THE SPECTRUM OF A PARAMETRIZED PARTIAL DIFFERENTIAL OPERATOR OCCURRING IN
HYDRODYNAMICS
R. DENK, M. M ¨OLLER, AND C. TRETTER
Abstract. A partial differential operator associated with natural oscillations of an incompressible fluid in the neighbourhood of an elliptical flow is con- sidered. The differentiation is only taken with respect to the angular vari- able, and thus the operator becomes a family of ordinary differential operators parametrized by the radial variable. It is shown that the spectra of these or- dinary differential operators completely determine the spectrum of the given operator which turns out to have a kind of skeleton structure.
1. Introduction
The analysis of natural oscillations of an incompressible fluid in the neighbour- hood of an elliptical flow leads to a spectral problem for an operator A of the form (see Lifschitz [3])
(1.1) A=
⎛
⎜⎝
∂
∂ψ 0
0 ∂
∂ψ
⎞
⎟⎠+ 2N
inL2((0,∞)×(0,2π), ρ)2 with (see [3, (20)])
N(ρ, ψ) =
⎛
⎜⎜
⎝
0 − 1−δ2
ρ2(1−δcos(2ψ)) + 1−δ2
1 δρ2sin(2ψ)
ρ2(1−δcos(2ψ)) + 1−δ2
⎞
⎟⎟
⎠, ρ∈(0,∞), ψ∈(0,2π),
where (ρ, ψ) are polar coordinates. The domain of A is given by D(A) =
f ∈L2((0,∞)×(0,2π), ρ)2 :f(ρ, ·)∈H1(0,2π)2 for almost all ρ, f(·,0) =f(·,2π)
.
The ellipticity parameterδ∈[0,1) characterizes the geometry of the flow; for the special case δ= 0 the flow becomes circular.
The operator A can also be considered in the space L2((0,∞), ρ, L2(0,2π)2) consisting of allL2–functions on (0,∞) with respect to the weightρhaving values inL2(0,2π)2. In this space the domain ofA is given by
D(A) = {f ∈L2((0,∞), ρ, H1(0,2π)2) :f(ρ)(0) =f(ρ)(2π) for almost allρ}.
AMS Subj. Class.: Primary 47F05; secondary 47E05, 76B99 1
Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5066/
URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-50661
In this setting, the partial differential operatorAturns out to be related to the family of ordinary differential operators (compare [3, pp. 1627–1628])
(1.2) A(ρ) =
⎛
⎜⎝ d
dψ 0
0 d
dψ
⎞
⎟⎠+ 2N(ρ, ·), ρ ∈(0,∞),
acting in the space L2(0,2π)2 with domain
D:=D(A(ρ)) ={f(ρ)∈H1(0,2π)2 :f(ρ)(0) =f(ρ)(2π)} not depending on ρ. This relation is given by
(1.3) (Af)(ρ) =A(ρ)f(ρ), ρ∈(0,∞).
Hence A can be considered as a generalized multiplication operator. However, since A is not self-adjoint, the spectrum of A cannot be determined using the results of Reed and Simon [5, Section XIII.16].
In the next section we are going to show that the spectrum of A is the union of the spectra of A(ρ) taken over ρ ∈ [0,∞]. In the following Section 3 we calculate the spectrum of A. For fixed ρ the spectrum of A(ρ) consists only of eigenvalues of finite algebraic multiplicity. We prove that the spectrum of A is purely essential and has a certain skeleton structure, i.e., it lies on the imaginary axis and on infinitely many horizontal segments crossing the imaginary axis. In Section 4 numerical calculations show, in accordance with results in [3], that the spectrum of A is connected (see Figure 2). Finally, in Section 5, we consider a slight modification of the physical problem, changing the sign of one of the entries in N. In this case we show that the spectrum of the corresponding operator has a similar structure, but it has infinitely many components.
