Scaling inequalities and limits for Robin and Dirichlet eigenvalues
Scott Harman



§ ABSTRACT 
   For the Laplacian in spherical and hyperbolic spaces, Robin eigenvalues in two dimensions and Dirichlet eigenvalues in higher dimensions are shown to satisfy scaling inequalities analogous to the standard scale invariance of the Euclidean Laplacian. These results extend work of Langford and Laugesen to Robin problems and to Dirichlet problems in higher dimensions.
   In addition, scaled Robin eigenvalues behave exotically as the domain expands to a 2-sphere, tending to the spectrum of an exterior Robin problem.

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§ INTRODUCTION



 §.§ From Euclidean to spherical and hyperbolic

In Euclidean space, we have the following scaling identity of eigenvalues for every t ∈:

    t^2λ(tΩ) = λ(Ω)

where λ is a Dirichlet or Neumann Laplacian eigenvalue on a sufficiently regular domain Ω. Robin eigenvalues λ(Ω, α) with parameter α∈ also satisfy a scale invariance provided one also scales the parameter, namely

    t^2 λ(tΩ, α/t) = λ(Ω, α).

Equivalently, one can interpret these identities as the two functionals being constant for every value of t. However, when space is curved, the above identities fail, even after making sense of linear dilation on the new spaces. Our goal is to recover information about these functionals on spherical and hyperbolic spaces.

We will analyze the functionals

    λ_k(Ω_t)S(t)^2

where λ_k is a Laplace–Beltrami eigenvalue, Ω_t is a spherical cap or geodesic ball, and S(t) is some natural geometric scaling factor. For Robin eigenvalues, we will also look for natural normalizing factors N(t) and their corresponding functionals

    λ_k(Ω_t, α / N(t))S(t)^2.


The three main results in this paper are Robin monotonicity in spherical and hyperbolic space in two dimensions, Dirichlet monotonicity in spherical and hyperbolic space in higher dimensions, and computations of the limits for the Robin functionals. The limiting cases exhibit exotic behavior: as the spherical caps expand to fill up the entire sphere, the scaled Robin functional converges to a negative eigenvalue of the exterior Robin problem on the complement of a Euclidean disk.


 §.§ Unifying spherical and hyperbolic notation


Let ^n be the unit sphere of positive curvature 1. We will employ the coordinate system (θ, ξ) where θ∈ (0, π) is the angle of aperture from the north pole and ξ∈^n-1 represents the other angular components. Observe that any point (θ, ξ) has geodesic distance θ from the north pole. The domain of interest is

    C(Θ) := {(θ, ξ) : 0 ≤θ < Θ, ξ∈^n-1}

where 0 < Θ < π, or geometrically, the spherical cap of aperture Θ. We will consider the eigenvalue problem on C(Θ):

    -_^nu = λ u

subject to Dirichlet, Neumann, or Robin boundary conditions.

We also consider hyperbolic space ^̋n of negative curvature -1. In this space, for Θ < 0 we put C(Θ) as the geodesic disk of radius |Θ|. There is the analogous coordinate system (θ, ξ) where θ∈ (0, ∞) and ξ∈^n-1, and the corresponding eigenvalue problem

    -_^̋nu = λ u.

Frequently we will denote the Laplace–Beltrami operator on these sets as

    _C(Θ) := 
        _^̋n,     -∞ < Θ < 0, 
    _^n,     0 < Θ < π.




 §.§ Two-dimensional Robin results


Our goal is to quantify the effect that curving space has on the scaled eigenvalues, that is, how the geometry of the space interacts with its underlying analytic properties. In Euclidean ^n, taking ^n as the unit ball, the constant function t^2 λ(t ^n) takes the appealing geometric form as an eigenvalue times the radius squared. In curved spaces such as ^n and ^̋n, it is unclear what the “scaling factor” should be, i.e., with what geometric quantity one should replace t^2.

In two dimensions, Langford and Laugesen in <cit.> present results and scaling factors for Dirichlet and Neumann eigenvalues in two dimensions. A natural question to ask is whether analogous scaling results apply to Robin eigenvalues. We recall that the Robin problem is

    {    -_C(Θ) u = λ u   in  C(Θ),
       ∂ u/∂ n + α u = 0 on ∂ C(Θ),
        .

where ∂/∂ n is the outer unit normal derivative and α is a real-valued parameter. This problem induces a discrete spectrum of eigenvalues

    λ_1(Θ, α) < λ_2(Θ, α) ≤λ_3(Θ, α) ≤…→∞

where λ_k(Θ, α) is the k^th Robin eigenvalue on C(Θ) with parameter α.

The Euclidean scale invariance (<ref>) suggests that the Robin parameter needs to be suitably scaled in the hyperbolic and spherical cases. Additionally, if the Robin parameter is negative, the spectrum now has negative eigenvalues. The Robin problem thus contrasts sharply with Neumann and Dirichlet problems, which have non-negative spectra. The negative eigenvalues of course tend to reverse the monotonicity of the functional, but the main interest for the negative eigenvalues is their peculiar limiting behavior. 

First, we handle positive Robin parameters (and hence positive Robin eigenvalues). A straightforward generalization of <cit.> holds. The main adaptation for the Robin problem is to scale the Robin parameter α by the perimeter of the geodesic disk (sinh |Θ|) or by the perimeter of the spherical cap (sinΘ). At Θ = 0, we define λ_k(, α) to be the k^th Robin eigenvalue of the Euclidean unit disk ⊂^2 with parameter α.


    Fix n = 2, k ≥ 1, and α > 0.

    
    
  * The function 
    Θ↦λ_k(Θ, α/sinh |Θ|)   4 tanh^2(Θ/2),    Θ∈ (-∞,0) , 
    λ_k(𝔻, α) ,    Θ=0 , 
    λ_k(Θ, α/sinΘ)   4 tan^2(Θ/2) ,    Θ∈ (0,π),

    increases strictly and continuously from 0 to ∞. See Figure <ref>.
    
  * The function 
    Θ↦λ_k(Θ, α/sinh |Θ|)  sinh^2(Θ),    Θ∈ (-∞,0) , 
    λ_k(𝔻, α),    Θ=0 , 
    λ_k(Θ, α/sinΘ)  sin^2(Θ) ,    Θ∈ (0,π),

    decreases strictly and continuously from ∞ to 0. See Figure <ref>.
    


The function (<ref>) has the geometric interpretation of an eigenvalue times stereographic radius squared, where the stereographic transformation is detailed in (<ref>). The function (<ref>) has the interpretation of an eigenvalue times perimeter squared.

Langford and Laugesen showed the Neumann case α = 0 in <cit.>. For the Dirichlet case (i.e., α→∞) the theorem was also proven in <cit.>; in that case the function (<ref>) increases from 1 to ∞.

For negative Robin parameters in the next theorem, we encounter more obstacles. The eigenvalue can turn negative, which reverses monotonicity and necessitates different techniques for the limiting values. In particular, a negative eigenvalue causes strange limiting behavior towards a seemingly unrelated unbounded Robin problem. 

We define λ_k^ext(, α)
as the k^th Robin eigenvalue on the exterior of the Euclidean unit disk . (The full definition of the PDE system is (<ref>). More information on exterior Robin problems can also be found in <cit.>.) We show that this exterior eigenvalue arises as a limit when the Robin eigenvalue is negative.


    Fix n = 2, k ≥ 1, and α < 0 where α∈ [-m - 1, -m) and m is a non-negative integer.

    
    
  * If k ≤ 2m + 1, then the eigenvalue  function (<ref>) is negative and decreases strictly and continuously from 0 to -∞ as a function of Θ∈ (-∞, π). If k = 2m+2, 2m+3, and α = -m-1, then λ_2m+2 = λ_2m+3 = 0 for all Θ and (<ref>) is identically zero. Otherwise, λ_k > 0 for all Θ and Theorem <ref>(i) continues to hold. See Figure <ref>.
    

    
  * If k ≤ 2m + 1, then the eigenvalue function (<ref>) is negative and increases strictly and continuously from -∞ to λ_k^ext(, α) as a function of Θ∈ (-∞, π). If k = 2m+2, 2m+3, and α = -m-1, then λ_2m+2 = λ_2m+3 = 0 for all Θ and (<ref>) is identically zero. Otherwise, λ_k > 0 for all Θ and Theorem <ref>(ii) continues to hold. See Figure <ref>.
    

The appearance of the exterior Robin eigenvalue problem is quite surprising, but we will provide some intuition when we prove the limiting values. Compared to the previous theorem, monotonicity and limits reverse sign when the eigenvalue is negative. Another fact that emerges from the theorem is that the normalized eigenvalues cannot change sign as a function of Θ: they remain negative or positive (or zero) for all Θ.

We will often make use of the following function definitions:

    (Θ) = 
    sinh(|Θ|)    for Θ < 0, 
    sin(Θ)    for Θ≥ 0,
    (Θ) = 
    tanh(|Θ|)    for Θ < 0, 
    tan(Θ)    for Θ≥ 0.

These definitions differ from <cit.> in using sinh (|Θ|) for Θ < 0 rather than sinh( Θ).
 These functions make our statements more succinct, e.g., the first part of Theorem <ref> reads that λ_k(Θ, α/(Θ))4^2(Θ/2) is an increasing function. 








Using this notation, we provide graphs of the first spectral curve (k = 1) in terms of α and Θ. See Figures <ref>, <ref>, <ref> and <ref>. These plots help visualize Theorems <ref> and <ref>. In addition, Table <ref> gives an overview of the limiting values. 






 §.§ Higher-dimensional results


Higher dimensions pose completely different challenges to two dimensions. The Dirichlet integral is no longer conformally invariant and the proof techniques break down. In addition, previously natural choices of scaling factors in lower dimensions are no longer so in higher dimensions. The initial goal then is to determine which geometric quantities do suitably generalize to higher dimensions and scale the eigenvalue appropriately.

Table <ref> gives examples of natural quantities on ^n which could potentially work, along with their geometric interpretation. The quantity sin(Θ) is one such geometric factor that does not experience a monotonicity defect in higher dimensions, and it is this scaling quantity we focus on for n ≥ 3. Volume also appears to be a monotonic scaling factor by the numerics in the Dirichlet case, but we postpone discussing this quantity for a brief moment.

