Key words. Dirac operator, Laplace operator, eigenvalue estimates, warped products

Abstract In this paper we show that the space of spinors over a warped product over S^{1} has a certain splitting Γ Σ = ⊕W (k, n) in spaces of spinors of weight k and winding number n which is respected by the Dirac operator. The same holds for the space of functions and the Laplace operator. We give eigenvalue estimates for the eigenvalues λ_{k,n} of the Dirac operator with eigenspinors of weight k and winding number n and eigenvalues μ_{m,n} of the Laplace operator with eigenfunctions of weight m and winding number n. In particular, we show that μ_{k^{2},n} ≥ λ_{k,n}^{2} holds for large n on S^{1} ×_{f} T^{l} where T^{l} is a flat torus with the trivial spin structure.

2  Margarita Kraus

manuscripta mathematica manuscript No. (will be inserted by the editor)

Margarita Kraus

Asymptotic estimates of Dirac and Laplace eigenvalues on warped products over S^{1}

24th June 2003

1. Introduction

On a Riemannian manifold (M, g), the Laplace operator is a second order elliptic differential operator acting on the space of functions on M. If the manifold is spin, the Dirac operator is an elliptic first order differential operator which acts on sections of the spinor bundle. If M is closed, its spectrum is discrete and real. Recently, there is great interest in the case of the two dimensional torus T^{2} with S^{1}-symmetry (cf. [1], [2], [3], [6]). On the two dimensional torus, there are four spin structures, and exactly one of these – which we call the trivial one – has a nontrivial kernel. For the other spin structures on T^{2}, there exists a constant C depending on the conformal structure and the spin structure such that the first eigenvalue satisfies (cf. [10])

λ_{1}^{2}(g) ≥ \frac{C}{vol(T^{2}, g)} .

However, this constant is not explicitly known. The case of the two dimensional torus with S^{1}- symmetry and the trivial spin structure has been studied intensively in [2]. In this paper, it has been shown that for a certain class of small deformations of the flat metric, the first eigenvalues of the Laplace and Dirac operator satisfy μ_{1} ≤ λ_{1}^{2}. We consider the more general case of spin manifolds which are warped products over S^{1}, M = S^{1} ×_{f} F := (S^{1} × F, dt^{2} + f^{2}(t)g_{F}), where (F, g_{F}) is a Riemannian spin manifold and f : S^{1} → ℝ^{+} a positive function. Clearly, the two dimensional torus with S^{1}-symmetry is an example for such a warped product T^{2} := S^{1} ×_{f} S^{1}. On a warped product M = S^{1} ×_{f} F, we consider the spin structure which is the product of the given spin structure on F and the trivial spin structure on S^{1}. The relation between the eigenvalues of this spin structure and the spin structure which is the product of the given spin structure on F and the non-trivial spin structure on S^{1} has been studied intensively in [6]. The eigenspaces of the Dirac as well as the eigenspaces of the Laplace operator on a warped product split:

E_{D}(λ) = ⨁_{k∈Spec_{D}(F)} E(λ, k)
E_{L}(μ) = ⨁_{m∈Spec_{L}(F)} E(μ, m)

and we call an eigenvalue λ of the Dirac operator an eigenvalue of weight k if E(λ, k) ≠ 0 and the same for the eigenvalues of the Laplace operator. One obtains from the eigenvalue equation for fixed weight a Sturm-Liouville equation or equivallently a two dimensional system of first order differential equations over [0, 2π] with periodic boundary conditions. Therefore we can

Fakultät für Mathematik Universität Regensburg D-93040 Regensburg Germany e-mail: Margarita.Kraus@mathematik.uni-regensburg.de Mathematics Subject classification: 34L20, 53C21, 58J50, 58J60

Asymptotic estimates of Dirac and Laplace eigenvalues on warped products over S^{1}  3

speak of the winding number of an eigenspinor, respectively of an eigenfunction and call it the winding number of the corresponding eigenvalue. In Section 4 we will show by means of a generalized Prüfer transformation:

Theorem 4.1. For every fixed k and winding number n, there exist two eigenvalues λ_{k,n} and λ̃_{k,n} of the Dirac operator on S^{1} ×_{f} F of weight k and winding number n such that

… λ_{k,−1} ≤ λ̃_{k,−1} < λ_{k,0} ≤ 0 ≤ λ̃_{k,0} < λ_{k,1} ≤ λ̃_{k,1} < …

Theorem 4.2. For every fixed m, there exists one eigenvalue μ_{m,0} ≥ 0 of weight m and winding number 0 and two eigenvalues μ_{m,n} and μ̃_{m,n} of weight m and winding number n for all n > 0 of the Laplace operator on S^{1} ×_{f} F such that

0 ≤ μ_{m,0} < μ_{m,1} ≤ μ̃_{m,1} < ….

In Section 5, we get estimates for the eigenvalues of fixed weight k and winding number n (Lemma 5.1 and Lemma 5.4) which improve in the case of the torus the well known estimates given by comparison with the eigenvalues of the conformal equivalent flat metric (cf. [3], p. 74, p.77):

\frac{λ_{n}^{2}(g_{0})}{h_{max}^{2}} ≤ λ_{n}^{2}(g) ≤ \frac{λ_{n}^{2}(g_{0})}{h_{min}^{2}}
\frac{μ_{n}^{2}(g_{0})}{h_{max}^{2}} ≤ μ_{n}^{2}(g) ≤ \frac{μ_{n}^{2}(g_{0})}{h_{min}^{2}},

where g = h^{2}g_{0}. In the last section, we show an asymptotic estimate for the eigenvalues of the Dirac and Laplace operator. The eigenvalues of the Dirac operator of weight k and winding number n satisfy for big n:

λ_{k,n} = 2πn + \frac{1}{4πn} ∫_{0}^{1} \frac{k^{2}}{f^{2}(t)}dt + O(n^{−2}),

and the eigenvalues of weight m and winding number n of the Laplace operator satisfy

\sqrt{μ_{m,n}} = 2πn + \frac{1}{4πn} ∫_{0}^{1}(\frac{m}{f^{2}(t)} + \frac{l^{2}}{4} \frac{ḟ^{2}(t)}{f^{2}(t)})dt + O(n^{−2}),

where l is the dimension of the fiber F. In particular, the Laplace eigenvalue is always bigger than the square of the corresponding eigenvalue of the Dirac operator on the two dimensional torus with the trivial spin structure

μ_{k^{2},n} ≥ λ_{k,n}^{2}

for n sufficiently large. Note that the eigenvalues m of the Laplace operator on the fiber S^{1} are equal to the squares of the eigenvalues k of the Dirac operator on the fiber, m = k^{2}. The question whether or not there exists a Riemannian metric g on the two dimensional torus such that for the first positive eigenvalue of the Laplace and Dirac operator μ_{1} > λ_{1}^{2} holds, remains open.

