us result W / ∆ = ((cid:15)2(cid:115)), we have (cid:90) g0 (cid:90) (cid:18) (cid:19) − 1 − − 1(cid:115) W (cid:90) c2 dζ O2((t τ)ζ) = (cid:112) c2 dζ ∆g0 1 + ∆ (390) 0 0 g0 − (cid:32) 1 (cid:82) W (cid:33) = (cid:112) c2∆ 1 + dζ (391) g0 ∆ 0 g0 − − 1 c2 1 dζ W(ζτ) O = c2∆ 1 + −0 + ((cid:15)2), (392) g0 2 c2∆ g0 √ (cid:39) O where we only work up to order (cid:15)2 in the familiar Taylor expansion of 1 + x 1 + 1/2x + (x2). (cid:107) (cid:107) Functional calculus assures that in the . -norm, we can actually perform this expansion. Next, we need 2,2 two further identities. The addition theorem of the sine function and the geometric series. For the operator sine function in the integral equation for p, we use the following addition theorem and the power series ∈ R O representation of the sine and cosine function excluding orders from (cid:15)4 on. Let x, y +, y/x = ((cid:15)2) and use the addition theorem, sin(x + y) = sin x cos y + sin y cos x, (393) to obtain O sin(x + y) = sin x + y cos x + ((cid:15)4). (394) Next, we use the geometric series for x, y as in the above identity, (cid:88) (cid:16) (cid:17) (cid:112) ∞ √ 1 √1 1 √1 1 √1 − y k √1 − 1 √y O = = − − = = + ((cid:15)4), (395) x + y x 1 + (y/x) x 1 ( y/(2x)) x 2x x 2 x3 k=0 O since y/x = ((cid:15)2) by assumption. By functional calculus, we can make the substitutions (x , y ) for 1 1 the the operator sine function and the substitutions (x , y ) for the denominator involving the square root 2 2 of the integral under consideration, (cid:82) (cid:112) (cid:112) − − − − t τ · c2 (cid:82)1 dζ W x = (t τ)(cid:112) c2∆ and y = −0 , (396) 1 g0 1 2 (cid:112) c2∆g0 − − √y c2 1 dζ W x2 = c2∆g0 and y2 = x223 = −c02∆g03 . (397) 111 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

text: us

text: now

text: investigate

text: the

text: integral

text: in

text: the

text: integral

text: equation

text: more

text: closely.

text: By

text: deﬁnition

text: of

text: and

text: O

text: using

text: the

text: result

text: W

text: we

text: have

text: W

text: dζ

text: τ)ζ)

text: dζ

text: W

text: dζ

text: dζ

text: W(ζτ)

text: O

text: −0

text: O

text: where

text: we

text: only

text: work

text: up

text: to

text: order

text: in

text: the

text: familiar

text: Taylor

text: expansion

text: of

text: x

text: Functional

text: calculus

text: assures

text: that

text: in

text: the

text: -norm,

text: we

text: can

text: actually

text: perform

text: this

text: expansion.

text: Next,

text: we

text: need

text: two

text: further

text: identities.

text: The

text: addition

text: theorem

text: of

text: the

text: sine

text: function

text: and

text: the

text: geometric

text: series.

text: For

text: the

text: operator

text: sine

text: function

text: in

text: the

text: integral

text: equation

text: for

text: p,

text: we

text: use

text: the

text: following

text: addition

text: theorem

text: and

text: the

text: power

text: series

text: R

text: O

text: representation

text: of

text: the

text: sine

text: and

text: cosine

text: function

text: excluding

text: orders

text: from

text: on.

text: Let

text: x,

text: y

text: y/x

text: and

text: use

text: the

text: addition

text: theorem,

text: sin(x

text: y)

text: sin

text: x

text: cos

text: y

text: sin

text: y

text: cos

text: x,

text: to

text: obtain

text: O

text: sin(x

text: y)

text: sin

text: x

text: y

text: cos

text: x

text: Next,

text: we

text: use

text: the

text: geometric

text: series

text: for

text: x,

text: y

text: as

text: in

text: the

text: above

text: identity,

text: √1

text: √1

text: √1

text: y

text: k

text: √1

text: √y

text: O

text: x

text: y

text: x

text: (y/x)

text: x

text: x

text: x

text: O

text: since

text: y/x

text: by

text: assumption.