Let λ_k(Θ) represent the k^th Dirichlet eigenvalue of the Laplace–Beltrami operator on C(Θ). The scaling factor ^2(Θ) is shown to produce monotonicity in higher dimensions, thus generalizing Langford and Laugesen's result to all dimensions, in the Dirichlet case. At Θ = 0, we define λ_k(^n) to be the k^th Dirichlet eigenvalue of the Euclidean ball ^n ⊂^n.

The function

    Θ↦λ_k(Θ)  sinh^2(Θ),    Θ∈ (-∞,0) , 
    λ_k(^n) ,    Θ=0 , 
    λ_k(Θ)  sin^2(Θ) ,    Θ∈ (0,π) ,

decreases strictly from ∞ to 0, for all k ≥ 1 and n ≥ 3. See Figure <ref>.


The result does not exclude other geometric scaling factors from working, nor does it state that other eigenvalues such as Neumann or Robin do not behave monotonically when scaled. Indeed, we will briefly discuss open problems below.


 §.§ Open monotonicity problems

The statement analogous to Theorem <ref> for the Neumann spectrum is supported by numerical evidence. However, the proof of the theorem does not translate well to the Neumann case, and so we state the analog as a conjecture. Let μ_k(Θ)  be the k^th Neumann eigenvalue on C(Θ).
 
The function

    Θ↦μ_k(Θ)  sinh^2(Θ),    Θ∈ (-∞,0) , 
    μ_k(^n) ,    Θ=0 , 
    μ_k(Θ)  sin^2(Θ) ,    Θ∈ (0,π/2) ,

decreases strictly from ∞ to 0, for all k ≥ 2 and n ≥ 3. See Figure <ref>.

The lower-dimensional case n = 2 of Conjecture <ref> was shown to hold by Langford and Laugesen in <cit.>. One could further conjecture that the Robin case satisfies a monotonicity result in higher dimensions, but the proof would encounter similar obstacles to the Neumann case.

For another open problem, we look at the scaling factor volume. Write n-dimensional volume of the geodesic ball of radius |Θ| as

    V(Θ) := c_n∫_0^|Θ|(θ)^n-1  dθ,
where c_n is the appropriate normalizing factor, equal to the surface area of the Euclidean (n-1)-sphere ^n-1. Consider the scaled Dirichlet and Neumann eigenvalues 
    λ_k(Θ)V(Θ)^2/n,   μ_k(Θ)V(Θ)^2/n,
with λ_k(Θ)V(Θ)^2/n = λ_k(^n)|^n|^2/n at Θ = 0 by convention. This scaling factor is somewhat peculiar since Figure <ref> numerically supports that λ_k(Θ)V(Θ)^2/n is always decreasing whereas Figure <ref> numerically supports that μ_k(Θ)V(Θ)^2/n is always increasing.

We look at the Dirichlet case first, for k =1,2. Observe that after a change of variable t = θ/Θ, we have for Θ∈ (0, π) that

    V(Θ)/Θ^n = c_n ∫_0^1 (sin(Θ t)/Θ t)^ n-1 t^n-1 dt,
implying that V(Θ)^2/n/Θ^2 is a decreasing function since sin(x)/x is decreasing.  (The Θ < 0 case is proven similarly.) One may also verify that the displayed ratio converges to |^n| as Θ→ 0. Since λ_1(Θ)Θ^2 decreases from ∞ to 0 by <cit.>, we see in all dimensions n ≥ 2 that

    λ_1(Θ)V(Θ)^2/n

decreases strictly from ∞ to 0 for Θ∈ (-∞, π), passing through the value λ_k(^n)|^n|^2/n continuously at Θ = 0. Note that <cit.> also manifests as a consequence of a theorem by Cheng (see <cit.> and <cit.>.) Furthermore, for the second eigenvalue, from <cit.> we see for all n ≥ 2 that

    λ_2(Θ)V(Θ)^2/n

decreases strictly from ∞ to λ_k(^n)|^n|^2/n  for Θ∈ (-∞, 0]. However, we cannot make further deductions on the behavior of the function on the interval (0, π) using this theorem. 

For the Neumann case, <cit.> implies in dimension n =2

    μ_2(Θ)V(Θ)^2/n

increases strictly and continuously on (-∞, π) from 2π to 8π. Observe how the function converges to non-zero values, highlighting the peculiar behavior of this scaling factor.

We state monotonicity conjectures for these two functionals. 

 The function

    λ_k(Θ)   V(Θ)^2/n

decreases strictly for Θ∈ (-∞, π) for k ≥ 2 and n ≥ 2. See Figure <ref>.


 The function

    μ_k(Θ)   V(Θ)^2/n

increases strictly for Θ∈ (-∞, π) for k ≥ 2 and n ≥ 2. See Figure <ref>.

(The Neumann case where both k = 2 and n =2 has already been proven; see Langford and Laugesen <cit.>.) 

The limiting values of the functionals are not obvious. Certainly at the origin they should pass through λ_k(^n)|^n|^2/n and μ_k(^n)|^n|^2/n but the endpoint Θ = -∞ is not clear. The Neumann case k=2, n =2 has already been shown to converge to a non-zero limit as Θ→ -∞, and Figure <ref> supports that this phenomenon continues in higher dimensions. For the Dirichlet eigenvalues, since λ_1(Θ)V(Θ)^2/n→∞ as Θ→ -∞, it must also be the case that λ_k(Θ)V(Θ)^2/n→∞. For Θ→π, the Dirichlet and Neumann spectra should converge to the spectrum of the whole sphere; see <cit.> and <cit.>. More investigation and analysis is needed for this scaling factor. 






 §.§ Related results


Upper and lower bounds and some exact values are known for the Dirichlet eigenvalues in both hyperbolic and spherical spaces. For spherical caps in dimension n = 3, we have an exact formula for the first Dirichlet eigenvalue λ_1(Θ) = π^2/Θ^2 - 1, which one may verify is monotonically decreasing after multiplying by the scaling factor sin^2(Θ). From Borisov and Freitas <cit.>, we provide a two-sided estimate on the first eigenvalue, which we compute here for dimension n = 4 as

    λ_1(^4)/Θ^2 - 2    ≤λ_1(Θ) ≤λ_1(^4)/Θ^2 - 3/4(1/sin^2(Θ) - 1/Θ^2) for Θ > 0, 
    λ_1(^4)/Θ^2 + 2    ≤λ_1(Θ) ≤λ_1(^4)/Θ^2 + 3/4(1/sinh^2(Θ) - 1/Θ^2) for Θ < 0.

Such bounds cannot prove monotonicity. However, they are sharper estimates than the bounds acquired from Theorem <ref>, which are

    λ_1(Θ)    ≤λ_1(^4)/sin^2(Θ)for Θ > 0, 
    λ_1(Θ)    ≥λ_1(^4)/sinh^2(Θ)for Θ < 0.

There is other literature involving ratios of eigenvalues and Faber–Krahn type inequalities by Ashbaugh, Benguria, and Linde <cit.>. Estimates for higher Dirichlet eigenvalues can be found in a paper by Berge <cit.>.



§ TWO-DIMENSIONAL ROBIN MONOTONICITY — THEOREMS 1.1 AND 1.2

For this section, we will work exclusively in two dimensions and prove the monotonicity statements in Theorems <ref> and <ref>. The proofs of the limiting values are in the next section.

Let C(Θ) remain as defined above, either as a hyperbolic or spherical cap. 
We consider the Robin eigenvalue problem

    {    -_C(Θ) u = λ u   in  C(Θ), 
       ∂ u/∂ n + α u = 0 on ∂ C(Θ),
    .

with discrete spectrum

    λ_1(Θ, α) < λ_2(Θ, α) ≤λ_3(Θ, α) ≤…→∞.

The variational characterization of λ_k(Θ, α) is

    λ_k(Θ, α) = min_ℒmax_0 ≠ u ∈ℒ∫_C(Θ) |_^2 u|^2  dA + α∫_∂ C(Θ) |u|^2  dS/∫_C(Θ) |u|^2  dA

where ℒ varies over k-dimensional subspaces of H^1(C(Θ)), dA is the 2-dimensional area element, and dS is the appropriate 1-dimensional measure on the boundary. When α < 0, at least one negative eigenvalue exists, which impacts monotonicity calculations and proofs. We will separate then our monotonicity discussion into two cases: positive α parameters and negative α parameters.



 §.§ Positive Robin parameter — Theorem 1.1


 Fix α > 0. We perform the following conformal transformation to the plane under polar coordinates:


    (θ, ξ) ↦
     (tanθ/2, ξ)    if Θ > 0, 
    
     (tanhθ/2, ξ)    if Θ < 0.

This map is exactly stereographic projection in the spherical cap case, mapping the spherical cap C(Θ) to a disk in the plane.

The conformal map defined above transforms the Rayleigh quotient and eigenequation in 
a systematic way. Define the weight functions

    w_±(r) = 4/(1 ± r^2)^2.

Consider the eigenequation -_C(Θ) u = λ_k(Θ, α) u. Letting v be the pushforward of u under the above map, the equation transforms as

    -_^2 v = λ_k(Θ, α) w_± v

where the weight is w_+ for positively curved spherical caps and w_- for negatively curved geodesic disks. One can also consult <cit.> for more detail on the transformations.

We define R := tan(Θ/2) (or tanh(|Θ|/2) in the hyperbolic case) for the radius of the disk where v is defined. The Rayleigh quotient for λ_k(Θ, α) transforms by conformal invariance of the Dirichlet integral to


    ∫_R  | v(r, ϕ)|^2   dA + α√(w_±(R))∫_∂ (R ) v(R, ϕ)^2   dϕ/∫_R v(r, ϕ)^2 w_±  dA

where dA = r   dr dϕ.
The rescaling g(z) = v(Rz) further transforms the expression into

    ∫_ | g(r, ϕ)|^2  dA + α√(w_±(R))∫_∂ g(1, ϕ)^2   dϕ/∫_ g(r, ϕ)^2 w_± (Rr) R^2 dA.

After normalizing α by the factor present in Theorem <ref>, and noting that sinΘ = √(w_+(R)) and sinh |Θ| = √(w_-(R)), the Rayleigh quotient for λ_k(Θ, α/ (Θ)) becomes

    ∫_ | g(r, ϕ)|^2  dA + α∫_0^2π g(1, ϕ)^2  d ϕ/∫_ g(r, ϕ)^2 w_± (Rr) R^2  dA.