2. The eigenvalue equation on warped products over S^{1}

Let (F, g_{0}) be an l-dimensional Riemannian manifold, m_{j}(j ∈ ℕ) the eigenvalues of its Laplace operator with eigenspaces V (m_{j}). The eigenspaces E_{L}(μ) of the Laplace operator on the warped product

S^{1} ×_{f} F := (S^{1} × F, dφ^{2} + f^{2}(φ)g_{0})

with a smooth function f : S^{1} → ℝ^{+} decompose as

E_{L}(μ) = E_{L}(μ) ∩ ⊕(C^{∞}(S^{1}) ⊗ V (m_{j})) =: ⨁ E_{L}(μ, m_{j}) .

We call an element of C^{∞}(S^{1}) ⊗ V (m_{j}) a function on S^{1} ×_{f} F of weight m_{j} and an eigenvalue μ of the Laplace operator with E(μ, m_{j}) ≠ 0 eigenvalue of weight m_{j}. For g ∈ C^{∞}(S^{1}) and σ ∈ V (m), the product gσ is an eigenfunction for the eigenvalue μ of weight m, i.e. gσ ∈ E_{L}(μ, m), if and only if g is a periodic solution of

(f^{l}ẏ)˙ + f^{l}(μ − \frac{m}{f^{2}} )y = 0,  (1)

which is equivalent to the fact that gf^{\frac{l}{2}} is a periodic solution of

ÿ + (μ − \frac{m}{f^{2}} + \frac{(2l − l^{2})}{4} \frac{ḟ^{2}}{f^{2}} − \frac{lf̈}{2f})y = 0 .  (2)

The first order system associated to (1) is given by

ẋ = (\begin{matrix}0 & \frac{1}{f^{l}} \\ f^{l}(−μ + \frac{m}{f^{2}} ) & 0 \end{matrix} ) x  (3)

over [0, 2π] with periodic boundary conditions. We say that μ is an eigenvalue of (3) if (3) has a periodic solution. Now let (F, g_{0}) be a Riemannian spin manifold with fixed spin structure Q_{F} and spinor bundle Σ_{F}. We denote by … < k_{−1} < 0 < k_{1} < k_{2} < … the non zero eigenvalues of the Dirac operator on Σ_{F} with eigenspaces W (k_{j}) and by W (0) the space of harmonic spinors. Then the warped product S^{1} ×_{f} F is a spin manifold, too, and we equip it for the length of this paper with the spin structure, which is the product of the given spin structure on F and the trivial spin structure on S^{1}, namely Q ≅ S^{1} × (Q_{F} ×_{Spin(n)} Spin(n + 1)). If the dimension of F is odd, the spinor bundle splits into Σ_{M} ≅ S^{1} × (Σ_{F} ⊕ Σ̂_{F}), where Σ_{F} and Σ̂_{F} are the two spinor bundles given by the two representations of the n-dimensional Cliffordalgebra. In case the dimension of F is even, we have Σ_{M} ≅ S^{1} × Σ_{F}. The Dirac operator respects the decomposition

Γ (Σ_{M}) = ⨁(C^{∞}(S^{1}, ℂ) ⊗ (W (k_{j}) ⊕ W (−k_{j})))  (4)

of the space of spinors. The summation is over the spectrum of the fiber if the dimension of F is odd and over the non negative spectrum otherwise (for details, see [6], [9]). We say that a spinor has weight k_{j} if it is in the j-th summand of (4). The space of eigenspinors decomposes according to (4) and we call an eigenvalue λ of the Dirac operator an eigenvalue of weight k_{j} if there is an eigenspinor for λ of weight k_{j}. In [6], [9] we have shown that λ is an eigenvalue of the Dirac operator on S^{1} ×_{f} F of weight k if and only if the system

ẋ = (\begin{matrix}−\frac{k}{f} & λ \\ −λ & \frac{k}{f} \end{matrix}) x  (5)

admits a periodic solution. If this is the case, we shall just speak of an eigenvalue of (5). Under

the transformation T = \frac{\sqrt{2}}{2} (\begin{matrix}1 & 1 \\ −1 & 1 \end{matrix}), this system becomes (cf. [9])

ẋ = (\begin{matrix}0 & \frac{k}{f} − λ \\ \frac{k}{f} + λ & 0 \end{matrix}) x  (6)

and hence (5) has a periodic solution if and only if λ^{2} is an eigenvalue of the Sturm-Liouville equation

ẍ + (λ^{2} − \frac{k^{2}}{f^{2}} − \frac{kḟ}{f^{2}})x = 0  (7)

with periodic boundary conditions.

3. The argument equation

In [6], we applied the well known results of Sturm Liouville theory to the ordinary differential equation (7) to get results on the eigenvalues of the Dirac operator on bundles over S^{1}. From the associated first order systems for the Dirac and the Laplace operator, we will get more detailed information on the eigenspinors and the eigenfunctions of the Laplacian as well as further estimates for their eigenvalues. Consider a linear first order differential equation with traceless coefficient matrix

ẋ = (\begin{matrix}a & b \\ c & −a \end{matrix}) x  (8)

on [0, L] for L > 0. Then the Wronski-determinant is constant, and the determinant of the Poincaré map which assigns to every p ∈ ℝ^{2} the value γ_{p}(L), where γ_{p} is the solution of (8) with γ_{p}(0) = p, is one. The first order system (8) has periodic solutions if and only if the Poincaré map has eigenvalue one, and in this case we speak of the winding number ν(γ) of a solution. The Sturm Liouville equation (py)˙ + (Λs − q)y = 0 with p > 0 and periodic boundary conditions is equivalent to the first order system

ẏ = (\begin{matrix}0 & \frac{1}{p} \\ q − Λs & 0 \end{matrix}) y

and we call an eigenvalue of Λ of the Sturm Liouville equation an eigenvalue of winding number n if the associated first order system has a solution of winding number n. In particular, if the solution of (3) given by an eigenfunction of a Laplace eigenvalue μ of weight m has winding number ν(γ), we say the Laplace eigenvalue μ of weight m has winding number n = ν(γ) . If the solution of (5) given by an eigenspinor of a Dirac eigenvalue λ of weight k has winding number ν(γ), we call n = ν(γ) the winding number of the Dirac eigenvalue λ of weight k. We also speak of eigenfunctions and eigenspinors of winding number n. Under the modified polar

coordinate transformation (\begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix}) x = ρe^{iϑ} with strictly positive, L-periodic smooth functions

g_{1}, g_{2}, equation (8) gives the equation for the argument (cf. [11])