2. The relation between the spectra of A and A(ρ)
We consider the operators A and A(ρ) defined by (1.1) and (1.2). Using stan- dard techniques it is easy to see that the operatorsAandA(ρ) are closed. Indeed, it is well known that the first operator in (1.1) and in (1.2), respectively, is closed, and N andN(ρ, ·), respectively, are bounded multiplication operators on the re- spective Lebesgue spaces. Observe that since N(ρ, ψ) has a continuous extension to ρ= 0 andρ=∞,A(ρ) is also defined and closed forρ= 0 and ρ=∞.
In the following the spectrum of an operator T is denoted by σ(T) and σp(T) denotes its point spectrum, i. e., the set of its eigenvalues.
Theorem 2.1. The spectra of the operator A and the operator family A are related as follows:
σ(A) =
ρ∈[0,∞]
σ(A(ρ)).
Proof. For convenience we setH:=L2((0,∞), ρ, L2(0,2π)2) andH :=L2(0,2π)2. Let λ ∈
ρ∈[0,∞]σ(A(ρ)). Hence there is a ρ0 ∈ [0,∞] such that λ ∈ σ(A(ρ0)).
Then the range of A(ρ0)− λ is closed and a proper subspace of H, or there is a sequence (fn)∞1 ⊂ D, fnH = 1, with (A(ρ0)− λ)fnH < n1. We first consider the second case. SinceN is continuous, we can choose a sequence of open neighbourhoodsUnofρ0 in [0,∞) so thatA(ρ)−A(ρ0)< 1n for allρ∈Un. Let En ⊂Un∩(0,∞) be a compact interval with βn :=
Enρ dρ >0, αn := (βn)−12, and
fn(ρ) :=αnχEn(ρ)fn, ρ∈(0,∞).
Obviously, fn ∈H and fnH= 1. Furthermore, by Fubini’s theorem, (A−λ)fn2H=α2n
En
(A(ρ)−λ)fn2
Hρ dρ
≤α2nβn sup
ρ∈En
A(ρ)−A(ρ0)+(A(ρ0)−λ)fnH
2
< 4 n2, which provesλ∈σ(A).
Now assume that A(ρ0)−λ has a closed range which is a proper subspace of H. Then there is a g ∈ H, gH = 1, such that ((A(ρ0)−λ)f, g)H = 0 for all f ∈D. Suppose that λ∈σ(A). Choose En and αn as above and set
gn(ρ) :=αnχEn(ρ)g, ρ∈(0,∞).
Thengn ∈H, gnH= 1, but for fn:= (A−λ)−1gn we have 1 = ((A−λ)fn,gn)H
= ∞
0
((A−λ)fn(ρ),gn(ρ))Hρ dρ
=αn
En
((A(ρ)−λ)fn(ρ), g)Hρ dρ
=αn
En
((A(ρ)−A(ρ0))fn(ρ), g)Hρ dρ
≤αn
En
A(ρ)−A(ρ0) fn(ρ)H ρ dρ
< αn 1 n
En
fn(ρ)Hρ dρ
≤αn 1
nβn12 fnH
= 1
nfnH,
and hence fnH = (A−λ)−1gnH → ∞ as n → ∞. This is a contradiction because (A−λ)−1 is bounded by assumption.
Conversely, letλ∈σ(A). Ifλis an eigenvalue ofA, then there exists a non-zero f ∈ D(A) such that Af = λf. Hence, by (1.3), A(ρ)f(ρ) = λf(ρ) for almost all ρ ∈(0,∞). Since f = 0,f(ρ)= 0 for all ρ in some set of positive measure. Thus there is a ρ0 ∈(0,∞) such thatf(ρ0)= 0 andA(ρ0)f(ρ0) =λf(ρ0). This proves
σp(A)⊂
ρ∈(0,∞)
σ(A(ρ)).
Now assume that λ ∈σ(A) is not an eigenvalue of A. Then A−λ is injective but not surjective, and we can find an element g∈H such that (A−λ)f=g for allf∈D(A). Assume λ∈
ρ∈[0,∞]σ(A(ρ)). For (almost all)ρ∈(0,∞) we define h(ρ) := (A(ρ)−λ)−1g(ρ).