text: By

text: functional

text: calculus,

text: we

text: can

text: make

text: the

text: substitutions

text: (x

text: y

text: for

text: the

text: the

text: operator

text: sine

text: function

text: and

text: the

text: substitutions

text: (x

text: y

text: for

text: the

text: denominator

text: involving

text: the

text: square

text: root

text: of

text: the

text: integral

text: under

text: consideration,

text: t

text: τ

text: dζ

text: W

text: x

text: (t

text: and

text: y

text: −0

text: √y

text: dζ

text: W

text: and

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text: ##LTLine##

text: ##LTLine##


text: (cid:76) ∈ ⊕ conclude that A End((L2 (Ω )2) N (L2 (Γ ))2) is essentially self-adjoint since the set of analytic N,0 g0 0 j=1 0,g0 j vectors for A contains a subset which i(cid:77)s dense, (cid:77) N,0 ⊃ ⊕ N → ⊕ N V ω(A ) (H2,2(Ω ))2 (H2,4 (Γ ))2 d(cid:44)ense (L2 (Ω )2) (L2 (Γ ))2. N,0 g0 0 0,g0 k g0 0 0,g0 j i=1 j=1 C Thus, A generates a 0 semi-group. The perturbation lemma relating Sˆ to Sˆ now gives that, by N,0 0 ˆ boundedness of W, also S is an evolution family. The application of the Banach ﬁxed-point theorem to the three equations, the acoustic wave equation, the boundary vibrations equation and the equation for the C C∞ ∝ full S in terms of u(k) and S yields also the 0-property if p is . Since in our model p exp(iωt), 0 ex ex the smoothness of p in spatial and temporal arguments is clear. We emphasize that it is crucial that ex the source term is suitably well-behaved, ideally smooth but H2,2;2,4(∂Ω ) regularity suﬃces. Otherwise, g0 0 the Magnus series might not converge (in the norm sense) due to the radius of convergence of Bernoulli numbers being only < π, c.f. [2]! 6 Associated Piston Bundles Poincar´e’s inequality and piston bundles: In [98, 99], the piston approximation has been intro- duced arguing that for the ICE model, high vibration frequencies of the acoustic are physically negligible because they are outside the audible frequency range of the geckos. A physical back-of-the-envelope ar- gument suggested that indeed only plane-wave modes for p inside Ω are dominant. In [42], the spinning t mode expansion has been introduced as generalization of the modal cut-oﬀ criterion for evanescent modes [45, 44, 43, 46]. In the generalized setup, the spinning mode series expansion can be performed as well, but it is of little use when no empirical data for an investigation of the individual contributions are available. From a more formal viewpoint, the theory of partial diﬀerential equations [49, 51, 104] features a result due to H. Poincar´e which can be interpreted as a piston approximation in a more general context - the Poincar´e inequality. The versions that we will be interested in give an upper bound on the L2-deviation ∈ M ∈ of u H2,2(∂ ), interpreted u = u H2,2(∂Ω ) as an (almost) smooth 1-parameter family of functions 0 t 0 (cid:104) (cid:105) 0 deﬁned on ∂Ω , from its geometrical mean u deﬁned(cid:90)by 0 ∂Ω0 (cid:104) (cid:105) (cid:104) (cid:105) ≡ 1 u = u (t) dVol − (∂Ω ) u(t, y). (404) ∂Ω0 ∂Ω0 Vol − (∂Ω ) n 1 0 n 1 0 ∂Ω0 As shorthand notation, we will make use of the abbreviations dVol − (∂Ω ) = d(∂Ω ) and Vol − (∂Ω ) = | | n 1 0 0 n 1 0 ∂Ω in the following derivation. By the partial diﬀerential equation that u satisﬁes, we need to have ∈0 M (cid:40) { } u H2,2(∂ ) H1,2(Γ ). By our decomposition of the u in the ﬁnite family u ≤ ≤ which is non- 0 0 i i 1 i N constant regarded as a t-parameterized family of functions on Γ , we have, i 114 ##LTLine##

text: conclude

text: that

text: A

text: (Ω

text: N

text: (Γ

text: is

text: essentially

text: self-adjoint

text: since

text: the

text: set

text: of

text: analytic

text: j

text: vectors

text: for

text: A

text: contains

text: a

text: subset

text: which

text: dense,

text: N

text: N

text: V

text: ω(A

text: (Γ

text: (Ω

text: (Γ

text: k

text: j

text: C

text: Thus,

text: A

text: generates

text: a

text: semi-group.