As the numerator is independent of R and positive (recall α > 0), to show parts (i) and (ii) of the theorem we need only show monotonicity with respect to R in the denominator after multiplying by the appropriate scaling factor. The scaling factors transform as

    ^2(Θ) = 4R^2/(1± R^2)^2,      4^2(Θ/2) = 4R^2,

where (Θ) and (Θ) were defined in the remark after Theorem <ref>.
Let S(R) be equal to the appropriate scaling factor.
The variational characterization for the scaled eigenvalue is thus

    λ_k(Θ, α/(Θ)) S(R):= min_ℒmax_0 ≠ g ∈ℒ∫_ | g(r, ϕ)|^2   dA + α∫_0^2π g(1, ϕ)^2   d ϕ/∫_ g(r, ϕ)^2 w_± (Rr)/S(R) R^2   dA

where ℒ varies over k-dimensional subspaces of H^1(). The crucial point is that the only dependence on R in the expression is the quantity

    w_±(Rr)/S(R)R^2

in the denominator. Langford and Laugesen's proof of <cit.> analyzes this quantity and justifies the strict monotonicity statements in Theorem <ref>.

We still need to show the limiting values in the theorem are correct, but we will delay the proof until after analyzing the negative parameters.



 §.§ Negative Robin parameter — Theorem 1.2


Negative parameters require extra care as the eigenvalue could be negative, which reverses the monotonicity results. For spherical caps and geodesic disks, the behavior of negative eigenvalues is fortunately quite tractable, as seen in the following lemma.


    Let α < 0 and Θ≠ 0. If α∈ [-m-1, -m), then the first 2m+1 perimeter-normalized eigenvalues λ_k(Θ, α/ (Θ)) are negative. The other eigenvalues are positive, except that when α = -m-1 there are two zero eigenvalues, λ_2m+2 = λ_2m+3 = 0. The analogous results hold when Θ = 0 for λ_k(, α).



The proof relies on the fundamental connection between Robin and Steklov eigenvalues, and so we will mostly focus on integer α. Consider the Steklov problem on the unit disk , which is

    {    -_^2 g = 0  in ,
       ∂ g/∂ n = σ g on ∂.
        .

The eigenfunctions and eigenvalues of this problem are well-understood (see <cit.> and <cit.>.) The eigenvalues repeated according to multiplicity are

    σ∈{0, 1, 1, 2, 2, 3, 3, …}.

Suppose Θ > 0. We will transform the Steklov problem on the disk into a Steklov problem on a spherical cap by reversing the transformations performed in the preceding section. Letting R := (Θ/2), we perform the rescaling v(z) = g(z/R), so that v is defined on the disk R. Then, we take the pullback of v along the conformal map (<ref>) and call the pullback u(θ, ξ), whose domain is the spherical cap C(Θ). Under these conformal mappings, the Steklov problem on the disk transforms into the perimeter-normalized Steklov problem

    {    -_^2 u = 0 in  C(Θ),
       ∂ u/∂ n + (α /  (Θ)) u = 0 on ∂ C(Θ),
        .

where α = -σ. 

So, the perimeter-normalized Robin problem on a spherical cap or geodesic disk has a zero eigenvalue, λ(Θ, α/ (Θ)) = 0, precisely when α is a non-positive integer. Recall that Robin eigenvalues are strictly increasing and continuous as a function of α. This fact along with the systematic form of the Steklov spectrum on the disk implies there are at least 2m+1 negative eigenvalues when α = -m-1. If there were more than 2m+1, then the Neumann spectrum (α = 0) would have a negative eigenvalue, which is impossible. It is easy to see λ_2m+1 < λ_2m+2 = λ_2m+3 = 0 when α = -m -1, as one simply uses the Steklov eigenfunctions as Robin ones. Furthermore, λ_2m+1 < 0 < λ_2m+2 when α∈ (-m-1, m).

For Θ = 0, the argument is even simpler as one does not need to perform any conformal mappings and the result is immediate.

The graphs in <cit.> give a detailed picture of what is occurring with the Robin eigencurves on the disk. The perimeter normalization is crucial for us so that the number of negative eigenvalues is independent of Θ. That is, if λ_k(Θ, α/(Θ)) < 0 for some Θ, then it remains negative for all Θ values.

We can now show the monotonicity statements in Theorem <ref>.
All calculations resulting in (<ref>) are still valid, and Theorem <ref> continues to hold for positive eigenvalues. For negative eigenvalues, one notes that if we already know that λ_k(Θ, α/(Θ)) < 0, then we can modify the family of subspaces ℒ in (<ref>) to only be those for which the maximizer of the Rayleigh quotient is negative. Using the lemma above, the negativity of the eigenvalue is independent of Θ, and so we can use this modified variational characterization for each value of Θ.

Now, since the numerator is uniformly negative in (<ref>), the monotonicity results reverse direction, and we arrive at the desired statements in Theorem <ref>.


§ POSITIVE ROBIN LIMITING VALUES FOR THEOREM <REF>


We move forward to proving the limits for the scaled Robin eigenvalues. These proofs are long and involved, so we split them over three sections. A plethora of disparate techniques will be employed, with the negative eigenvalues in the next two sections requiring the most interesting and challenging ones. In this section, we prove the limits for positive eigenvalues in Theorems <ref>. Each functional has three limits that we need to compute: Θ→ -∞, Θ→ 0, and Θ→π. 

The first and simplest case is when Θ→ 0. We want to show it converges to the Euclidean eigenvalue. Observe that in (<ref>), simple computations show the quantity w_±(Rr)R^2/S(R) → 1 uniformly as R → 0  and we recover the Euclidean Rayleigh quotient and eigenvalues.

There are quite a few cases to consider for the two other limiting values of Θ, so we will be doing casework. Before doing so, we state the following Lemma, a Robin generalization of <cit.> which states that certain weighted Euclidean eigenvalues have an infinite limit. The Laplacian  in the following lemma is interpreted as the Euclidean Laplacian _^2.

(Pointwise vanishing of the weight)
    Let Ω⊂^2 be a bounded, regular domain. Fix α∈. For each R > 0, let w_R: Ω→ (0,1] be a measurable function with ess inf w_R > 0. Let λ_k(w_R, α) be the k^th Robin eigenvalue of -w_R^-1. Then, there exists an index N independent of R such that λ_k(w_R, α) > 0 if and only if k ≥ N. Furthermore, if lim_R →∞ w_R = 0 a.e., then 
    
    lim_R →∞λ_k(w_R, α) = ∞  for each k ≥ N.




First, we show the existence of the index N. The variational characterization reads

    λ_k(m_R, α) = min_ℒmax_0 ≠ f ∈ℒ∫_Ω | f|^2  dA + α∫_∂Ω|f|^2  dS/∫_Ω |f|^2 w_R  dA

where ℒ varies over k-dimensional subspaces of H^1(Ω), dA = r   dr dϕ and dS is the appropriate 1-dimensional Hausdorff measure. 

For each R, let N_R be the first index k such that λ_N_R(w_R,α) > 0. Let R_1, R_2  be arbitrary positive R-values. If N_R_1 = 1, then the spectrum is positive, and so α > 0 and N_R_1 = N_R_2 = 1. So, assume N_R_1≥ 2 and k < N_R_1. Then, λ_k(w_R_1, α) ≤ 0.  By homogeneity of the Rayleigh quotient, we may assume that each trial function f in (<ref>) is L^2-normalized so that ∫_Ω |f|^2   dA = 1. Then, the denominator of the Rayleigh quotient is bounded above and below:

    ess inf w_R_1≤∫ |f|^2 w_R_1  dA ≤ 1.
Since the denominator is bounded and λ_k(w_R_1, α) ≤ 0, the variational characterization (<ref>) implies there exists a k-dimensional subspace ℳ of H^1(Ω) such that

    max_0 ≠ f ∈ℳ(∫_Ω | f|^2  dA + α∫_∂Ω|f|^2  dS) ≤ 0.

Using ℳ as a trial subspace in the variational characterization of λ_k(w_R_2, α) implies λ_k(w_R_2, α) ≤ 0. It follows that N_R_1≤ N_R_2, and N_R_1=N_R_2 by symmetry. Thus, the index N_R is indeed independent of R and we may write it simply as N.

We move on to the second statement in the lemma. Fix 0 < δ < 1 and define a new weight m_R := max (δ, w_R) so that m_R is uniformly bounded away from zero. The k^th Robin eigenvalue of the operator -m_R^-1 has variational characterization

    λ_k(m_R, α) = min_ℒmax_0 ≠ f ∈ℒ∫_Ω | f|^2  dA + α∫_∂Ω|f|^2  dS/∫_Ω |f|^2 m_R  dA.

Assume k ≥ N. Then, the numerator of the maximizer is positive, and by comparing Rayleigh quotients, we see that λ_k(w_R, α) ≥λ_k(m_R, α) for each R. Furthermore, since m_R is a function into (0,1] with ess inf m_R > 0, we see that λ_k(m_R, α) > 0 as well.

By the variational characterization, we see that

    δ^-1λ_k(1, α) ≥λ_k(m_R, α) ≥λ_k(1, α)

where λ_k(1, α) is the k^th Robin eigenvalue of the unweighted Laplacian. We will show that lim inf_R →∞λ_k(m_R, α) ≥δ^-1λ_N(1, α) which implies λ_k(w_R, α) →∞ as R →∞.

For each R, take a k^th Robin eigenfunction f_R of -m_R^-1. The eigenfunction satisfies the weak equation

    ∫_Ωϕ· f_R   dA + α∫_∂Ωϕ f_R   dS= λ_k(m_R, α) ∫_Ωϕ f_R m_R   dA

for all ϕ∈ H^1(Ω). We normalize the eigenfunction f_R in L^2(Ω) so that ∫_Ω |f_R|^2 = 1. Now take a sequence of R values tending to ∞. We see that

    ∫_Ω  | f_R|^ 2  dA + α∫_∂Ω  f_R^2   dS = λ_k(m_R, α) ∫_Ω f_R^2 m_R   dA ≤δ^-1λ_k(1, α) ∫_Ω f_R^2   dA

where the inequality used (<ref>) and that m_R ≤ 1.
Applying the L^2 normalization of f_R, we derive an upper bound δ^-1λ_k(1, α) which is independent of R.