ϑ̇ = [a + \frac{1}{2}(\frac{ġ_{1}}{g_{1}} − \frac{ġ_{2}}{g_{2}} )] sin 2ϑ + \frac{1}{2}(b\frac{g_{1}}{g_{2}} + c\frac{g_{2}}{g_{1}} ) cos 2θ + \frac{1}{2}(b\frac{g_{1}}{g_{2}} − c\frac{g_{2}}{g_{1}} )  (9)

and a periodic solution of (8) with winding number n gives a solution Θ of (9) with Θ(L) = Θ(0) + 2nπ. For the first order system

ẏ = (\begin{matrix}0 & \frac{1}{p} \\ q − Λs & 0 \end{matrix}) y

associated to a Sturm Liouville equation, the two cases g_{1} ≡ g_{2} ≡ 1 and g_{1} = g_{2}\sqrt{p(Λs − q)} for Λ > (\frac{q}{s})_{max} are of special interest (cf. [4]). The first case is the Prüfer transformation. In the second case equation (9) gives the equation

ϑ̇ = \sqrt{\frac{Λs − q}{p}} + \frac{1}{4} \frac{(p(Λs − q))}{˙ p(Λs − q)} sin 2ϑ .  (10)

The polar coordinate transformation with g_{1} = g_{2} applied to the first order system (3) associated to the Laplace equation yields

ϑ̇ = \frac{1}{f^{l}} cos^{2} ϑ + f^{l}(μ − \frac{m}{f^{2}} ) sin^{2} ϑ,  (11)

and, for μ > \frac{m}{f_{min}^{2}} , equation (9) with g_{1} = g_{2}f^{l−1}\sqrt{μf^{2} − m} applied to (3) reduces to

ϑ̇ = \sqrt{μ − \frac{m}{f^{2}}} + \frac{ḟ(lμf^{2} + (1 − l)m)}{2f(μf^{2} − m)} sin 2ϑ.  (12)

Similarly, applied to first order system (5) associated to the Dirac operator, the equation (9) with g_{1} = g_{2} gives

ϑ̇ = −\frac{k}{f} sin 2ϑ + λ,  (13)

and for λ > \frac{k}{f_{min}} equation (9) with g_{1} = g_{2}\sqrt{\frac{λf+k}{λf−k}} implies

ϑ̇ = \sqrt{λ^{2} − \frac{k^{2}}{f^{2}}} − \frac{1}{2} \frac{λkḟ}{λ^{2}f^{2} − k^{2}} sin 2ϑ.  (14)

4. Consequences of the argument equation

According to Sections 2 and 3, the eigenspace of every eigenvalue λ of the Dirac operator on \frac{1}{2π} S^{1} ×_{f} F is a direct sum ⨁ W (k, n) of spaces W (k, n) of eigenspinors of weight k and winding number n, which means that every Ψ ∈ W (k, n) is of the form ρ(t) cos ϑ(t)σ_{k} + ρ(t) sin ϑ(t)σ_{−k} with σ_{±k} ∈ W (±k) and a solution ϑ of (13) with ϑ(1) = ϑ(0) + 2nπ. Even in the case of odd dimension of F, we can concentrate on the case k > 0: For if ϑ is a solution of (13), then ϑ − \frac{π}{2} is a solution of ϑ̇ = \frac{k}{f} sin 2ϑ + λ. Furtheron if λ is an eigenvalue of the Dirac equation with winding number n, then −λ is an eigenvalue with winding number −n and the argument function ϑ̃ given by the eigenspinor for −λ is ϑ̃ = −ϑ, where ϑ is the argument function for the eigenfunction given by the eigenspinor for the eigenvalue λ. In a similar way, the space of eigenfunctions of an eigenvalue μ of the Laplace operator splits. In this section we will show the Theorems 4.1. and 4.2.

Theorem 1. For every fixed k and winding number n, there exist two eigenvalues λ_{k,n} and λ̃_{k,n} of the Dirac operator on S^{1} ×_{f} F of weight k and winding number n such that

… λ_{k,−1} ≤ λ̃_{k,−1} < λ_{k,0} ≤ 0 ≤ λ̃_{k,0} < λ_{k,1} ≤ λ̃_{k,1} < …

Theorem 2. For every fixed m, there exists one eigenvalue μ_{m,0} ≥ 0 of weight m and winding number 0 and two eigenvalues μ_{m,n} and μ̃_{m,n} of weight m and winding number n for all n > 0 of the Laplace operator on S^{1} ×_{f} F such that

0 ≤ μ_{m,0} < μ_{m,1} ≤ μ̃_{m,1} < ….

In the following, we consider the Dirac and the Laplace equation on (\frac{1}{2π} S^{1}) ×_{f} F with f : [0, 1] → ℝ^{+}. This does not change the results, because it is well known how the eigenvalues change under rescaling of the metric: The Laplace eigenvalues μ̃ and the Dirac eigenvalues λ̃ on a Riemannian manifold (M, g̃) with g̃ = cg satisfy μ̃ = \frac{1}{c^{2}} μ and λ̃^{2} = \frac{1}{c^{2}} λ^{2}, where μ and λ are the Laplace and the Dirac eigenvalues on (M, g).

Notation 41 We denote by Ψ(ϑ, μ, t) the solution of (11) with starting point ϑ, and by ϕ(ϑ, λ, t) the solution of (13) with starting point ϑ.

Lemma and Definition 41 The functions

l_{0} : [0, π] ⟶ [−|k|/f_{min}, |k|/f_{min}], ϕ(ϑ, l_{0}(ϑ), 1) = ϑ
l_{n} : [0, π] ⟶ ℝ\(−|k|/f_{max}, |k|/f_{max}), ϕ(ϑ, l_{n}(ϑ), 1) = ϑ + 2πn for n ∈ ℤ\0
u_{0} : (0, π) ⟶ (−∞, m/f_{min}^{2}], Ψ(ϑ, u_{0}(ϑ), 1) = ϑ
u_{n} : [0, π] ⟶ [m/f_{max}^{2}, ∞), Ψ(ϑ, u_{n}(ϑ), 1) = ϑ + 2nπ for n ∈ ℕ

are well defined for every k and m.

Proof. For fixed ϑ ∈ [0, π], the function μ ↦ Ψ(ϑ, μ, 1) is monotonously increasing and (cf. [5], p. 246)

lim_{μ→−∞} Ψ(ϑ, μ, 1) = 0 , lim_{μ→∞} Ψ(ϑ, μ, 1) = ∞.