text: The

text: perturbation

text: lemma

text: relating

text: Sˆ

text: to

text: Sˆ

text: now

text: gives

text: that,

text: by

text: ˆ

text: boundedness

text: of

text: W,

text: also

text: S

text: is

text: an

text: evolution

text: family.

text: The

text: application

text: of

text: the

text: Banach

text: ﬁxed-point

text: theorem

text: to

text: the

text: three

text: equations,

text: the

text: acoustic

text: wave

text: equation,

text: the

text: boundary

text: vibrations

text: equation

text: and

text: the

text: equation

text: for

text: the

text: C

text: C∞

text: full

text: S

text: in

text: terms

text: of

text: u(k)

text: and

text: S

text: yields

text: also

text: the

text: if

text: p

text: is

text: Since

text: in

text: our

text: model

text: p

text: exp(iωt),

text: ex

text: ex

text: the

text: smoothness

text: of

text: p

text: in

text: spatial

text: and

text: temporal

text: arguments

text: is

text: clear.

text: We

text: emphasize

text: that

text: it

text: is

text: crucial

text: that

text: ex

text: the

text: source

text: term

text: is

text: suitably

text: well-behaved,

text: ideally

text: smooth

text: but

text: regularity

text: suﬃces.

text: Otherwise,

text: the

text: Magnus

text: series

text: might

text: not

text: converge

text: (in

text: the

text: norm

text: sense)

text: due

text: to

text: the

text: radius

text: of

text: convergence

text: of

text: Bernoulli

text: numbers

text: being

text: only

text: π,

text: c.f.

text: Associated

text: Piston

text: Bundles

text: Poincar´e’s

text: inequality

text: and

text: piston

text: bundles:

text: In

text: the

text: piston

text: approximation

text: has

text: been

text: intro-

text: duced

text: arguing

text: that

text: for

text: the

text: ICE

text: model,

text: high

text: vibration

text: frequencies

text: of

text: the

text: acoustic

text: are

text: physically

text: negligible

text: because

text: they

text: are

text: outside

text: the

text: audible

text: frequency

text: range

text: of

text: the

text: geckos.

text: A

text: physical

text: back-of-the-envelope

text: ar-

text: gument

text: suggested

text: that

text: indeed

text: only

text: plane-wave

text: modes

text: for

text: p

text: inside

text: Ω

text: are

text: dominant.

text: In

text: the

text: spinning

text: t

text: mode

text: expansion

text: has

text: been

text: introduced

text: as

text: generalization

text: of

text: the

text: modal

text: cut-oﬀ

text: criterion

text: for

text: evanescent

text: modes

text: In

text: the

text: generalized

text: setup,

text: the

text: spinning

text: mode

text: series

text: expansion

text: can

text: be

text: performed

text: as

text: well,

text: but

text: it

text: is

text: of

text: little

text: use

text: when

text: no

text: empirical

text: data

text: for

text: an

text: investigation

text: of

text: the

text: individual

text: contributions

text: are

text: available.

text: From

text: a

text: more

text: formal

text: viewpoint,

text: the

text: theory

text: of

text: partial

text: diﬀerential

text: equations

text: features

text: a

text: result

text: due

text: to

text: H.

text: Poincar´e

text: which

text: can

text: be

text: interpreted

text: as

text: a

text: piston

text: approximation

text: in

text: a

text: more

text: general

text: context

text: the

text: Poincar´e

text: inequality.

text: The

text: versions

text: that

text: we

text: will

text: be

text: interested

text: in

text: give

text: an

text: upper

text: bound

text: on

text: the

text: M

text: of

text: u

text: interpreted

text: u

text: u

text: as

text: an

text: (almost)

text: smooth

text: family

text: of

text: functions

text: t

text: deﬁned

text: on

text: ∂Ω

text: from

text: its

text: geometrical

text: mean

text: u

text: ∂Ω0

text: u

text: u

text: (t)

text: dVol

text: (∂Ω

text: u(t,

text: y).