From the continuity of the inversion, see [2, Theorem IV.1.16], it follows that the mapping ρ → (A(ρ)−λ)−1 from [0,∞] into the bounded operators on H is continuous. Hence h is measurable and
∞
0 h(ρ)2Hρ dρ 12
≤ sup
ρ∈[0,∞](A(ρ)−λ)−1 gH<∞.
Thush∈H, andh(ρ)∈Dclearly impliesh∈D(A). This yields the contradiction
(A−λ)h=g.
3. The calculation of σ(A)
Theorem 3.1. The operator Ahas no eigenvalues and there are constants a≥0 and b ≥0 such that
σ(A) = iR∪
λ∈C:|(λ)| ≤a, (λ)∈Z
∪
λ∈C:|(λ)| ≤b, (λ)∈1 2+Z or constants c≥0 and d < 12 such that
σ(A) =
k∈Z
i [k−c, k+c]∪ {λ∈C:|(λ)| ≤d, (λ)∈Z}.
Proof. To find the spectrum ofA, by Theorem 2.1 it is sufficient to knowσ(A(ρ)) for ρ ∈[0,∞]. We will show that the eigenvalue problem for A(ρ) is equivalent to an eigenvalue problem for a second order differential equation.
It is well known from Floquet theory, see [1, Chapter 1] or [6, Chapter II], that the spectrum of A(ρ) is discrete. Let λ ∈σ(A(ρ)) and let y be an eigenfunction of A(ρ) at λ. Put u(ψ) := e−λψy(ψ). Then u(2π) = e−2λπy(2π) = e−2λπy(0) = e−2λπu(0) and
u(ψ) + 2N(ρ, ψ)u(ψ) =−λe−λψy(ψ) + e−λψy(ψ) + 2N(ρ, ψ)e−λψy(ψ)
=−λe−λψy(ψ) +λe−λψy(ψ)
= 0.
Since these steps can be reversed, λ∈σ(A(ρ)) if and only if the problem (3.1) y+ 2N(ρ,·)y= 0, y(2π) = e−2λπy(0)
has a nontrivial solution. Now letρ∈[0,∞). With w(ψ) := ρ2(1−δcos(2ψ)) + 1−δ2
2(1−δ2) we have
2N =
⎛
⎜⎝
0 −1 w
2 w
w
⎞
⎟⎠.
Then, withy = (y1, y2), the equationy + 2N y = 0 can be written in the form y1 − 1
wy2 = 0, y2 + 2y1 +w
wy2 = 0.
This is equivalent to
w2y1 −wy2 = 0, (wy2)+ 2wy1 = 0.
Differentiating the first equation and inserting into the second one gives
(3.2) (w2y1)+ 2wy1 = 0.
Sincew is 2π-periodic, the boundary conditions for y1 are (3.3) y1(π) = e−2λπy1(0), y1(π) = e−2λπy1(0).
If now y1 is a (nonzero) solution of this problem, then y2 =wy1 gives a nonzero solution y = (y1, y2) of (3.1). If, conversely, (3.1) has a nonzero solution, then y1 = 0 sincey1 = 0 would implyy2 = 0. This shows thatλ∈σ(A(ρ)) if and only if (3.2), (3.3) has a nontrivial solution.
Let y11, y12 be the fundamental system of (3.2) with y11(0) = 1, y11 (0) = 0, y12(0) = 0, y12 (0) = 1. It is a well-known fact that the Wronski determinant w2(y11y12 −y11 y12) is constant, and in particular
(3.4) det
y11(2π) y12(2π) y11 (2π) y12 (2π)
= 1.
The existence of a nonzero solution of the boundary value problem (3.2), (3.3) means that
det
y11(2π)−e−2λπ y12(2π) y11(2π) y12 (2π)−e−2λπ
= 0.
This leads to
(3.5) e−4λπ−e−2λπ(y11(2π) +y12(2π)) + 1 = 0.