The function λ ↦ ϕ(ϑ, λ, 1) is also monotonously increasing and

lim_{λ→−∞} ϕ(ϑ, λ, 1) = −∞, lim_{λ→∞} ϕ(ϑ, λ, 1) = ∞

For |λ| > \frac{|k|}{f_{min}} , equation (13) implies ϑ̇(t) ≠ 0 for all t, therefore (13) admits no periodic solution in this case, and similarly one shows that equation (11) has no periodic solutions in case μ > \frac{m}{f_{min}^{2}} . For k > 0 and |λ| < \frac{k}{f_{max}} , every solution ϕ(ϑ, λ, t) of (13) with ϕ(ϑ, λ, t_{0}) = \frac{π}{4} +aπ

for t_{0} ∈ [0, 1] satisfies ϕ̇(ϑ, λ, t_{0}) < 0 and with ϕ(ϑ, λ, t_{1}) = −\frac{π}{4} + aπ satisfies ϕ̇(ϑ, λ, t_{1}) > 0 for t_{1} ∈ [0, 1] and a ∈ ℤ. Therefore for λ ≥ 0 every solution with starting point ϑ ∈ [0, \frac{π}{4} ] satisfies ϕ(ϑ, λ, 1) ∈ [0, \frac{π}{4} ) and with ϑ ∈ [\frac{π}{4} , π) satisfies ϕ(ϑ, λ, 1) ∈ [0, \frac{5π}{4} ) and for λ < 0 every solution with ϑ ∈ [0, \frac{3π}{4} ] satisfies ϕ(1, λ, ϑ) ∈ (−\frac{π}{4} , π) and every solution with ϑ ∈ (\frac{3π}{4} , π) satisfies ϕ(1, λ, ϑ) ∈ (\frac{3π}{4} , π). The proof for the Laplace equation works in a similar way.

Lemma 1. For every n ∈ ℤ (resp. n ∈ ℕ), the extrema λ_{n} := min l_{n} and λ̃_{n} := max l_{n} (resp. μ_{n} := min u_{n} and μ̃_{n} := max u_{n} for n ∈ ℕ and μ_{0} := max u_{0}) are the eigenvalues of equation (5) (resp. (3)) with winding number n.

Proof. Denote by P_{λ} the Poincaré map for (5). If l_{n}(ϑ_{1}) = ξ_{1}, then e^{iϑ_{1}} is an eigenvector of P_{ξ_{1}}, therefore either there is at last one ϑ_{2} ∈ [0, π) with l_{n}(ϑ_{2}) = ξ_{1} or l_{n} ≡ ξ_{1}. Hence the extrema of l_{n} are the values that l_{n} takes exactly once or l_{n} is constant. If l_{n}(ϑ_{1}) = ξ and ξ is not an eigenvalue of (5), then 1 is not an eigenvalue of P_{ξ}, therefore P_{ξ} has two eigenvectors e^{iϑ_{1}} ≠ e^{iϑ_{2}} and ϑ_{1} is no extremum on l_{n}. On the other hand, if ϑ_{1} is not an extremum, there exists a ϑ_{2} with l_{n}(ϑ_{1}) = l_{n}(ϑ_{2}) and either P_{l_{n}(ϑ_{1})} = id or e^{iϑ_{1}} and e^{iϑ_{2}} are eigenvectors of P_{l_{n}(ϑ_{1})} for eigenvalues not equal to 1. The proof for the functions u works in the same way.

Lemma 2. For every ϑ_{1}, ϑ_{2} ∈ [0, π) and n ∈ ℤ, one has l_{n}(ϑ_{1}) > l_{n−1}(ϑ_{2}), and for n ∈ ℕ, it holds u_{n}(ϑ_{1}) > u_{n−1}(ϑ_{2}).

Proof. Let ξ_{1} ≥ ξ_{2} and ϑ_{1} > ϑ_{2}, then clearly ϕ(ϑ_{1}, ξ_{1}, 1) > ϕ(ϑ_{2}, ξ_{2}, 1). For ϑ_{2} < ϑ_{1} < ϑ_{2} + π it is ϕ(ϑ_{2}, l_{n+1}(ϑ_{2}), 1) > ϕ(ϑ_{1}, l_{n}(ϑ_{1}), 1) and therefore l_{n}(ϑ_{1}) < l_{n+1}(ϑ_{2}). For 0 ≤ ϑ_{1} < ϑ_{2} ≤ π we have ϕ(ϑ_{1} + 2π, l_{n}(ϑ_{1}), 1) ≤ ϕ(ϑ_{2}, l_{n+1}, 1) and therefore l_{n}(ϑ_{1}) ≤ l_{n+1}(ϑ_{2}), too. The proof for the functions u works in the same way.

Proof (Proof of Theorem 4.1. and 4.2.). The theorems follow from Lemmas 1 and 2 and the fact that l_{0}(0) = 0 and l_{0}(\frac{π}{2} ) = 0, therefore λ_{0} ≤ 0 ≤ λ̃_{0} and u(\frac{π}{2} ) ≥ 0, therefore μ_{0} ≥ 0.

As a corollary from the theorem, we obtain the estimates for the first eigenvalue of the Dirac operator shown in [6]. We give here the new proof and state the analogous assertion for the Laplace operator.

Corollary 1 ([6]). For the eigenvalues of winding number zero, λ̃_{k,0} ≥ \frac{|k|}{f_{max}} and μ_{m,0} ≥ \frac{m}{f_{max}^{2}} . Equality holds only for constant warping function f.

Proof. For k > 0 from ϕ̇(ϑ, \frac{k}{f_{max}} , t) ≤ 0 for every t with ϕ(ϑ, \frac{k}{f_{max}} , t) = \frac{π}{4} follows l_{0}(\frac{π}{4} ) ≥ \frac{k}{f_{max}} , and therefore λ̃_{k,0} := max l_{0} ≥ \frac{k}{f_{max}} . From Ψ̇(ϑ, \frac{m}{f_{max}^{2}} , t) ≤ 0 for every t with Ψ(ϑ, \frac{m}{f_{max}^{2}} , t) = \frac{π}{4} follows u_{0}(\frac{π}{2} ) ≥ \frac{m}{f_{max}^{2}} and therefore μ_{m,0} = max u_{0} ≥ \frac{m}{f_{max}^{2}} .