When α≥ 0, the above estimate gives us a bound on the H^1 norm of f_R that is independent of R. For α < 0, some more work has to be done. From the calculations done in <cit.>, justified by the proof of the trace theorem in Evans <cit.>, one deduces that

    ∫_Ω | f_R|^2   dA + α∫_∂Ω |f_R|^2   dA ≥1/2∫_Ω | f_R|^2   dA + Cα∫_Ω |f_R|^2   dA

for some constant C depending only on Ω. Since f_R is normalized in L^2, we again derive a uniform upper bound for the H^1 norm of f_R that is independent of R.

After applying the Rellich–Kondrachov theorem, we pass to a subsequence of f_R and yield a function f ∈ H^1(Ω) such that f_R ⇀ f weakly in H^1(Ω) and f_R → f strongly in L^2(Ω). Furthermore, by <cit.>, the trace operator T: H^1(Ω) → L^2(∂Ω) is compact, and since f_R ⇀ f weakly in H^1, Tf_R → T_f strongly in L^2(∂Ω).

We may further assume λ_k(m_R, α) converges to a finite limit as R →∞, after extracting a subsequence. As the weak Robin eigenequation (<ref>) is preserved under weak limits, the calculations in <cit.> still apply, and we deduce

    -  f = (δlim_R →∞λ_k(m_R, α)) f

in the weak sense. (This step uses the fact that w_R → 0 a.e. to imply m_R →δ a.e.)
Therefore, f is an eigenfunction of the unweighted Robin Laplacian on Ω. The eigenvalue δlim_R →∞λ_k(m_R, α) must be positive by (<ref>) since k ≥ N. It follows that δlim inf_R →∞λ_k(m_R, α) ≥λ_N(1, α), concluding the proof.




 §.§ Scaling factor 4^2(Θ/2)


Now we prove the limits for Theorem <ref>(i). Suppose either α > 0 and k ≥ 1 or α∈ [-m-1, -m) and k ≥ 2m+2 (≥ 2m+4 if α = -m-1.) By Lemma <ref>, we know λ_k(Θ, α / (Θ)) > 0. 

First consider Θ→π. We apply the lemma above to the weighted eigenvalue λ_k(Θ, α/sinΘ)4tan^2(Θ/2). The variational characterization (<ref>) reads (recalling that 4tan^2(Θ/2) = 4R^2)

    λ_k(Θ, α/sinΘ ) 4tan^2(Θ/2):= min_ℒmax_0 ≠ g ∈ℒ∫_ | g(r, ϕ)|^2  dA + α∫_0^2π g(1, ϕ)^2  d ϕ/∫_ g(r, ϕ)^2 (1+R^2r^2)^-2 dA.

The weights w_R := (1+R^2r^2)^-2 satisfy the hypotheses of Lemma <ref>, and it follows that λ_k(Θ, α/sinΘ)4tan^2(Θ/2) →∞ as R →∞, that is, as Θ→π.

Let Θ→ -∞. A modified proof of the limiting case R → 1 and Ω = in <cit.> will be applied. Refer there for more details on the following proof. Choose h ∈ C^1([0,1]) such that h is increasing with h = 0 on [0, 1/4] and h = 1 on [3/4, 1]. Take the subspace spanned by the H^1 functions h(r) cos jϕ, 1 ≤ j ≤ k, where (r, ϕ) are the polar coordinates. By <cit.>, these functions are L^2-orthogonal to the constant and to each other, with respect to the weight (1-R^2r^2)^-2. Further recall {cos(jϕ)} is orthogonal within L^2([0, 2π]). Using this subspace as a trial function space in (<ref>), and by homogeneity of the Rayleigh quotient, we deduce the upper bound (recalling that tanh(|Θ|/2) = R)

    0 ≤λ_k(Θ, α/ sinh |Θ|) 4tanh^2(Θ/2) 
    ≤max_|c| = 1∑_j = 1^k c_j^2  ∫_ | (h(r)cos jϕ)|^2   dA + α∑_j = 1^k c_j^2 ∫_0^2 π |cos j ϕ|^2   d ϕ/∑_j = 1^k c_j^2 ∫_ |h(r) cos j ϕ|^2 (1-R^2r^2)^-2  dA

where c = (c_1, …, c_j) is the coefficient vector.
As R → 1, each integral in the denominator tends to ∞, and the numerator is independent of R. It follows that λ_k(Θ, α/ sinh |Θ|) 4tanh^2(Θ/2) → 0 when R → 1 (i.e. when Θ→ -∞).



 §.§ Scaling factor ^2(Θ)


Now consider the limits for Theorem <ref>(ii). Let Θ→π. The inequality 0 ≤λ_k(Θ, α / sinΘ)sin^2(Θ) ≤λ_k(Θ) sin^2(Θ), where λ_k(Θ) is the k^th Dirichlet eigenvalue, always holds for the positive eigenvalues. By <cit.>, λ_k(Θ) sin^2(Θ) → 0 and thus so does λ_k(Θ, α / sinΘ)sin^2(Θ).

Let Θ→ -∞. We will apply Lemma <ref> again. The variational characterization of the weighted eigenvalue λ_k(Θ, α/ sinh |Θ|)sinh^2(Θ) (recalling that sinh |Θ| = 2R/(1-R^2)) is

    λ_k(Θ, α/sinh|Θ|) sinh^2(Θ):= min_ℒmax_0 ≠ g ∈ℒ∫_ | g(r, ϕ)|^2  dA + α∫_0^2π g(1, ϕ)^2  d ϕ/∫_ g(r, ϕ)^2 (1-R^2/1-R^2r^2)^2 dA.

Taking Θ→ -∞ is equivalent to taking R → 1, and one sees the weights (1-R^2/1-R^2r^2)^2 satisfy the hypotheses of Lemma <ref>. (Although the lemma is stated for R →∞, a simple relabeling lets it work for R → 1.) We conclude that λ_k(Θ, α/ sinh |Θ|)sinh^2(Θ) →∞.



§ NEGATIVE EIGENVALUE LIMITS FOR THEOREM <REF>

The negative eigenvalues are more delicate and require a thorough analysis. We need some stronger tools than those used in the previous section.

Dirichlet eigenvalues satisfy domain monotonicity, where shrinking the domain causes the eigenvalues to increase. Furthermore, for weighted Laplacian operators with positive eigenvalues, decreasing the weight also increases the eigenvalues. For negative Robin eigenvalues, these phenomena tend to reverse. That is, the Robin eigenvalues have a tendency to decrease (become more negative) for these processes, letting us derive tractable lower bounds on the eigenvalues.

To be precise, we consider the weighted eigenvalue problem


    {    - u = η w u in Ω, 
       ∂ u/∂ n + α u = 0 on Σ,
       ∂ u/∂ n = 0 on ∂Ω∖Σ,
    .

where Ω is a bounded domain with piecewise smooth boundary, Σ⊂∂Ω is a piecewise smooth, non-empty portion of the boundary, and w is some smooth positive weight function on Ω. Then, η_k(Ω, Σ, α, w) := η_k forms a discrete spectrum

    η_1 < η_2 ≤η_3 ≤⋯

(In the case Σ = ∂Ω, we recover a standard weighted Robin problem.) We will typically suppress notation when it is clear from context. If α and w are clear from context, or are unchanging, then we usually denote the eigenvalues as η_k(Ω, Σ). If the Robin boundary portions are clear from context, or are unchanging, then we usually write η_k(Ω, w). 

The weighted Robin-Neumann eigenvalue problem allows us to control negative eigenvalues from below by shrinking the domain or decreasing the weight:


    Fix the Robin parameter α < 0, and let k ≥ 1.
    
    
    
    
  * (Domain monotonicity.) Let Ω_1, Ω_2 ⊂^n be bounded domains with piecewise smooth boundaries. Suppose Σ_1 ⊂∂Ω_1 and Σ_2 ⊂∂Ω_2 are piecewise smooth, non-empty portions of the boundaries and w is positive and smooth on Ω. Assume Ω_1 ⊃Ω_2 and Σ_1 ⊂Σ_2. If η_k(Ω_1, Σ_1) < 0, then 
    
    η_k(Ω_2, Σ_2) ≤η_k(Ω_1, Σ_1) < 0.

    
  * (Weight monotonicity.) Let Ω⊂^n be a bounded domain with piecewise smooth boundary. Suppose w_1, w_2 are two positive smooth weight functions on Ω. Assume w_1 ≥ w_2. If η_k(Ω, w_1) < 0, then 
    
    η_k(Ω, w_2) ≤η_k(Ω, w_1) < 0.


    




We will focus on (i) first.
The variational characterization of (<ref>) reads

    η_k(Ω, Σ) = min_ℒmax_0 ≠ f ∈ℒ∫_Ω | f|^2   dA + α∫_Σ |f|^2   dS/∫_Ω |f|^2 w  dA

where ℒ varies over k-dimensional subspaces of H^1(Ω). Suppose η_k(Ω_1, Σ_1) < 0. We allow ℒ to vary only over k-dimensional subspaces of H^1(Ω_1) such that the quantity

    ∫_Ω_1 | f|^2  dA + α∫_Σ_1 |f|^2   dS

is negative for all f ∈ℒ∖{0}. These subspaces must exist since the eigenvalue is negative. Call the class of such subspaces 𝔑. It follows immediately that if Ω_1 ⊃Ω_2 and Σ_1 ⊂Σ_2 then

    η_k(Ω_1, Σ_1)    = 
    min_ℒ∈𝔑max_0 ≠ f ∈ℒ∫_Ω_1 | f|^2   dA + α∫_Σ_1 |f|^2   dS/∫_Ω_1 |f|^2 w  dA
       ≥min_ℒ∈𝔑max_0 ≠ f ∈ℒ∫_Ω_2 | f|^2   dA + α∫_Σ_2 |f|^2   dS/∫_Ω_2 |f|^2 w  dA≥η_k(Ω_2, Σ_2),

since the denominator is being made less positive and the numerator more negative. 

The proof of (ii) is similar: one restricts to negative numerators, and then the integrals decrease as the weight is decreased.

The above lemma gives a tractable lower bound on negative Robin eigenvalues (perhaps with a weight) by taking Σ_1 = Σ_2 = ∂Ω_2 and then comparing Robin eigenvalues of the disk Ω_1 with Robin-Neumann eigenvalues of an annulus Ω_2 as in Figure <ref>. 