Observe that we can find the spectrum of A(0) explicitly since w(0, ψ) = 12. Here y11(ψ) = cos(2ψ) and y12(ψ) = 12sin(2ψ), and the solutions of (3.5) are λ= ik,k ∈Z.
The two solutions of (3.5), in terms of e−2λπ, are two positive numbers whose product is 1, two negative numbers whose product is 1, or two conjugate complex numbers with modulus 1. This means that λ lies in R+ iZ, in R+ i(1/2 +Z), or in iR. The continuous dependence of the spectrum on ρ and the fact that the spectrum for ρ = 0 lies on the imaginary axis shows that the parts of the spectrum on the parallels to the real axis must be intervals. The two different representations given in the statement of this theorem depend on whether or not there is ρ∈[0,∞] such that y11(2π) +y12(2π)≤ −2.
To show that σp(A) =∅, observe that the characteristic determinant ∆ is not unique, but one can choose ∆ such that it depends analytically on the spectral parameter λ. Since N depends analytically on ρ, ∆ can be chosen so that it depends analytically on both parameters. If λ were an eigenvalue of A, then λ∈σ(A(ρ)) forρin a set of positive Lebesgue measure, see the proof of Theorem 2.1. In that case, ∆(ρ, λ) = 0 for this particularλ and a set ofρ with limit point in (0,∞). By the identity theorem for analytic functions, this would imply that
∆(ρ, λ) = 0 for allρ ∈(0,∞). This contradicts the fact that the eigenvalues are not constant as functions of ρ, which will follow from the considerations in the
next section.
4. Numerical results for σ(A)
Below we will show by numerical calculations, in accordance with [3], that for the spectrum of A described in Theorem 3.1 the first case with b = 0 applies, i. e. ,
σ(A) = iR∪ {λ∈C:|(λ)| ≤a, (λ)∈Z}.
We want to investigate how the solutions of (3.2) depend on ρ and thus find information about the dependence of the eigenvalues of A(ρ) on ρ. First let us consider ρnear 0. For this it is easier to introduce τ =ρ2. With
w1(ψ) := 1−δcos(2ψ) 2(1−δ2)
we have w(ψ) =τ w1(ψ) +12. Lety1 be a solution (depending on τ) of (3.2) such that y1(0) and y1(0) are independent of τ. Then we can differentiate (3.2) with respect to τ and obtain (with ∂τ∂ y1 =:y1τ) that
∂
∂τw2
y1
+ (w2y1τ ) + 2 ∂
∂τw
y1 + 2wy1τ = 0.
Evaluation at τ = 0 gives 1
4y1τ +y1τ =−(w1y1) −2w1y1.
Note that y11(ψ) = cos(2ψ), y12(ψ) = 12sin(2ψ) at τ = 0 and that y1τ(0) = 0, y1τ (0) = 0 since the initial conditions do not depend on τ. An evaluation ofy11τ and y12τ (e.g., with MAPLE) shows that y11τ(π) +y12τ(π) = 0 at τ = 0. Hence we need the second derivative with respect toτ. This gives
(w2y1ττ )+ 4
w ∂
∂τw
y1τ
+ 2(w21y1)+ 4w1y1τ + 2wy1ττ = 0, and thus, at τ = 0,
1
4y1ττ +y1ττ =−2(w1y1τ )−2(w12y1) −4w1y1τ. Another calculation (e.g., with MAPLE) gives
y11ττ(2π) +y12ττ(2π) = − 8π2 (1−δ2)2
atτ = 0. Hence −2< y11(2π) +y12 (2π)<2 for sufficiently small positiveρ, and the eigenvalue equation (3.5) shows that the eigenvalues of A(ρ) for these ρ lie on the imaginary axis.