5. Some estimates for the eigenvalues

In the case of a torus, S^{1} ×_{f} T^{l}, the eigenvalues m_{j} of the Laplacian on the fiber and the squares of the eigenvalues k_{j} of the Dirac operator on the fiber T^{l} = S^{1} × ⋯ × S^{1} a flat torus with the trivial spin structure coincide: m_{α} = k_{α}^{2} = ∑ α_{j}^{2} for α ∈ ℕ_{0}^{l}. Therefore the lower bound of Corollary 1 is the same for the first eigenvalue of the Laplacian and the square of the Dirac operator. For l = 1 the well known upper bound for the first eigenvalue of the Laplacian is lower than for the square of the Dirac operator: Inserting the constant function into the Rayleigh quotient gives an upper estimate for the first eigenvalues. For the Dirac operator on (\frac{1}{2π} S^{1}) ×_{f} F, the first eigenvalue of weight k satisfies

λ_{k,0}^{2} ≤ ∫_{0}^{1} \frac{k^{2}}{f^{2}(t)}dt.

For the Laplace operator, the first eigenvalue of weight m satisfies

μ_{k,0} ≤ ∫_{0}^{1} f^{l−2}(t)mdt [∫_{0}^{1} f^{l}(t)dt]^{−1} .

In case l = 1 and m = k^{2}, it is

∫_{0}^{1} \frac{k^{2}}{f^{2}(t)}dt ≥ [∫_{0}^{1} \frac{k}{f(t)}dt]^{2} ≥ ∫_{0}^{1} \frac{k^{2}}{f(t)}dt [∫_{0}^{1} f(t)dt]^{−1} .

In the following, we consider not only winding number zero, but an arbitrary winding number. For the Dirac operator on (\frac{1}{2π} S^{1}) ×_{f} F, Lemma 1 implies by integration of equation (13):

Lemma 3. The two eigenvalues of the Dirac operator on (\frac{1}{2π} S^{1}) ×_{f} F of weight k and winding number n satisfy

|λ_{k,n} − 2nπ| ≤ ∫_{0}^{1} \frac{k}{f(t)}dt.  ⊓ ⊔  (15)

The next lemma shows that λ_{k,n} = 2πn + O(1/n). We state here the direct proof but in the next section we will improve this estimate.

Lemma 4. For the eigenvalues of the Dirac operator on (\frac{1}{2π} S^{1}) ×_{f} F of weight k and winding number n such that λ_{k,n} ≥ \frac{k}{f_{min}} , one has

\frac{2n}{\sqrt{λ^{2} − \frac{k^{2}}{f_{max}^{2}}}} [\frac{π}{2} + arcsin \frac{k}{λf_{max}} ] + \frac{2n}{\sqrt{λ^{2} − \frac{k^{2}}{f_{min}^{2}}}} [\frac{π}{2} − arcsin \frac{k}{λf_{min}} ] ≤ 1  (16)

and

1 ≤ \frac{2n}{\sqrt{λ^{2} − \frac{k^{2}}{f_{min}^{2}}}} [\frac{π}{2} + arcsin \frac{k}{λf_{min}} ] + \frac{2n}{\sqrt{λ^{2} − \frac{k^{2}}{f_{max}^{2}}}} [\frac{π}{2} − arcsin \frac{k}{λf_{max}} ] .  (17)

Proof. Equation (13) implies for nπ ≤ ϑ ≤ nπ + \frac{1}{2}π

−\frac{k}{f_{min}} sin 2φ + λ ≤ φ̇ ≤ −\frac{k}{f_{max}} sin 2φ + λ

and for nπ − \frac{1}{2}π ≤ ϑ ≤ nπ

−\frac{k}{f_{max}} sin 2φ + λ ≤ φ̇ ≤ −\frac{k}{f_{min}} sin 2φ + λ.

This implies

1 ≤ 2n [∫0^{\frac{π}{2}} \frac{dφ}{λ − \frac{k}{f_{min}} sin 2φ} + ∫ π_{\frac{π}{2}} \frac{dφ}{λ − \frac{k}{f_{max}} sin 2φ}]

and

1 ≥ 2n [∫0^{\frac{π}{2}} \frac{dφ}{λ − \frac{k}{f_{max}} sin 2φ} + ∫ π_{\frac{π}{2}} \frac{dφ}{λ − \frac{k}{f_{min}} sin 2φ}] .

Integration then yields the result.

Remark 51 Remark that with Lemma 3 from 2πn ≥ k(\frac{1}{f_{min}} + ∫_{0}^{1} \frac{1}{f(t)}dt) it follows λ_{k,n} ≥ \frac{k}{f_{min}} and the assumption of Lemma 4 is satisfied.

For the Laplace operator on (\frac{1}{2π} S^{1}) ×_{f} F^{l}, we get from the usual comparison theorem of the Sturm Liouville theory (cf. [4], p. 23) by comparison with the equations for l = 1

(f_{max}ẏ)˙ + (μf_{min} − \frac{m}{f_{min}} )y = 0 and (f_{min}ẏ)˙ + (μf_{max} − \frac{m}{f_{max}} )y = 0

and for l ≥ 2:

(f_{max}^{l}ẏ)˙ + (μf_{min}^{l} − mf_{max}^{l−2})y = 0 and (f_{min}^{l}ẏ)˙ + (μf_{max}^{l} − mf_{min}^{l−2})y = 0

Lemma 5. For the eigenvalues of the Laplace operator on (\frac{1}{2π} S^{1})×_{f} F^{l} of weight m and winding number n for l = 1

4π^{2}n^{2} \frac{f_{min}}{f_{max}} + \frac{m}{f_{max}^{2}} ≤ μ_{m,n} ≤ 4π^{2}n^{2} \frac{f_{max}}{f_{min}} + \frac{m}{f_{min}^{2}}  (18)

and for l ≥ 2

(\frac{f_{min}}{f_{max}} )^{l}(4π^{2}n^{2} + \frac{m}{f_{min}^{2}} ) ≤ μ_{m,n} ≤ (\frac{f_{max}}{f_{min}} )^{l}(4π^{2}n^{2} + \frac{m}{f_{max}^{2}} )  (19)

holds. ⊓ ⊔

Therefore we may decide in some special cases which of the eigenvalues μ_{01} or μ_{10} is the first positive in the case of the two dimensional torus S^{1} ×_{f} S^{1}. Analogous results for the eigenvalues of the Dirac operator have been derived in [6].

Corollary 2. If f_{max}^{3} ≤ f_{min}, the first non zero eigenvalue of the Laplace operator on S^{1} ×_{f} S^{1} is an eigenvalue of weight 0 and winding number 1. If f_{min}^{3} ≥ f_{max} then the first positive eigenvalue of the Laplace operator is the eigenvalue of weight 1 and winding number 0.

We now show the result analogous to Lemma 3 for the Laplace operator on (\frac{1}{2π} S^{1}) ×_{f} F. We will use this in the next section to show an asymptotic estimate for the eigenvalues.