 §.§ Scaling factor 4^2(Θ/2)


Let Θ→π. We will show λ_k(Θ, α/sinΘ)4tan^2(Θ/2) → -∞. Since the eigenvalue is negative for all Θ by Lemma <ref>, it is negative at Θ = 0 with eigenfunction span {f_1, …, f_k}. Computing (<ref>), we have the upper bound

    λ_k(Θ, α/sinΘ)4tan^2(Θ/2) ≤max_f ∈span{f_1, …, f_k}∫_ | f|^2  dA + α∫_∂ f^2  dS/∫_ |f|^2 (1+R^2r^2)^-2 dA.

The numerator is negative for all linear combinations f by negativity of the eigenvalue at R = 0. Additionally, the only dependency of this expression on R is in the denominator, and tending R →∞ (i.e., Θ→π) causes the denominator to decrease to 0 for all r > 0. Thus, the left-hand side must go to -∞.

Now let Θ→ -∞. To prove λ_k(Θ, α/sinh|Θ|)4tanh^2(Θ/2) → 0, it is sufficient to show this is the case for the first eigenvalue. Since the Robin ground state is radial on the geodesic disk, the variational characterization (<ref>) can be written (recalling R = tanh(|Θ|/2))

    λ_1(Θ, α/sinh|Θ|)4tanh^2(Θ/2) = min_g ∈ C^1([0,1])∫_0^1 g'(r)^2 r  dr +  α g(1)^2/∫_0^1 g(r)^2 (1-R^2r^2)^-2r  dr

where g varies over the space of C^1 functions on [0,1].
We may assume the numerator is negative since the eigenvalue is. From Lemma <ref>, we may shrink the domain from [0,1] to [R,1] and decrease the variational characterization. We thus see that

    λ_1(Θ, α/sinh|Θ|)4tanh^2(Θ/2) ≥min_g ∈ C^1([R,1])∫_R^1 g'(r)^2 r  dr +  α g(1)^2/∫_R^1 g(r)^2 (1-R^2r^2)^-2r  dr.

By making the numerator more negative and the denominator less positive, we further deduce

    ∫_R^1 g'(r)^2 r  dr + α g(1)^2/∫_R^1 g(r)^2 (1-R^2r^2)^-2r  dr≥∫_R^1 g'(r)^2 R  dr +  α g(1)^2/∫_R^1 g(r)^2 (1-R^4)^-2R  dr,

where we used that r ≥ R and (1-R^2r^2)^-2≥ (1-R^4)^-2 for r ∈ [R,1].
After performing the linear rescaling r ↦r-1/R-1 and doing some algebra on the R-terms, we conclude that

    λ_1(Θ, α/sinh|Θ|)4tanh^2(Θ/2) ≥(1-R^4)^2/(1-R)^2 min_g ∈ C^1([0,1])∫_0^1 g'(r)^2   dr + (1-R)/Rα g(0)^2/∫_0^1 g(r)^2  dr.

The variational characterization on the right-hand side represents the lowest eigenvalue of a mixed Robin-Neumann problem on the interval [0,1] with Robin parameter (1-R)α/R. By letting R → 1 (which is Θ→ -∞), the parameter tends to 0 and the eigenvalue converges to the lowest pure Neumann eigenvalue, which is 0. (One can justify that this limit is accurate by bounding below by a pure Robin problem, or by formally letting R → 1 in the Rayleigh quotient.) Since (1-R^4)^2/(1-R)^2→ 16, we conclude that λ_1(Θ, α/sinh |Θ|)4tanh^2(Θ/2) → 0 and the claim follows.



 §.§ Scaling factor ^2(Θ)


The case Θ→π is long and involved, and we place it in the next section

Let Θ→ -∞. To see that λ_k(Θ, α/sinh|Θ|)sinh^2(Θ) → -∞, let f_1, …, f_k be the first k Robin eigenfunctions for the Euclidean unit disk (Θ = 0). By (<ref>), we have the upper bound (recalling sinh|Θ| = 2R/(1-R^2))

    λ_k(Θ, α/sinh|Θ|)sinh^2(Θ)≤max_|c| = 1 ∑_j=1^k c_j^2( ∫_ | f_j|^2   dA + α∫_∂ |f_j|^2 dS)/∑_j=1^k c_j^2  ∫_ |f_j|^2 ( 1-R^2/1-R^2r^2)^ 2 dA,

where c = (c_1, …, c_k) is the coefficient vector. The numerator is independent of R and negative, so we wish to show the quantities ∫_ |f_j|^2 ( 1-R^2/1-R^2r^2)^ 2 dA tend towards 0 as R → 1. Note that the weight is bounded above by 1 so that each |f_j|^2 (1-R^2/1-R^2r^2)^2 is bounded above by the integrable function |f_j|^2, and the weights tend pointwise to 0 as R→ 1. The claim follows from the dominated convergence theorem.



§ EXOTIC LIMIT TOWARDS EXTERIOR EIGENVALUE


Consider the scaled spherical eigenvalue λ_k(Θ, α/sinΘ)sin^2(Θ) and let Θ→π. In this case, we are claiming the scaled eigenvalue approaches the eigenvalue for a Robin exterior problem. Namely, we define the eigenvalue problem

    {    - u = λ_k^ext(, α)u    in ℝ^2 ∖𝔻, 
       ∂_ν u - α u = 0 on ∂𝔻,
        .

where ∂_v is the inner normal derivative, i.e., pointing into ^2 ∖. We call λ_k^ext(, α) the k^th exterior Robin eigenvalue, and we claim for negative eigenvalues that 

    λ_k(Θ, α/sinΘ)sin^2(Θ) →λ_k^ext(, α)

as Θ→π.

At first glance, the eigenvalue problem does not seem well-posed since the embedding H^1(^2∖) into L^2(^2 ∖) is not compact. Indeed, the problem has essential spectrum equal to [0, ∞) by <cit.>. The negative eigenvalues, however, are discrete and satisfy a variational characterization by <cit.>. If there are k negative eigenvalues, then

    λ_k^ext(, α) = min_ℒmax_0 ≠ f ∈ℒ∫_^2 ∖ | f|^2  dA + α∫_∂ |f|^2  dS/∫_^2 ∖ |f|^2  dA

where ℒ varies over k-dimensional subspaces of H^1(^2∖).

Consider the variational characterization (<ref>) of the spherical eigenvalue with scaling factor sin^2(Θ) = 4R^2/(1+R^2)^2. After one performs the conformal map z ↦ 1/z on the Rayleigh quotient in (<ref>), the variational characterization reads

    λ_k(Θ, α/sinΘ)sin^2(Θ)= min_ℒmax_0 ≠ f ∈ℒ∫_^2 ∖ | f|^2  dA + α∫_∂ |f|^2  dS/∫_^2 ∖ |f|^2 ( 1+R^2/r^2+R^2)^ 2  dA

where ℒ now varies over k-dimensional subspaces of the function space

    H_ext(^2 ∖) := { f : ∫_^2 ∖ |f|^2 |z|^-4  dA + ∫_^2 ∖ | f|^2   dA < ∞}.

The function space H_ext(^2 ∖) is not the same as H^1(^2 ∖).  Note  that  H^1(^2 ∖) ⊂ H_ext(^2 ∖).

Formally, if one takes the variational characterization (<ref>) and lets R →∞, without regard for interchanging limits or different function spaces, the variational characterization (<ref>) is recovered. Of course, we do care about interchanging limits and different function spaces, and so we must proceed quite delicately.

There are quite a few steps to the proof, and so we give a brief overview of the attack plan:


    
  * Show that the number of discrete negative eigenvalues is the same for both problems, so that the convergence makes sense. 

    
  * Bound the spherical eigenvalues above by the exterior eigenvalues, and below by the eigenvalues of a problem related to the exterior problem, namely a Robin-Neumann problem on a large annulus.

    
  * Justify that separation of variables applies to the exterior and annular eigenvalues, and reduce the problem to convergence of their respective radial modes.

    
  * Prove the radial modes of the annular eigenvalues converge to the radial modes of the exterior eigenvalues as the outer radius of the annulus goes to ∞, by analyzing the implicit eigenvalue equations. 

Each step is proven in their respectively numbered subsections: <ref>, <ref>, <ref>, <ref>. 



 §.§ Step (<ref>): counting negative eigenvalues

We first show that the dimension of the negative eigenspace is equal for both the exterior problem and the spherical cap problem, and thus there is a variational characterization to converge to as R →∞. We computed the dimension of the negative eigenspace for the spherical cap previously in Lemma <ref>. The following lemma shows the dimension remains the same for the exterior problem.


   If α∈ [-m-1, -m) where m is a non-negative integer, then there exist 2m+1 negative discrete exterior eigenvalues λ_k^ext(, α), k = 1, …, 2m+1.


Our proof will be a modified version of <cit.>. Denote the exterior Laplacian with parameter α as -_α^ext. By <cit.>, -_α^ext is diagonalized into an orthogonal sum of fiber operators 𝖧_α, j acting on L^2((1, ∞), r  dr ) for each j ≥ 0. Each 𝖧_α, j is of the form

    𝖧_α, j f:= -f”(r) - f'(r)/r + j^2f(r)/r^2, 
    dom 𝖧_α, j = {f : f, f” + f'/r∈ L^2((1,∞), r   dr), f'(1) = α f(1)}.

Let λ_1(𝖧_α, j) be the infimum of the spectrum of 𝖧_α, j. By <cit.>, there exists a constant j_*(α) such that λ_1(𝖧_α, j) is negative and a discrete eigenvalue of -_α^ext if and only if 0 ≤ j ≤ j_*(α). Furthermore, by <cit.>, each 𝖧_α, j has at most one negative eigenvalue, λ_1(𝖧_α, 0) is a simple eigenvalue of -_α^ext, and λ_1(𝖧_α, j) is a double eigenvalue for 1 ≤ j ≤ j*(α). The exterior problem thus has exactly 2j_*(α) + 1 negative eigenvalues and so our goal is to show j_*(α) = m.