text: ∂Ω0

text: ∂Ω0

text: Vol

text: (∂Ω

text: n

text: n

text: ∂Ω0

text: As

text: shorthand

text: notation,

text: we

text: will

text: make

text: use

text: of

text: the

text: abbreviations

text: dVol

text: (∂Ω

text: d(∂Ω

text: and

text: Vol

text: (∂Ω

text: n

text: n

text: ∂Ω

text: in

text: the

text: following

text: derivation.

text: By

text: the

text: partial

text: diﬀerential

text: equation

text: that

text: u

text: satisﬁes,

text: we

text: need

text: to

text: have

text: ∈0

text: M

text: u

text: By

text: our

text: decomposition

text: of

text: the

text: u

text: in

text: the

text: ﬁnite

text: family

text: u

text: which

text: is

text: non-

text: i

text: i

text: i

text: N

text: constant

text: regarded

text: as

text: a

text: t-parameterized

text: family

text: of

text: functions

text: on

text: Γ

text: we

text: have,

text: i

text: ##LTLine##


text: (cid:90) S [S] = dVol − (U)(∂ S + 2D HS) = min.! (420) mcf n 1 t mcf U¯ By non-negativity of S and positivity as well as monotony of the integral, we receive the following diﬀerential equation, − ∂ S = 2D HS. (421) t mcf Deﬁning j = 2n , this can be recast in the form of a conservation equation, graph(S) − ∇ ∂ S = D j. (422) t The integral formulation states that the velocity that S changes in time is proportional to the net ﬂux of the curvature stream density j through the boundary ∂graph(S). The proportionality constant D is mcf called the mean curvature diﬀusion constant in the following. For weakly curved hypersurfaces graph(S), (cid:107)∇ (cid:107) (cid:28) i.e., hypersurfaces with S 1, the mean curvature ﬂow equation reduced to a diﬀusion equation for 2 S, − ∂ S = D∆S. (423) t The equation has been extensively studied by mathematicians and physicists alike such that we restrict ourselves to discussing two special cases. The ﬁrst special case is when ∂ S = 0 such that the surface does t not evolve in time. Then the mean curvature ﬂow model tells us that 2DHS = 0, i.e., H[S] = 0. This means that S is a so-called minimal surface and S = S(x, y). On the other hand, if 2HS = 2γ¯S for a (cid:54) ∅ (positive because ∂graphS = by assumption) γ¯ > 0, we have the ordinary diﬀerential equation − ∂ S = 2Dγ¯S, (424) t − Then, the mean curvature ﬂow states that S = S(t, x, y) = S (x, y) exp( 2Dγ¯t) where we have chosen 0 the dummy initial condition S(0, x, y) = S (x, y). In other words, a surface with constant mean curvature → ∞ 0 γ¯ reduces in the limit t to a ﬂat surface with mean curvature 0. Mean curvature vibrations: The boundary vibrations satisfy a damped wave equation such that → → ∞ for constant mean curvature H[u] = γ¯ limit, we would have u 0 as t as well. This agrees with ∞ the -periodicity from above. Furthermore, it is known experimentally [96, 98, 99, 18, 20, 105, 19] that curvature eﬀects in bio-membranes are typically negligibly small. Physically this is due to large curvature eﬀects costing the membranes a large amount of energy, contradicting the principle of least equation. As a matter of fact, [98, 99, 101, 42] only considered a special case of the model for our boundary vibra- O tions, namely a model where H[u] = ((cid:15)). As a consequence, we can ask what happened if we deﬁned 118

text: S

text: [S]

text: dVol

text: (U)(∂

text: S

text: HS)

text: min.!

text: mcf

text: n

text: t

text: mcf

text: U¯

text: By

text: non-negativity

text: of

text: S

text: and

text: positivity

text: as

text: well

text: as

text: monotony

text: of

text: the

text: integral,

text: we

text: receive

text: the

text: following

text: diﬀerential

text: equation,

text: S

text: HS.

text: t

text: mcf

text: Deﬁning

text: j

text: this

text: can

text: be

text: recast

text: in

text: the

text: form

text: of

text: a

text: conservation

text: equation,

text: graph(S)

text: S

text: D

text: j.