Lemma 6. For the eigenvalues μ_{m,n} of the Laplace operator of weight m and winding number n with μ_{m,n} ≥ \frac{2m}{f_{min}^{2}} it is

|\sqrt{μ_{m,n}} − 2nπ| ≤ \frac{(l + 1)}{2} ∫_{0}^{1} |\frac{ḟ}{f}|dt + \frac{1}{2nπ} ∫_{0}^{1} \frac{m}{f^{2}} dt.  (20)

Proof. From (12) for μ ≥ \frac{2m}{f_{min}^{2}} follows

| ∫_{0}^{1} \sqrt{μ − \frac{m}{f^{2}}} dt − 2nπ| ≤ \frac{1}{2} ∫_{0}^{1} \frac{|ḟ|}{f}(\frac{1}{1 − \frac{m}{μf^{2}}} + (l − 1))dt ≤ \frac{(l + 1)}{2} ∫_{0}^{1} |\frac{ḟ}{f}|dt, hence

|\sqrt{μ} − 2nπ| ≤ ∫_{0}^{1} [\frac{|μ − \frac{m}{f^{2}} − 4n^{2}π^{2}|}{\sqrt{μ − \frac{m}{f^{2}}} + 2nπ} + \frac{\frac{m}{f^{2}}}{\sqrt{μ − \frac{m}{f^{2}}} + 2nπ} ] dt

≤ \frac{(l + 1)}{2} ∫_{0}^{1} |\frac{ḟ}{f}|dt + \frac{1}{2nπ} ∫_{0}^{1} \frac{m}{f^{2}} dt.

The next lemma gives in some special cases an estimate for the winding number n and an improvement of Lemma 5 for functions with sharp peaks.

Lemma 7. Suppose that μ_{m,n} is an eigenvalue of the Laplace operator of weight m and winding number n.

(1) If μ_{m,n} ≥ \frac{1+mf_{max}^{2l−2}}{f_{min}^{2l}}, then ∫_{0}^{1} \frac{1}{f^{l}(t)}dt < 2πn and μ_{m,n} > \frac{2πn + m ∫_{0}^{1} f^{l−2}(t)dt}{∫_{0}^{1} f^{l}(t)dt};

(2) If μ_{m,n} ≤ \frac{1+mf_{min}^{2l−2}}{f_{max}^{2l}}, then ∫_{0}^{1} \frac{1}{f^{l}(t)}dt > 2πn and μ_{m,n} < \frac{2πn + m ∫_{0}^{1} f^{l−2}(t)dt}{∫_{0}^{1} f^{l}(t)dt}.

Proof. In the first case, (11) gives \frac{1}{f^{l}(t)} ≤ φ̇(t) ≤ μ_{m,n}f^{l}(t)−mf^{l−2}(t), in the second μ_{m,n}f^{l}(t)− mf^{l−2}(t) ≤ φ̇(t) ≤ \frac{1}{f^{l}(t)} and integration and Lemma 1 gives the result.

Especially for m = 0 we obtain from Lemma 5 and Lemma 7:

Corollary 3. If 2πn < (\frac{f_{min}}{f_{max}^{3}} )^{\frac{l}{2}} , then μ_{0,n} ≤ _{\frac{2πn}{∫_{0}^{1} f^{l}(t)dt}}. If 2πn > (\frac{f_{max}}{f_{min}^{3}} )^{\frac{l}{2}} , then

μ_{0,n} ≥ _{\frac{2πn}{∫_{0}^{1} f^{l}(t)dt}}. ⊓ ⊔

The first part of the corollary is an improvement of Lemma 5 in the case that

(f_{max}^{l} ∫_{0}^{1} f^{l}(t)dt)^{−1} f_{min}^{l} < 2πn < (f_{max}^{− \frac{3}{2}}fmin^{\frac{1}{2}})^{l},

which is possible for (f_{min}f_{max})^{\frac{l}{2}} < ∫_{0}^{1} f^{l}(t)dt, the second part of the corollary is an improvement in the case that

f_{min}^{− \frac{3l}{2}} fmax^{\frac{l}{2}} < 2πn < (f_{min}^{l} ∫_{0}^{1} f^{l}(t)dt)^{−1} f_{max}^{l},

which is possible for (f_{max}f_{min})^{\frac{l}{2}} > ∫_{0}^{1} f^{l}(t)dt. From the general theory [[4],p.45] follows for every weight

μ_{m,n} ≥ \frac{4π^{2}n^{2}}{f_{max}^{2l}(∫_{0}^{1} f^{−l}(t)dt)^{2}} .

From Lemma 5.5 ensues a related result.

Lemma 8. For the eigenvalues of weight 0 one has

\frac{4π^{2}n^{2}f_{min}^{2l}}{(∫_{0}^{1} f^{l}(t)dt)^{2}} ≤ μ_{0,n} ≤ \frac{4π^{2}n^{2}f_{max}^{2l}}{(∫_{0}^{1} f^{l}(t)dt)^{2}}  (21)

Proof. If μ is an eigenvalue of (1) with m = 0 for a function f, then also for every function cf with a constant c. Now let c := f_{min}^{−1}μ^{− \frac{1}{2l}} and f̃:= cf. Then _{\frac{1}{f̃_{min}^{2l}}} = μ and therefore Lemma

7 can be applied and gives μ > _{\frac{2πn}{∫_{0}^{1} f̃^{l}(t)dt}} from which the first inequality follows. The same

argument with c = \frac{1}{\sqrt{μ}f_{max}} shows the second inequality.

6. Asymptotic estimates

In this section we will compare for large n the eigenvalues of the Dirac operator and the Laplace operator on warped products over S^{1}. For the Sturm Liouville equation

y′′ + (Λ − q)y = 0  (22)

over [0, 1] with periodic boundary conditions an asymptotic estimate for the eigenvalues is given in [4], p. 58. We will give here a new proof in our special situation and additionally an error estimation.

Theorem 3. Let Λ be an eigenvalue of (22) of winding number n and let n_{0} ≥ 0 and a > 0 exist, such that for n ≥ n_{0}

|\sqrt{Λ} − 2nπ| ≤ a.