If α < 0, the two eigenvalues λ_1^ext(, α) and λ_1(Θ, α/ sinΘ) are both simple and negative. Our goal is then to detect when the exterior problem begins to accrue further negative eigenvalues. Let j ≥ 1. Modifying <cit.>, the operator 𝖧_α, j has a negative eigenvalue if and only if the system

    {    -f”(r) - f'(r)/r + j^2 f(r)/r^2 = σ f(r)    in  (1, ∞), 
        f'(1) = α f(1),
        .

has a non-trivial eigenfunction in L^2((1, ∞), r   dr) with eigenvalue σ < 0. Using <cit.> and the fact an eigenfunction of the above system belongs to L^2((1,∞), r dr), we obtain the solution

    f(r) = K_j(μ r)

where K_j is a modified Bessel function of the second kind and μ := √(-σ). Thus, σ is a discrete negative eigenvalue of 𝖧_α, j if and only if μ satisfies the Robin boundary condition at r = 1:

    μ K_j'(μ) = α K_j(μ).
 
By using the recurrence K_j'(z) = -K_j-1(z) - j/zK_j(z) <cit.>, we may rewrite the equation as

    -μ K_j-1(μ)/K_j(μ) = α + j.

Define

    h_j(x) := K_j-1(x)/K_j(x)

so that (<ref>) reads -xh_j(x) = α + j. By <cit.> and <cit.> we have the bounds

    x/j - 1/2 + √((j- 1/2)^2 + x^2) < h_j(x) ≤ 1

    for x > 0, j ≥ 1. It follows that x h_j(x) tends to 0 as x → 0 and tends to ∞ as x →∞. We want to show that xh_j(x) is strictly increasing so that the solution of (<ref>) is unique. From <cit.>, we see h_j(x) is strictly increasing in a neighborhood of the origin. The proof of <cit.> shows h_j'(x) > 0 for all x > 0. It follows that xh_j(x) is strictly increasing.

We conclude (<ref>) has a unique positive solution μ if and only if α + j < 0. Since α∈ [-m-1, -m), the condition α + j < 0 means j ≤ m and so j_*(α) = m. Hence, the dimension of the negative eigenvalues on the spherical cap and on the exterior of the unit disk are equal in the case α∈ [-m-1, -m). 


We have managed to show step (<ref>), justifying that the spherical eigenvalues can converge to the exterior ones. Now, we must show that they do converge. We start with some upper and lower bounds.



 §.§ Step (<ref>): upper and lower bounds.

Henceforth, we assume k ≤ 2m+1 so that λ_k(Θ, α) and λ_k^ext(, α) are both negative. Perform the conformal map z ↦ 1/z on (<ref>) to yield the characterization (<ref>) of λ_k(Θ, α/sinΘ)sin^2(Θ). Observe in (<ref>) that the weight in the denominator is bounded by 1, and we may assume the numerator is negative. Furthermore, we may restrict ℒ to vary over k-dimensional subspaces of H^1(^2 ∖) rather than H_ext(^2 ∖) (defined in (<ref>)), in light of the inclusion H^1(^2 ∖) ⊂ H_ext(^2 ∖). Hence, we obtain  the upper bound

    lim_Θ→πλ_k(Θ, α/sinΘ)sin^2(Θ) ≤λ_k^ext(, α) < 0.


We cannot directly bound below by the exterior eigenvalues nor take the limit of (<ref>) as R →∞. Instead, we will chop off the unbounded portion of the domain and consider a large annulus with a Robin boundary on the inner radius and Neumann boundary on the outer annulus. We are able to bound the scaled spherical eigenvalue below by these annular eigenvalues.  

Define A(M) as the annulus of inner radius 1 (the Robin portion) and outer radius M (the Neumann portion), and define w as the weight (1+R^2/r^2+R^2)^ 2.
By reworking Lemma <ref> slightly for unbounded domains and different function spaces, we see that

    λ_k(Θ, α/sinΘ)sin^2(Θ)     = min_ℒmax_0 ≠ f ∈ℒ∫_^2 ∖ | f|^2  dA + α∫_∂ |f|^2  dS/∫_^2 ∖ |f|^2 ( 1+R^2/r^2+R^2)^ 2  dA
       ≥min_ℒmax_0 ≠ f ∈ℒ∫_A(M) | f|^2  dA + α∫_∂ |f|^2  dS/∫_A(M) |f|^2 ( 1+R^2/r^2+R^2)^ 2  dA

where ℒ varies over k-dimensional subspaces of H_ext(^2 ∖). The key point here is that if f is in H_ext(^2 ∖), the restriction of the function to A(M) must be in L^2(A(M)) since |z|^4 is a bounded function on the annulus. That is, when we restrict the function space to exist on A(M), we can revert to the standard H^1 space.
We thus generate the lower bound

    λ_k(Θ, α/ sinΘ) sin^2(Θ) ≥η_k(A(M), ∂, α, w).

Since the weight w converges uniformly to 1 on the annulus, taking R →∞ yields the bound

    lim_Θ→πλ_k(Θ, α/sinΘ) sin^2(Θ) ≥η_k(A(M), ∂,  α, 1).

For the right-hand side, we will suppress notation and denote the unweighted Robin-Neumann eigenvalue on the annulus as η_k(M, α). Take care to note the inner radius at R = 1 has the Robin boundary condition for this eigenvalue problem. Again, using Lemma <ref>, this eigenvalue is an increasing function in M. By combining (<ref>) and (<ref>), we see the limit lim_M →∞η_k(M, α) is a number η_k^*(α) < 0. We will show η_k^*(α) = λ_k^ext(, α).

Step (<ref>) is now complete. To analyze further, we will look at the implicit equation that characterizes the exterior eigenvalues and the one that characterizes the annular eigenvalues. In order to do this, we will separate variables and reduce our analysis to the radial modes. The radial modes satisfy certain well-behaved implicit equations characterizing the eigenvalues completely, and we will show uniform convergence of these equations.



 §.§ Step (<ref>): reduction to the radial modes


First, we state properties of the exterior eigenvalues. Observe the domain ^2 ∖ is radially symmetric. Then, referring to Subsection <ref>, we separate variables into polar coordinates and find the angular components of the modes are cos(jϕ), sin(jϕ) and the radial components are found by solving the modified Bessel ODE system


    {    -f”(r) - f'(r)/r + j^2 f(r)/r^2 = σ f(r)    in  (1, ∞), 
        f'(1) = α f(1). 
    .

For α∈ [-m-1, -m), the system (<ref>) has a unique eigenvalue σ_j(α) < 0 for each 0 ≤ j ≤ m from the proof of Lemma <ref>, and by <cit.>, we have that σ_k(α) < σ_j(α) < 0 if 0 ≤ k < j ≤ m.  The eigenvalue σ_j(α) has associated eigenfunction

    K_j(μ_j(α) r)

where μ_j(α) := √(-σ_j(α)). So, separation of variables is quite quickly seen to be justified in the exterior case.

For annular eigenvalues, there is more work to be done, since less literature exists on this subject. Freitas and Krejčiřík provide a treatment of the first Robin eigenvalue with negative parameter on disks and annuli in <cit.>. Here, we will be considering all negative eigenvalues, including the first. 

Rotational symmetry of the annulus again allows us to separate variables into polar coordinates, for the mixed Robin-Neumann problem. The angular components are of the form cos(j ϕ) and sin (j ϕ) while the radial components are found by solving the modified Bessel ODE system


    {    -f”(r) - f'(r)/r + j^2 f(r)/r^2 = ρ f(r)    in  (1, M), 
        f'(1) = α f(1), 
        f'(M) = 0.
        .

We need to prove for each j = 1, …, m that the system possesses a unique eigenvalue ρ_j(M, α) < 0 satisfying the eigenfunction equation and boundary conditions. Furthermore, we also want to prove that increasing j increases the eigenvalue ρ_j(M,α). After proving these properties, we can justify separation of variables in the annular case.


    Let α∈ [-m-1, -m) where m is a non-negative integer. If 0 ≤ j ≤ m, then there exists a unique eigenvalue ρ_j(M,α) < 0 of system (<ref>). Furthermore, if 0 ≤ k < j ≤ m, then ρ_k(M,α) < ρ_j(M, α) < 0.


System (<ref>) induces a discrete spectrum of eigenvalues

    ρ_j,1(M, α) < ρ_j,2(M,α) ≤ρ_j,3(M,α) ≤⋯→∞

and we claim ρ_j,1(M,α) < 0 while ρ_j,2(M,α) ≥ 0. 

The variational characterization for the lowest eigenvalue reads

    ρ_j,1(M,α) = min_f ≠ 0∫_1^M (f'(r)^2 + j^2/r^2f(r)^2) r  dr + α f(1)^2/∫_1^M f(r)^2r  dr

where f varies over H^1([1,M], r dr). Furthermore, also consider the variational characterization for the lowest eigenvalue σ_j(α) of System (<ref>):

    σ_j(α) = min_f ≠ 0∫_1^∞(f'(r)^2 + j^2/r^2f(r)^2) r  dr + α f(1)^2/∫_1^∞ f(r)^2r  dr

where f varies over H^1([1, ∞], r  dr). 
As α∈ [-m-1, -m) by assumption, the eigenvalue σ_j(α) is negative from the proof of Lemma <ref>, so we may restrict its variational characterization to functions f such that the numerator is negative. By reducing the domain of integration in the numerator and denominator (such as in Lemma <ref>), we see immediately that

    ρ_j,1(M, α) ≤σ_j(α) < 0.


We show non-negativity for the rest of the spectrum. Let f_1 and f_2 be eigenfunctions for ρ_j,1(M,α) and ρ_j,2(M,α) respectively. Since f_1 cannot change sign on [1,M] and f_2 is orthogonal to f_1 in L^2([1, M], r dr), we see that f_2 must change sign at some point x^* ∈ (1,M). Then, the restriction of f_2 to the interval [x^*, M] is an eigenfunction of the system 

    {    -f”(r) - f'(r)/r + j^2 f(r)/r^2 = ρ f(r)    in  (x^*, M), 
        f(x^*) = 0, 
        f'(M) = 0.
        .
 
The system above represents an eigenvalue problem for a mixed Dirichlet-Neumann operator on the interval [x^*, M]. The variational characterization for this system implies the operator's eigenvalues must be non-negative, and it follows that ρ_j,2(M,α) ≥ 0. Hence, ρ_j,1(M, α) is the unique negative eigenvalue of (<ref>), and we denote it ρ_j(M, α).

By comparing the variational characterizations (<ref>), we see ρ_k(M, α) < ρ_j(M,α) if k < j and the claim follows.

Note that Lemma <ref> is not stating there are exactly 2m+1 negative eigenvalues, rather that there are at least 2m+1 negative eigenvalues. 