text: t

text: The

text: integral

text: formulation

text: states

text: that

text: the

text: velocity

text: that

text: S

text: changes

text: in

text: time

text: is

text: proportional

text: to

text: the

text: net

text: ﬂux

text: of

text: the

text: curvature

text: stream

text: density

text: j

text: through

text: the

text: boundary

text: ∂graph(S).

text: The

text: proportionality

text: constant

text: D

text: is

text: mcf

text: called

text: the

text: mean

text: curvature

text: diﬀusion

text: constant

text: in

text: the

text: following.

text: For

text: weakly

text: curved

text: hypersurfaces

text: graph(S),

text: i.e.,

text: hypersurfaces

text: with

text: S

text: the

text: mean

text: curvature

text: ﬂow

text: equation

text: reduced

text: to

text: a

text: diﬀusion

text: equation

text: for

text: S,

text: S

text: D∆S.

text: t

text: The

text: equation

text: has

text: been

text: extensively

text: studied

text: by

text: mathematicians

text: and

text: physicists

text: alike

text: such

text: that

text: we

text: restrict

text: ourselves

text: to

text: discussing

text: two

text: special

text: cases.

text: The

text: ﬁrst

text: special

text: case

text: is

text: when

text: S

text: such

text: that

text: the

text: surface

text: does

text: t

text: not

text: evolve

text: in

text: time.

text: Then

text: the

text: mean

text: curvature

text: ﬂow

text: model

text: tells

text: us

text: that

text: i.e.,

text: H[S]

text: This

text: means

text: that

text: S

text: is

text: a

text: so-called

text: minimal

text: surface

text: and

text: S

text: S(x,

text: y).

text: On

text: the

text: other

text: hand,

text: if

text: 2γ¯S

text: for

text: a

text: (positive

text: because

text: ∂graphS

text: by

text: assumption)

text: γ¯

text: we

text: have

text: the

text: ordinary

text: diﬀerential

text: equation

text: S

text: t

text: Then,

text: the

text: mean

text: curvature

text: ﬂow

text: states

text: that

text: S

text: S(t,

text: x,

text: y)

text: S

text: (x,

text: y)

text: exp(

text: where

text: we

text: have

text: chosen

text: the

text: dummy

text: initial

text: condition

text: x,

text: y)

text: S

text: (x,

text: y).

text: In

text: other

text: words,

text: a

text: surface

text: with

text: constant

text: mean

text: curvature

text: γ¯

text: reduces

text: in

text: the

text: limit

text: t

text: to

text: a

text: ﬂat

text: surface

text: with

text: mean

text: curvature

text: Mean

text: curvature

text: vibrations:

text: The

text: boundary

text: vibrations

text: satisfy

text: a

text: damped

text: wave

text: equation

text: such

text: that

text: for

text: constant

text: mean

text: curvature

text: H[u]

text: γ¯

text: limit,

text: we

text: would

text: have

text: u

text: as

text: t

text: as

text: well.

text: This

text: agrees

text: with

text: the

text: -periodicity

text: from

text: above.

text: Furthermore,

text: it

text: is

text: known

text: experimentally

text: that

text: curvature

text: eﬀects

text: in

text: bio-membranes

text: are

text: typically

text: negligibly

text: small.

text: Physically

text: this

text: is

text: due

text: to

text: large

text: curvature

text: eﬀects

text: costing

text: the

text: membranes

text: a

text: large

text: amount

text: of

text: energy,

text: contradicting

text: the

text: principle

text: of

text: least

text: equation.

text: As

text: a

text: matter

text: of

text: fact,

text: only

text: considered

text: a

text: special

text: case

text: of

text: the

text: model

text: for

text: our

text: boundary

text: vibra-

text: O

text: tions,

text: namely

text: a

text: model

text: where

text: H[u]