We have
N(∞, ·) =
⎛
⎝ 0 0 2 w∞
w∞
⎞
⎠,
wherew∞(ψ) = 1−δcos(2ψ). Putting W∞(ψ) :=
ψ
0
1 w∞(t)dt, it is easy to see that
Y(ψ) =
eλψ 0 2w∞(ψ)W∞(ψ)eλψ w∞(ψ)eλψ
is a fundamental matrix ofy+ 2N(∞,·)y=λy, and therefore
∆(λ) = det
e2λπ−1 0
2w∞(2π)W∞(2π)e2λπ w∞(2π)(e2λπ−1)
is a characteristic determinant of A(∞). Obviously, ∆(λ) is zero if and only if λ∈iZ. Hence, at ρ=∞, y11(2π) +y12 (2π) = 2.
In order to investigate the behaviour of the eigenvalues forρ near ∞, we now make the substitutionτ =ρ−2, so that, after multiplication by τ2, (5.1) becomes (we already know that the eigenvalues depend continuously onρatρ=∞, so any formulation of the eigenvalue problem which is continuous atτ = 0 is acceptable)
w1+τ
2 2
y1
+τ(2w1+τ)y1 = 0.
Differentiation with respect to τ leads to
w1+ τ 2
2 y1τ
+
w1+τ 2
y1
+ 2(w1+τ)y1 +τ(2w1+τ)y1τ = 0.
At τ = 0 we obtain
(w12y1τ ) =−(w1y1)−2w1y1, where
(w21y1) = 0.
Hence w21y1 is constant. From y11 (0) = 0 it therefore follows that y11 = 0, and hence y11(ψ) = 1. Fory12 we get w21y12 =w21(0), and hence
y12(ψ) = ψ
0
w12(0) w21(t) dt.
Thus
(w21y11τ ) =−2w1 and
w21(ψ)y12τ (ψ) =−w21(0)
w1(ψ)+w1(0)−2 ψ
0
w1(t)y12(t)dt.
This gives
y12τ(2π) = −2 2π
0
w1(ψ) ψ
0
1
w12(t)dt dψ=−2 2π
0
1 w21(t)
2π
t
w1(ψ)dψ dt and
w21(t)y11τ (t) =−2 t
0
w1(ψ)dψ, which leads to
y11τ(2π) =−2 2π
0
1 w21(t)
t
0
w1(ψ)dψ dt.
Therefore
y11τ(2π) +y12τ (2π) =−2 2π
0
1 w21(t)
2π
0
w1(ψ)dψ dt < 0,
and hence −2 < y11(2π) +y12(2π) < 2 for sufficiently large ρ. This shows that the eigenvalues of A(ρ) for these ρ lie on the imaginary axis.
To summarize, we have shown that the eigenvalues ofA(ρ) lie on the imaginary axis for ρ near 0 and∞.
For δ= 0 we can solve (3.2) explicitly. Here y11(2π) +y12 (2π) = 2 cos
4π ρ2+ 1
,
and the eigenvalue equation (3.5) becomes
exp(−2λπ)−exp
4πi
ρ2+ 1 exp(−2λπ)−exp
− 4πi ρ2+ 1
= 0,
and thereforeσ(A(ρ)) = i
±√2
ρ2+1 +Z
, whence σ(A) = iZ.
For δ > 0 it seems to be impossible to obtain more exact information on the spectrum ofA(ρ). Therefore we present some numerical results.
Figure 1 shows the values ofy11(2π) +y12 (2π). The different curves correspond to the values of δ = 0,0.1, . . . ,0.6 and are plotted as functions of ρ2ρ+12 . The curves are “decreasing” near ρ = 0 and ρ = ∞ with increasing δ, but their maxima increase with δ.
Note that the eigenvalues ofA(ρ) lie on the imaginary axis ify11(2π) +y12(2π) lies between −2 and 2, whereas they move onto the lines ik +R, k ∈ Z, if y11(2π) +y12(2π) becomes greater than 2.
1 0
1 2 3 4 5 6 7 8 9 10 11
−1
−2
Figure 1: The values ofy11(2π) +y12(2π) as function of ρ2ρ+12
The differential equations were solved with the Bulirsch-Stoer method, using the implementation [4, Section 15.4]. The above curves suggest that the spectrum of A is the union of the imaginary axis and lines ik+ [−a, a], where a is strictly
increasing with δ and a = 0 at δ = 0. Numerical calculations of the part of the spectrum of A with imaginary parts between −2 and 2 for δ = 0.6 and δ = 0.9 are shown in Figure 2.