For the exterior and annular radial modes, we see that increasing the parameter j increases both eigenvalues σ_j(α) and ρ_j(M, α). Furthermore, for j ≥ 1, each of the eigenvalues is repeated twice in the unseparated form of their respective PDE systems due to the existence of the two angular components cos(jϕ) and sin(jϕ). These facts, along with σ_j(α) and ρ_j(M,α) being unique negative eigenvalues for the separated Bessel ODEs, imply that

    σ_j(α)    = λ_2j^ext(, α) = λ_2j+1^ext(, α) , 
    ρ_j(M, α)    = η_2j (M, α) = η_2j+1(M, α),

for 1 ≤ j ≤ m, where we recall λ_k^ext(, α) is the k^th exterior eigenvalue and η_k(M,α) is the k^th Robin-Neumann eigenvalue on the annulus A(M). For j = 0, the exterior and annular eigenvalues are simple and radial so that they coincide with their radial modes. Thus, proving lim_M→∞η_k(M, α) = λ_k^ext(, α), 0 ≤ k ≤ 2m+1, is equivalent to proving the radial modes converge:

    lim_M →∞ρ_j(M, α) = σ_j(α),    0 ≤ j ≤ m.


We have successfully completed step (<ref>) of the proof. We will finish the final step of the proof by showing the implicit equations corresponding to the radial eigenvalues converge uniformly.



 §.§ Step (<ref>): convergence of the radial modes
 By Subsection <ref>, μ_j(α) = √(-σ_j(α)) is the unique root in (0, ∞) of the implicit function

    A(x, α) := x K_j'(x) - α K_j(x).

Similarly to how Equation (13) is derived in <cit.>, we find ω_j(M, α) := √(-ρ_j(M, α)) is the unique root in (0, ∞) of the implicit function

    G_M(x, α) := A(x, α)D_M(x, α) - B(x, α)C_M(x, α)

with coefficients given in terms of the modified Bessel functions I_j and K_j as

    A(x, α)    := x K_j'(x) - α K_j(x), 
    
        B(x, α)    := x I_j'(x) - α I_j(x), 
    
        C_M(x, α)     := x K_j'(Mx), 
    
        D_M(x, α)     := x I_j'(Mx).

(Despite the lack of dependence of C_M and D_M on α, we keep the notation for consistency.)

Since D_M is non-vanishing on (0, ∞), we may divide G_M by D_M and preserve the location of the unique root of G_M. Furthermore, recall that η_k(M,α) is increasing as a function of M and bounded above by λ_k^ext(, α) (see the paragraph after Equation (<ref>)). It follows that ω_j(M, α) is decreasing as a function of M and bounded below by μ_j(α) > 0, and so we may restrict our analysis to the compact interval 

    S = [μ_j(α)/2, ω_j(M_0, α)]
where M_0 is large and finite. We will show that
G_M(x,α)/D_M(x, α) converges uniformly to A(x,α) on the interval S as M →∞, or, equivalently:

    -B(x, α)C_M(x, α)/D_M(x, α) 0   on  S  as M →∞.

Since the convergence is uniform, the roots ω_j(M, α) must converge to a root of A(x,α), and as A(x, α) has μ_j(α) ∈ S as its unique root, we will be able to conclude the proof.



By <cit.>, we have the recurrence relations


    I_j' = 1/2(I_j-1 + I_j+1),         -K_j' = 1/2(K_j-1 + K_j+1),         (j ≥1)

    I_0' = I_1,         -K_0' = K_1,         (j = 0).

From these relations we see that I_j' increases strictly from 0 to ∞ (or 1/2 to ∞ if j = 1) and -K_j' decreases strictly from ∞ to 0. Thus, it follows the ratio

    -C_M(x,α)/D_M(x,α) = -K_j'(Mx)/I_j'(Mx)

is positive and decreases to 0 as M increases. Furthermore, the ratio achieves its maximum value at the left endpoint of S for each fixed M. Since B is positive and does not depend on M, we deduce that

    sup_x ∈ S-B(x,α)C_M(x,α)/D_M(x,α)≤ B(μ_j(α)/2, α) -K_j'(M μ_j(α)/2)/I_j'(M μ_j(α)/2)→ 0   as  M →∞.

It follows that the convergence is uniform, and we conclude that

    lim_Θ→πλ_k(Θ, α/sinΘ)sin^2(Θ) = lim_M →∞η_k(M, α) = λ_k^ext(, α).




§ HIGHER-DIMENSIONAL DIRICHLET MONOTONICITY — THEOREM 1.3

For this section, we jump up to dimensions three and higher, and the reader needs to keep in mind we now analyze higher-dimensional Dirichlet eigenvalues, not two-dimensional Robin eigenvalues. One may find it helpful to refer to Appendix <ref> for a refresher on the Laplace–Beltrami operator in our local coordinate system. The techniques here are quite different from the proofs of Theorems <ref> and <ref>, as we use direct trial functions instead of conformal maps. 

There are three main steps in the proof of Theorem <ref>:

    
  * Transform an eigenspace on C(Θ_1) to a trial subspace on C(Θ_2).

    
  * Rescale the Rayleigh quotient on C(Θ_2) back to C(Θ_1), incurring the cost of certain coefficient functions.

    
  * Bound the coefficient functions and recover the standard Rayleigh quotient for λ_k(Θ_1).

The main point of interest in the proof is in the first step, where the transformation is performed to preserve the area elements of C(Θ_1) and C(Θ_2). We will first consider the spherical case corresponding to Θ > 0, as the proof is slightly more demanding there.



 §.§ Spherical case

We first note that λ_k(Θ) is a positive decreasing function of Θ by Dirichlet domain monotonicity, and as sin^2 decreases strictly on [π/2, π] and is positive, we need only show that λ_k(Θ)sin^2(Θ) is decreasing strictly on the  interval (0, π/2). 

For any Θ we have the variational characterization.

    λ_k(Θ) = min_ℒmax_0 ≠ f ∈ℒ∫_C(Θ) |_^n f|^2   dV /∫_C(Θ)  |f|^2   dV  =: min_ℒmax_0 ≠ f ∈ℒℛ_Θ(f)

where ℒ varies over k-dimensional subspaces of H_0^1(C(Θ)). In coordinates, the Rayleigh quotient takes the form

    ℛ_Θ(f) =∫_^n-1∫_0^Θ(|∂_1 f|^2 + 1/sin^2θ|_^n-1 f|^2)sin^n-1θ  dθ dξ/∫_^n-1∫_0^Θ |f|^2sin^n-1θ  dθ dξ,

where we write ∂_1 f to denote the derivative in the first component, the aperture coordinate. Since we frequently pass between different spherical caps, this notation is employed for clarity. 

Let 0 < Θ_1 < Θ_2 ≤π/2. Our goal is to show λ_k(Θ_1)sin^2(Θ_1) > λ_k(Θ_2)sin^2(Θ_2). Let {ϕ_1, …, ϕ_k} be the first k Dirichlet eigenfunctions on C(Θ_1). Call the subspace generated by these functions 𝒦. We want to extend these functions suitably to C(Θ_2) to be trial functions in the Rayleigh quotient for λ_k(Θ_2). It is unclear what such an extension ϕ̃_j should be. A simple linear extension in the θ variable would not interact well with the Rayleigh quotient. The volume element would transform into sin^n-1(Θ_1θ/Θ_2) in both the numerator and denominator and not cancel out with any factors, crushing our hopes of proving monotonicity. However, if we normalize the trial function by the volume element, we can defeat this obstacle. We are thus motivated to define the following transformation: let f ∈ H_0^1(C(Θ_1)) and define

    f̃ (θ, ξ) := f(Θ_1 θ/Θ_2, ξ) √(sin^n-1(Θ_1 θ/Θ_2))/√(sin^n-1(θ))∈ H_0^1(C(Θ_2)).

(At the origin, the ratio of sines is understood as its limit lim_θ→ 0sin(Θ_1 θ/Θ_2)/sin(θ) = Θ_1/Θ_2.) In essence, this transformation is the spherical equivalent of linear dilation in the plane. Linear dilation in the plane stretches a function uniformly to fit into a larger domain while only incurring a constant multiplicative cost to the L^2 norm. Here one sees the same phenomenon and visualizes it suitably adjusted to a spherical cap scenario. One stretches the function along the spherical cap as one might stretch rubber, while scaling the L^2 norm by a constant, with 

    ∫_C(Θ_2) |f̃|^2   dV = Θ_2/Θ_1∫_C(Θ_1) |f|^2   dV.

We will state the slightly stronger lemma:

Let 0 < Θ_1 < Θ_2 ≤π/2 and f ∈ H_0^1(C(Θ_1)). Then, as defined above, f̃∈ H_0^1(C(Θ_2)). Further, if 𝒦 = span{f_1, …, f_k} is linearly independent in H_0^1(C(Θ_1)), then 𝒦̃ := span{f̃_̃1̃, …, f̃_̃k̃} is linearly independent in H_0^1(C(Θ_2)), and the map f ↦f̃ is linear.


The calculations done below resulting in Equation (<ref>) show that _^nf̃_L^2(C(Θ_2))^2 is bounded above by sin^2(Θ_1)/sin^2(Θ_2)f̃_L^2(C(Θ_2))^2ℛ_Θ_1(f). One can also clearly see f̃_L^2(C(Θ_2))^2 is finite. The linear independence and linearity are trivial and stated for completeness.

Consider the trial functions on C(Θ_2) defined to be ϕ̃_j for 1 ≤ j ≤ k, where the ϕ_j are Dirichlet eigenfunctions on C(Θ_1). The ϕ̃_j are linearly independent, so let their span be our trial subspace 𝒦̃ in the variational characterization for λ_k(Θ_2). The maximizing function is thus achieved for some linear combination g̃ = ∑ a_j ϕ̃_j so that

    λ_k(Θ_2) ≤∫_^n-1∫_0^Θ_2(|∂_1 g̃|^2 + 1/sin^2θ|_^n-1g̃|^2)sin^n-1(θ)   dθ dξ/∫_^n-1∫_0^Θ_2 |g̃|^2sin^n-1(θ)   dθ dξ.