text: As

text: a

text: consequence,

text: we

text: can

text: ask

text: what

text: happened

text: if

text: we

text: deﬁned


text: with the separation Laplacian ∆∂,N on ∂Ω , including the corresponding boundary conditions inherited g0 0 from ∆ on Ω . Since there are only ﬁnitely many Γ ’s, we can use a sequence of molliﬁcation functions g0 0 i C∞ deﬁned on ∂Ω and denoted by (m (t, y)) ∈N ≤ ≤ with regularity properties such that 0 k,i n ,1 i N ∈ R × lim m (t, y) = δ((t, y) + Γ ) (457) →∞ n,i 0 i n M ≤ ≤ and mollify u on ∂ by con(cid:34)volution over Γ for 1(cid:35) i N, 0 i (cid:88) (cid:88) N N ∈ R × u = lim (m (cid:63) u )(t, y) = u δ(t, y + Γ ) (458) n→∞ n,i Γi i i 0 i i=1 i=1 The boundary vibrations u are then deﬁned globally on ∂Ω and suﬃciently regular. We can then 0 carry out the procedure in the derivation above with the new u because Lichernowicz’ theorem ensures the existence of a complete and orthonormal set of eigenfunctions for ∆∂,N on ∂Ω . (cid:104)M(cid:105) g0 (cid:104) (cid:105) 0 (ii) The second issue is that still needs to be obtained in terms of u . By deﬁnition, we have ∂Ω0 (cid:104) (cid:105) ∂Ω0 to replace the boundary vibrations u with their geometrical mean, u . More haptically, the procedure → ⊂ R ∂Ω0 works as follows: Take the diﬀeomorphisms φ → : ∂Ω ∂Ω n and use the componentwisely in terms { } R 0 t 0 t of the canonical basis eˆ ≤ ≤ on n deﬁned geome(cid:90)trical mean, µ 1 µ n (cid:104) (cid:105) 1 φ → = dVol − (∂Ω ) φ → . (459) 0 t ∂Ω0 Vol − (∂Ω ) n 1 0 0 t n 1 0 ∂Ω0 (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) This maps ∂Ω to ∂ Ω = ∂Ω . Then, we use the Gaussian map to extrapolate from ∂Ω by 0 ∈ t ∂Ω0 t ∂Ω0(cid:104) (cid:105) t a radial coordinate s [0, 1] to the whole Ω , t (cid:104) (cid:105) × (cid:104) (cid:105) → (cid:104) (cid:105) → − (cid:104) (cid:105) σ : [0, 1] ∂Ω Ω , (s, y) y s y, n(cid:104) (cid:105) R , (460) t ∂Ω0 t ∂Ω0 t ∂Ω0 ∂Ωt ∂Ω0 g0, n (cid:104) (cid:105) (cid:104) (cid:105) and deﬁne Ω = Im( σ ). Last, we deﬁ(cid:91)ne the piston bundle t ∂Ω0 t ∂Ω0 (cid:104)M(cid:105) { } × (cid:104) (cid:105) = t Ω . (461) ∂Ω0 t ∂Ω0 ≥ t 0 (cid:104)M(cid:105) M Obviously, the piston bundle associated to a perturbation bundle is again a perturbation ∂Ω0 M bundle with the same perturbation strength (cid:15) as the original perturbation bundle to the reference M bundle . The perturbation theory for general perturbation bundle then recovers for the geometrical (cid:104) (cid:105) 0 (cid:104) (cid:105) mean u the perturbation equations with u replaced by u , i.e., the equations that we already ∂Ω0 ∂Ω0 obtained in the preceding derivation. Piston bundle theorems: In total our considerations have shown the validity of the piston bundle M theorems which give criteria for when one may approximate the perturbation bundle via the associated piston bundle. We give the geometrical and analytic formulation of the theorems. 125 ##LTLine##

text: with

text: the

text: separation

text: Laplacian

text: ∆∂,N

text: on

text: ∂Ω

text: including

text: the

text: corresponding

text: boundary

text: conditions

text: inherited

text: from

text: on

text: Ω

text: Since

text: there

text: are

text: only

text: ﬁnitely

text: many

text: Γ

text: ’s,

text: we

text: can

text: use

text: a

text: sequence

text: of

text: molliﬁcation

text: functions

text: i

text: C∞

text: deﬁned

text: on

text: ∂Ω

text: and

text: denoted

text: by

text: (m

text: (t,

text: y))

text: ∈N

text: with

text: regularity

text: properties

text: such

text: that

text: k,i

text: n

text: i

text: N

text: R

text: lim

text: m

text: (t,

text: y)

text: δ((t,

text: y)

text: Γ

text: n,i

text: i

text: n

text: M

text: and

text: mollify

text: u

text: on

text: by

text: over

text: Γ

text: for

text: i

text: N,

text: i

text: N

text: N

text: R

text: u

text: lim

text: (m

text: u

text: )(t,

text: y)

text: u

text: δ(t,

text: y

text: Γ

text: n→∞

text: n,i

text: Γi

text: i

text: i

text: i

text: The

text: boundary

text: vibrations

text: u

text: are

text: then

text: deﬁned

text: globally

text: on

text: ∂Ω

text: and

text: suﬃciently

text: regular.