−2 −1 0 1 2
−2
−1 0 1 2
−2 −1 0 1 2
−2
−1 0 1 2
Figure 2: σ(A) forδ= 0.6 andδ= 0.9
5. A modified problem As in Sections 2–4 we consider an operator of the form
A+ =
⎛
⎜⎝
∂
∂ψ 0
0 ∂
∂ψ
⎞
⎟⎠+ 2N+, where now
N+(ρ, ψ) =
⎛
⎜⎜
⎝
0 1−δ2
ρ2(1−δcos(2ψ)) + 1−δ2
1 δρ2sin(2ψ)
ρ2(1−δcos(2ψ)) + 1−δ2
⎞
⎟⎟
⎠,
that is, in the right upper entry of N+ the negative of the entry in N considered so far is taken. It is not difficult to see that also the operator A+ is closed.
Theorem 5.1. The operator A+ has no eigenvalues and there is a constanta≥2 such that
σ(A+) ={λ∈C:|(λ)| ≤a, (λ)∈2Z}.
Proof. The proof is very similar to that of Theorem 3.1, and we only point out the differences. Here w has to be replaced by −w, and therefore (3.2) becomes (5.1) −(w2y1) + 2wy1 = 0,
and again (3.5) becomes the eigenvalue equation.
From (5.1) it follows thatw2y1j is strictly increasing at a point ify1j is positive there (j = 1,2). Then the initial conditions imply that bothy11and y12together with their derivatives are positive on (0,2π]. In particular, (3.4) implies that
y11(2π)y12 (2π)>1, and the arithmetic-geometric mean inequality yields that the positive numbers y11(2π) and y12 (2π) satisfy y11(2π) +y12 (2π) >2. This shows that the two solutions of (3.5) as a quadratic equation in e−2λπ are positive and inverses of each other. Hence there is a positive function α : [0,∞) → (0,∞) such that the eigenvalues of A(ρ) are {±α(ρ) + ik : k ∈ R}. Performing the differentiation in (5.1) and dividing by w2 one obtains a differential equation whose coefficient aty is 1 and whose other coefficients are continuous functions ofρ∈[0,∞). Hence alsoy11(2π) andy12 (2π) depend continuously onρ∈[0,∞).
Therefore,α is continuous and has a continuous extension to ρ∈[0,∞].
Since A(∞) is as in Section 4, it follows that α(∞) = 0. Thus α([0,∞]) is a (compact) interval containing the points 0 and 2. Henceα([0,∞])∪(−α([0,∞])) is a compact interval of the form [−a, a], and the statement on σ(A) follows.
ThatA+ has no eigenvalues follows as in Section 3 from the fact that α is not
constant.
References
[1] Eastham, M. S. P., The spectral theory of periodic differential equations, Scottish Aca- demic Press, Edinburgh, 1973.
[2] Kato, T., Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin, 1980.
[3] Lifschitz, A., Exact description of the spectrum of elliptical vortices in hydrodynamics and magnetohydrodynamics, Phys. Fluids 7:7 (1995), 1626–1636.
[4] Press, W. H., et al., Numerical Recipes in PASCAL, Cambridge University Press, 1989.
[5] Reed, M., Simon, B., Analysis of operators, IV, Academic Press, New York, 1978.
[6] Yakubovich, V. A., Starzhinskii, V. M., Linear differential equations with periodic coeffi- cients, Part 1, John Wiley & Sons, New York, 1975.
NWF I – Mathematik, University of Regensburg, D-93040 Regensburg, Germany, robert.denk@mathematik.uni-regensburg.de
Dept. of Mathematics, University of the Witwatersrand, 2050 WITS, South Africa, 036man@cosmos.wits.ac.za
Dept. of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom,c.tretter@mcs.le.ac.uk