Define g := ∑ a_j ϕ_j ∈ H_0^1(C(Θ_1)) and note that 

    |_^n-1g̃|^2 = sin^n-1(Θ_1 θ/ Θ_2)/sin^n-1θ |_^n-1g|^2

as the transformation only affects θ and not ξ. After the linear change of variables θ = Θ_2 t /Θ_1, we find an upper bound for λ_k(Θ_2) of


    ∫_^n-1∫_0^Θ_1(|(∂_1 g̃)(Θ_2 t/Θ_1, ξ)|^2 sin^n-1(Θ_2 t/Θ_1)/sin^n-1(t)+ 1/sin^2(Θ_2 t/Θ_1)|_^n-1 g(t, ξ)|^2)sin^n-1(t)   dt dξ/∫_^n-1∫_0^Θ_1 |g(t, ξ)|^2sin^n-1(t)   dt dξ.


Our goal is to compare this last displayed formula with a Rayleigh quotient over C(Θ_1) involving the function g. A routine calculation shows in the numerator that

    |(∂_1 g̃)(Θ_2 t/Θ_1, ξ)|^2 sin^n-1(Θ_2 t/ Θ_1)/sin^n-1(t) = 1/4 Θ_2^2( (  n-1)^2( (t)Θ_1 -   ( Θ _2 t/Θ_1)Θ_2 )^2g(t, ξ)^2 
     + 4Θ_1^2∂_1 g(t, ξ)^2  + 4 (n-1)Θ_1 ( (t) Θ_1 -  (Θ_2 t/ Θ_1)Θ_2 )g(t, ξ) ∂_1 g(t, ξ) ).

At this point, we would like to perform a direct comparison of Rayleigh quotients, but g∂_1 g need not be positive. So, we write 2g∂_1 g = (g^2)_1 and integrate by parts. The boundary terms vanish by the Dirichlet boundary conditions and the fact that sin^n-2(t)  (t) = 0 when t = 0 and n ≥ 3.

We thus obtain an upper bound for λ_k(Θ_2)sin^2(Θ_2) (observe here we multiply the Rayleigh quotient by the scaling factor sin^2(Θ_2)) of

    λ_k(Θ_2)sin^2(Θ_2)≤∫_^n-1∫_0^Θ_1( P|∂_1 g|^2 +  Q|_^n-1g|^2 + R|g|^2 )sin^n-1(t)  dt dξ/∫_^n-1∫_0^Θ_1 |g|^2 sin^n-1(t)   dt dξ

where the coefficient functions are

    P :=  sin^2(Θ_2)Θ_1^2/Θ_2^2, 
    
    Q:= sin^2(Θ_2)/sin^2(Θ_2 t / Θ_1), 
    
    R :=   sin^2(Θ_2)(n-1)/4 Θ _2^2(Θ_2^2
       ((n-1)  ^2(Θ_2 t/Θ _1)-2 
       ^2(Θ _2 t/Θ _1)) 
     -Θ _1^2 ((n-1)  ^2(t) - 2  ^2(t)) ).

We wish to recover the original form of the Rayleigh quotient by bounding P, Q, and R above by the proper quantities. Namely, we want to show P is bounded by sin^2(Θ_1), Q is bounded by sin^2(Θ_1)/sin^2(t), and R is non-positive.



  §.§.§ Bound on P

As sin^2(x)/x^2 is a strictly decreasing function up to π, we readily see that P < sin^2(Θ_1).



  §.§.§ Bound on Q

Consider the function sin^2(x)/sin^2(x t/Θ_1) where t < Θ_1. By taking a derivative in x, we see the function is strictly decreasing for x ∈ (0, π/2) if the inequality tan(tx/Θ_1) < t/Θ_1tan(x) holds. As tan(x)/x is a strictly increasing function, it follows that Q < sin^2(Θ_1)/sin^2(t).



  §.§.§ Bound on R

We will show that R is negative. Since sin^2(Θ_2)(n-1)/4Θ_2^2 is positive, we need only show negativity of the quantity within the parentheses. It is enough to show

    x^2 ((n-1)  ^2(x t/Θ _1)-2 
       ^2(x t/Θ _1))

is a decreasing function for x ∈ (0, π/2). By differentiating with respect to x, we want to show

    (n-1)^2y - (n-3)y y^2 y-2^2 y

is negative for y ∈ (0, π/2). Rewrite the above expression as

    (n-3)^2(y)(1 - y  y) - (n-1).
 
Since y  y < 1 on (0, π/2) and n ≥ 3, the expression is clearly negative and (<ref>) is decreasing.

With these bounds on the coefficient functions, it follows that

    λ_k(Θ_2)sin^2(Θ_2) < ℛ_Θ_1(g)sin^2(Θ_1).

Since g ∈span{ϕ_1, …, ϕ_k}, we know ℛ_Θ_1(g) ≤λ_k(Θ_1)
and thus λ_k(Θ)sin^2(Θ) is a strictly decreasing function on (0, π/2). 



 §.§ Hyperbolic case


The proof in this case is virtually identical. Hyperbolic trig functions replace their Euclidean counterparts.

For Θ < 0, the Rayleigh quotient is

    ℛ_Θ(f) =∫_^n-1∫_0^|Θ|(|∂_1 f|^2 + 1/sinh^2θ|_^n-1 f|^2)sinh^n-1θ  dθ dξ/∫_^n-1∫_0^|Θ| |f|^2sinh^n-1θ  dθ dξ.


Let Θ_1 < Θ_2 < 0. We wish to show that λ_k(Θ_1)sinh^2(Θ_1) > λ_k(Θ_2)sinh^2(Θ_2). We rework Lemma <ref> and define

    f̃ (θ, ξ) := f(|Θ_1 |θ/|Θ_2|, ξ) √(sinh^n-1(|Θ_1| θ/|Θ_2|))/√(sinh^n-1(θ)).


Let Θ_1 < Θ_2 < 0 and f ∈ H_0^1(C(Θ_1)). Then, f̃∈ H_0^1(C(Θ_2)) and the mapping f ↦f̃ is linear and preserves linear independence.

The proof of the theorem proceeds the same, making necessary modifications, and we arrive at the bound 

    λ_k(Θ_2)sinh^2(Θ_2) ≤∫_^n-1∫_0^|Θ_1|( P|∂_1 g|^2 +  Q|_^n-1g|^2 + R|g|^2 )sinh^n-1(t)  dt dξ/∫_^n-1∫_0^|Θ_1| |g|^2 sinh^n-1(t)   dt dξ

where the new hyperbolic coefficient functions are

    P :=  sinh^2(|Θ_2|)|Θ_1|^2/|Θ_2|^2, 
    
    Q:= sinh^2(|Θ_2|)/sinh^2(|Θ_2| t / |Θ_1|), 
    
    R :=   sinh^2(|Θ_2|)(n-1)/4 |Θ _2|^2(|Θ_2|^2
       ((n-1)  ^2(|Θ_2| t/|Θ _1|)-2 
       ^2(|Θ| _2 t/|Θ| _1))
    -|Θ _1|^2 ((n-1)  ^2(t) - 2  ^2(t)) ).

The bounds on these coefficient functions follow from the fact that all of sinh^2(x)/x^2, sinh^2(x)/sinh^2(xt/|Θ_1|) and y^2 ((n-1)  ^2y -2 
^2y)) are increasing functions on (0, ∞), proven similarly as in the spherical case. The rest of the proof is identical, and we conclude that λ_k(Θ)sinh^2(Θ) decreases strictly on (-∞, 0).




 §.§ Continuity and limits as Θ→ -∞, 0, π

Continuity away from Θ = 0 follows from the fact λ_k(Θ), sin^2(Θ) and sinh^2(Θ) are all continuous. Near Θ = 0, sinΘ is well-approximated by Θ, and so the Rayleigh quotient for λ_k(Θ)sin^2(Θ) is uniformly approximated near Θ = 0 by

    Θ^2 ∫_^n-1∫_0^Θ(|∂_1 f|^2 + 1/θ^2|_^n-1 f|^2)θ^n-1  dθ dξ/∫_^n-1∫_0^Θ |f|^2θ^n-1  dθ dξ.

If λ_k(^n) is the eigenvalue of the Euclidean Laplacian for ^n, then the above quantity is the Rayleigh quotient for Θ^2 λ_k(Θ^n) in spherical coordinates. By Euclidean scale invariance, Θ^2 λ_k(Θ^n) = λ_k(^n). Apply the corresponding argument to sinh |Θ| and continuity at the origin follows.

When Θ > π/2, then 0 ≤λ_k(Θ)sin^2(Θ) ≤λ_k(π/2)sin^2(Θ) by Dirichlet domain monotonicity and hence λ_k(Θ)sin^2(Θ) → 0 as Θ→π

As Θ→ -∞, it is sufficient to show λ_1(Θ)sinh^2(Θ) →∞. From Borisov and Freitas <cit.>, we have λ_1(Θ) = π^2/Θ^2 + 1 for n=3 and the statement follows easily there. For n ≥ 4, they also provide the lower bound

    λ_1(Θ) ≥j^2_n-2/2, 1/Θ^2 + n(n-1)/6

and again the result is clear.



 

§ ACKNOWLEDGMENTS

The author was supported by National Science Foundation award #2246537 to Richard Laugesen.



§ COORDINATE REPRESENTATIONS




   Spherical       Hyperbolic

 _ ^n = ∂_θê_θ + 1/sinθ_ ^n-1     _^̋n = ∂_θê_θ + 1/sinhθ_ ^n-1 
 [3ex]
_^ n = 1/sin^n-1θ∂_θ(sin^n-1θ ∂_θ) + 1/sin^2θ_ ^n-1     _^̋ n = 1/sinh^n-1θ∂_θ(sinh^n-1θ ∂_θ) + 1/sinh^2θ_ ^n-1 
 [3ex]
dV = sin^n-1θ  dθ dξ     dV = sinh^n-1θ  dθ dξ







§ NUMERICS

The plots in this section accomplish the following:


    
  * Illustrate Theorem <ref> in Figure <ref>.
    
    
  * Numerically support Conjectures <ref>, <ref>, and <ref> in Figures <ref>, <ref>, and <ref> respectively. 
    
    
  * Highlight non-monotonicity of certain eigenvalue functionals in Figures <ref> and <ref>. 

The plots were generated by using the  command in Wolfram Mathematica 14.1.














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AB01
M. S. Ashbaugh and R. D. Benguria.
A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of S^n.
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BKN10
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BKN19
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