text: We

text: can

text: then

text: carry

text: out

text: the

text: procedure

text: in

text: the

text: derivation

text: above

text: with

text: the

text: new

text: u

text: because

text: Lichernowicz’

text: theorem

text: ensures

text: the

text: existence

text: of

text: a

text: complete

text: and

text: orthonormal

text: set

text: of

text: eigenfunctions

text: for

text: ∆∂,N

text: on

text: ∂Ω

text: (ii)

text: The

text: second

text: issue

text: is

text: that

text: still

text: needs

text: to

text: be

text: obtained

text: in

text: terms

text: of

text: u

text: By

text: deﬁnition,

text: we

text: have

text: ∂Ω0

text: ∂Ω0

text: to

text: replace

text: the

text: boundary

text: vibrations

text: u

text: with

text: their

text: geometrical

text: mean,

text: u

text: More

text: haptically,

text: the

text: procedure

text: R

text: ∂Ω0

text: works

text: as

text: follows:

text: Take

text: the

text: diﬀeomorphisms

text: φ

text: ∂Ω

text: ∂Ω

text: n

text: and

text: use

text: the

text: componentwisely

text: in

text: terms

text: R

text: t

text: t

text: of

text: the

text: canonical

text: basis

text: eˆ

text: on

text: n

text: deﬁned

text: mean,

text: µ

text: µ

text: n

text: φ

text: dVol

text: (∂Ω

text: φ

text: t

text: ∂Ω0

text: Vol

text: (∂Ω

text: n

text: t

text: n

text: ∂Ω0

text: This

text: maps

text: ∂Ω

text: to

text: Ω

text: ∂Ω

text: Then,

text: we

text: use

text: the

text: Gaussian

text: map

text: to

text: extrapolate

text: from

text: ∂Ω

text: by

text: t

text: ∂Ω0

text: t

text: t

text: a

text: radial

text: coordinate

text: s

text: to

text: the

text: whole

text: Ω

text: t

text: σ

text: ∂Ω

text: Ω

text: (s,

text: y)

text: y

text: s

text: y,

text: R

text: t

text: ∂Ω0

text: t

text: ∂Ω0

text: t

text: ∂Ω0

text: ∂Ωt

text: ∂Ω0

text: n

text: and

text: deﬁne

text: Ω

text: Im(

text: σ

text: Last,

text: we

text: the

text: piston

text: bundle

text: t

text: ∂Ω0

text: t

text: ∂Ω0

text: t

text: Ω

text: ∂Ω0

text: t

text: ∂Ω0

text: t

text: M

text: Obviously,

text: the

text: piston

text: bundle

text: associated

text: to

text: a

text: perturbation

text: bundle

text: is

text: again

text: a

text: perturbation

text: ∂Ω0

text: M

text: bundle

text: with

text: the

text: same

text: perturbation

text: strength

text: as

text: the

text: original

text: perturbation

text: bundle

text: to

text: the

text: reference

text: M

text: bundle

text: The

text: perturbation

text: theory

text: for

text: general

text: perturbation

text: bundle

text: then

text: recovers

text: for

text: the

text: geometrical

text: mean

text: u

text: the

text: perturbation

text: equations

text: with

text: u

text: replaced

text: by

text: u

text: i.e.,

text: the

text: equations

text: that

text: we

text: already

text: ∂Ω0

text: ∂Ω0

text: obtained

text: in

text: the

text: preceding

text: derivation.

text: Piston

text: bundle

text: theorems:

text: In

text: total

text: our

text: considerations

text: have

text: shown

text: the

text: validity

text: of

text: the

text: piston

text: bundle

text: M

text: theorems

text: which

text: give

text: criteria

text: for

text: when

text: one

text: may

text: approximate

text: the

text: perturbation

text: bundle

text: via

text: the

text: associated

text: piston

text: bundle.

text: We

text: give

text: the

text: geometrical

text: and

text: analytic

text: formulation

text: of

text: the

text: theorems.

text: ##LTLine##