Then for Λ ≥ \frac{3}{2}q_{max} and n > max(\frac{a}{π} , n_{0})

\sqrt{Λ} = 2nπ + \frac{1}{4nπ} ∫_{0}^{1} q(t)dt + \frac{ρ_{q,n,a}}{4n^{2}π^{2}}

with

|ρ_{q,n,a}| ≤ a| ∫_{0}^{1} q(t)dt| + ∫_{0}^{1} |q̇(t)|dt + \frac{1}{nπ} ∫_{0}^{1} q^{2}(t)dt + \frac{3}{n^{2}π^{2}} ∫_{0}^{1} |q̇(t)q(t)|dt  (23)

Proof. We apply (10) with p = s = 1. For Λ > \frac{3}{2}q we have

\sqrt{Λ − q} = \sqrt{Λ} − \frac{q}{2\sqrt{Λ}} + \frac{r_{1}}{\sqrt{Λ^{3}}}

with |r_{1}| ≤ \frac{1}{4}q^{2} because r_{1} = −\frac{q^{2}}{4} (\sqrt{1 − \frac{q}{Λ}} + 1 − \frac{1}{2} \frac{q}{Λ})^{−1}. Furthermore it is \frac{1}{Λ−q} = \frac{1}{Λ} + \frac{r_{2}}{Λ^{2}} with |r_{2}| ≤ 3|q|. Therefore integrating (10) gives

2nπ = \sqrt{Λ} − \frac{1}{2\sqrt{Λ}} ∫_{0}^{1} q(t)dt + \frac{1}{Λ}r  (24)

with

|r| ≤ \frac{1}{4}(∫_{0}^{1} |q̇(t)|dt + \frac{1}{\sqrt{Λ}} ∫_{0}^{1} q^{2}(t)dt + \frac{3}{Λ} ∫_{0}^{1} |q̇(t)q(t)|dt),

for Λ > \frac{3q_{max}}{2} . Now from |\sqrt{Λ} − 2nπ| ≤ a it follows for n ≥ max(\frac{a}{π} , n_{0})

\frac{1}{\sqrt{Λ}} = \frac{1}{2πn} + \frac{α}{4π^{2}n^{2}} with |α| ≤ 2a

such that (24) gives \sqrt{Λ} = 2nπ + \frac{1}{4nπ} ∫_{0}^{1} q(t)dt + \frac{1}{4n^{2}π^{2}} ρ with

|ρ| ≤ a| ∫_{0}^{1} q(t)dt| + \frac{1}{4}(1 + \frac{a}{nπ})^{2}(∫_{0}^{1} |q̇(t)|dt + \frac{1}{2nπ} (1 + \frac{a}{nπ}) ∫_{0}^{1} q^{2}(t)dt +

+\frac{3}{4π^{2}n^{2}} (1 + \frac{a}{nπ})^{2} ∫_{0}^{1} |q̇(t)q(t)|dt)

≤ a| ∫_{0}^{1} q(t)dt| + ∫_{0}^{1} |q̇(t)|dt + \frac{1}{nπ} ∫_{0}^{1} q^{2}(t)dt + \frac{3}{n^{2}π^{2}} ∫_{0}^{1} |q̇(t)q(t)|dt

This yields for the Dirac and Laplace eigenvalues

Corollary 4. (1) For the eigenvalues μ_{m,n} of the Laplace operator on S^{1} ×_{f} F^{l} of weight m and winding number n holds for n → ∞

\sqrt{μ_{m,n}} = 2πn + \frac{1}{4πn} ∫_{0}^{1}(\frac{m}{f^{2}(t)} + \frac{l^{2}}{4} \frac{ḟ^{2}}{f^{2}(t)})dt + O(n^{−2})

(2) For the eigenvalues λ_{k,n} of the Dirac operator of weight k and winding number n holds for n → ∞

λ_{k,n} = 2πn + \frac{1}{4πn} ∫_{0}^{1} \frac{k^{2}}{f^{2}(t)}dt + O(n^{−2})

Now we will apply Theorem 3 to the Dirac and Laplace operator on the torus S^{1} ×_{f} T^{l} with T^{l} = S^{1} × ⋯ × S^{1} the flat torus with the trivial spin structure. In this case the eigenvalues m_{j} of the Laplace operator on the fiber and the eigenvalues k_{j}^{2} of the square of the Dirac operator on the fiber are the same, and we get:

Corollary 5. For the eigenvalues λ_{k,n} and μ_{k^{2},n} of the Dirac and Laplace operator on (\frac{1}{2π} S^{1})×_{f} T^{l} with the trivial spin structure of weight k^{2} and winding number n the inequality

\sqrt{μ_{k^{2},n}} ≥ λ_{k,n}

holds, if n ≥ max(γ_{1}, γ_{2}, γ_{3}, γ_{4}) with

γ_{1} := \frac{k}{2π} \sqrt{\frac{2f_{max}^{l}}{f_{min}^{l+2}} − \frac{1}{f_{min}f_{max}}}

γ_{2} := \frac{1}{2π} (\frac{1}{f_{min}} \sqrt{\frac{3}{2}k(k + ḟ_{max})} + ∫_{0}^{1} \frac{k}{f(t)}dt)

γ_{3} := \frac{1}{2π} \sqrt{k^{2}(\frac{3}{2} \frac{f_{max}^{l}}{f_{min}^{l+2}} − \frac{1}{f_{max}f_{min}} ) + \frac{3}{4} \frac{f̈_{max}}{f_{min}^{l+1}} f_{max}^{l}}

γ_{4} := \{ \begin{matrix}\frac{4}{l^{2}π} (∫_{0}^{1} \frac{ḟ^{2}}{f^{2}} dt)^{−1}(|ρ_{q_{L},1,a_{L}}| + |ρ_{q_{D},1,∫_{0}^{1} \frac{k}{f} dt}|) & for & k ≠ 0 \\ \frac{4}{l^{2}π} (∫_{0}^{1} \frac{ḟ^{2}}{f^{2}} dt)^{−1}|ρ_{q_{L},1,a_{L}}| & for & k = 0 \end{matrix}

a_{L} := \frac{(l + 1)}{2} ∫_{0}^{1} |\frac{ḟ}{f}|dt + \frac{1}{2π} ∫_{0}^{1} \frac{k^{2}}{f^{2}}
and q_{L} := \frac{k^{2}}{f^{2}} − \frac{2l − l^{2}}{4} \frac{ḟ^{2}}{f^{2}} + \frac{l}{2} \frac{f̈}{f} and q_{D} = \frac{k^{2}}{f^{2}} + \frac{kḟ}{f^{2}}

Proof. For n > γ_{1} from Lemma 5 follows μ_{k^{2},n} ≥ \frac{2k^{2}}{f_{min}^{2}} and therefore with Lemma 6 |\sqrt{μ}−2nπ| ≤ a_{L}. Furthermore it is γ_{4} ≥ \frac{a_{L}}{π}. For n > max(γ_{1}, γ_{2}, γ_{3}, \frac{a_{L}}{π}) follows from Lemma 3 and 5.3 that Theorem 3 can be applied to the Dirac and Laplace operator and gives by partial integration for n ≥ 1 :

\sqrt{μ_{k^{2},n}} − λ_{k,n} = \frac{l^{2}}{16πn} ∫_{0}^{1} \frac{ḟ^{2}}{f^{2}} dt + \frac{1}{4π^{2}n^{2}} (ρ_{q_{L},n,a_{L}} − ρ_{q_{D},n,∫_{0}^{1} \frac{k}{f} dt})

For k = 0 the corollary follows from λ_{0,n} = 2πn.

At last we consider the example of the two dimensional torus (\frac{1}{2π} S^{1}) ×_{f} (\frac{1}{2π} S^{1}) with the trivial spin structure and f : \frac{1}{2π} S^{1} → ℝ^{+}, f(e^{2πit}) = 1 + r cos 2πωt with ω ∈ ℕ and 0 < r < 1. In this case the eigenvalues of the Laplace operator on the fiber \frac{1}{2π} S^{1} are given by m_{j} := 4j^{2}π^{2} and from Lemma 6 follows for the eigenvalues of the Laplace operator of weight m_{j} and winding number n

|\sqrt{μ_{m_{j},n}} − 2nπ| ≤ 2ω ln \frac{1 + r}{1 − r} + \frac{2j^{2}π}{n} \frac{1}{(1 − r^{2})^{\frac{3}{2}}}  (25)

if n ≥ \frac{j}{1−r} \sqrt{\frac{1+r^{2}+6r}{1−r^{2}}} , and from Lemma 3 follows for the eigenvalues of the Dirac operator

|λ_{k_{j},n} − 2πn| ≤ \frac{2jπ}{\sqrt{1−r^{2}}} . Then

\sqrt{μ_{m_{j},n}} = 2πn + \frac{π}{n\sqrt{1 − r^{2}}} (\frac{j^{2}}{1 − r^{2}} + \frac{ω^{2}}{4} (1 − \sqrt{1 − r^{2}})) + O(n^{−2})
λ_{k_{j},n} = 2πn + \frac{πj^{2}}{n(1 − r^{2})^{\frac{3}{2}}} + O(n^{−2}).

For weight 0 it is λ_{0,n} = 2πn and

\sqrt{μ_{0,n}} = 2πn + \frac{πω^{2}(1 − \sqrt{1 − r^{2}})}{4n\sqrt{1 − r^{2}}} + \frac{ρ_{q_{L},n,a}}{4π^{2}n^{2}}

with a = 2ω ln \frac{1+r}{1−r} and ρ_{q_{L},n,a} can be estimated by the sum of the terms

a ∫_{0}^{1} q(t)dt = \frac{a}{4} ∫_{0}^{1} \frac{ḟ^{2}(t)}{f^{2}(t)}dt = 2ω^{3}π^{2} \frac{1 − \sqrt{1 − r^{2}}}{\sqrt{1 − r^{2}}} ln \frac{1 + r}{1 − r}
∫_{0}^{1} |q̇(t)|dt = 4(πω)^{3}r(1 − r^{2}) ∫_{0}^{1} |\frac{sin 2πωt}{(1 + r cos 2πωt)^{3}} |dt
= \frac{8π^{2}ω^{3}r}{1 − r^{2}}
\frac{1}{nπ} ∫_{0}^{1} q^{2}(t)dt ≤ \frac{1}{nπ} (q^{2}(t))_{max} = \frac{4π^{3}ω^{4}}{n} (\frac{r}{1 − r})^{2}
\frac{3}{n^{2}π^{2}} ∫_{0}^{1} |q̇(t) · q(t)|dt ≤ \frac{3}{n^{2}π^{2}} |q(t)|_{max} ∫_{0}^{1} |q̇(t)|dt
= \frac{48π^{2}ω^{5}r^{2}}{n^{2}(1 − r)^{2}(1 + r)}

To apply the corollary we have to calculate the constants γ_{3} and γ_{4}. For γ_{3} we get

γ_{3} = \frac{ω\sqrt{3r(1 + r)}}{2(1 − r)} ≥ \frac{ωr}{1 − r}

Therefore we may estimate n instead of by

γ_{4} ≤ \frac{2ω}{π} ln \frac{1 + r}{1 − r} + \frac{4ω(\sqrt{1 − r^{2}} + 1 − r^{2})}{πr(1 − r^{2})(1 − r)}(2(1 − r) + ωπr(1 + r) + 12ω^{2}r)

by

γ̃_{4} := \frac{4}{π} (∫_{0}^{1} \frac{ḟ^{2}}{f^{2}} dt)^{−1}|ρ_{q_{L},n_{0},a}| with n_{0} = \frac{ωr}{1 − r} ,

which improves the estimate by γ_{4} for r > \frac{1}{1+ω}, namely

γ̃_{4} ≤ \frac{2ω}{π} ln \frac{1 + r}{1 − r} + \frac{4ω(\sqrt{1 − r^{2}} + 1 − r^{2})}{πr^{2}(1 − r^{2})}(12 + r(π − 10) + r^{2}π) =: γ̄_{4}

and for n ≥ max(γ_{3}, min(γ_{4}, γ̄_{4})) we have

λ_{0,n} ≤ \sqrt{μ_{0,n}}.

Therefore in the case ω = 1 the estimate holds for n ≥ N with N given by

\begin{matrix}r & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 0.99 & 0.99999 \\ γ_{i} & γ_{4} & γ_{4} & γ_{4} & γ_{4} & γ_{4} & γ̄_{4} & γ̄_{4} & γ̄_{4} & γ̄_{4} & γ_{3} & γ_{3} \\ N & 96 & 77 & 78 & 87 & 104 & 73 & 56 & 47 & 46 & 122 & 122474 \end{matrix}

The second line shows the γ_{i} which has to be used for the estimate. For r ≤ 0.9 we can improve the estimate for n, namely

λ_{0,n} ≤ \sqrt{μ_{0,n}}

holds for n ≥ N, where N is the smallest integer with

N ≥ γ̃_{4} := \frac{4}{π} (∫_{0}^{1} \frac{ḟ^{2}}{f^{2}} dt)^{−1}|ρ_{q_{L},n_{0},a}| with n_{0} = 10

this gives

Corollary 6. On (\frac{1}{2π} S^{1}) ×_{f} (\frac{1}{2π} S^{1}) with f(t) = 1 + r cos 2πt for the eigenvalues of the Dirac and Laplace operator of weight zero λ_{0,n} ≤ \sqrt{μ_{0,n}} holds if n ≥ N, where N is given by:

\begin{matrix}r & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ N & 53 & 28 & 20 & 17 & 15 & 15 & 17 & 22 & 42 \end{matrix}

Acknowledgements. I am very grateful to K. Jänich for many helpful suggestions. I also thank I. Agricola for useful comments.

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