OON THE ROBIN SPECTRUM FOR THE HEMISPHERE
ZE´EV RUDNICK AND IGOR WIGMAN
Abstract.
We study the spectrum of the Laplacian on the hemi-sphere with Robin boundary conditions. It is found that the eigen-values fall into small clusters around the Neumann spectrum, andsatisfy a Szeg˝o type limit theorem. Sharp upper and lower boundsfor the gaps between the Robin and Neumann eigenvalues are de-rived, showing in particular that these are unbounded. Further, itis shown that except for a systematic double multiplicity, there areno multiplicities in the spectrum as soon as the Robin parameter ispositive, unlike the Neumann case which is highly degenerate. Fi-nally, the limiting spacing distribution of the desymmetrized spec-trum is proved to be the delta function at the origin. Introduction
The Robin problem.
Let Ω be the upper unit hemisphere (Fig-ure 1), with its boundary ∂ Ω the equator. Our goal is to study theRobin boundary problem on the hemisphere Ω:∆ F + λF = 0 , ∂F∂n + σF = 0where ∂/∂n is the derivative in the direction of the outward pointingnormal to the equator, and σ ≥ σ = 0 or σ = ∞ ) are classical [2, p. 243-244]: The eigenfunctions are restrictionsto Ω of the eigenfunctions on the sphere (spherical harmonics), deter-mined by the parity under reflection in the equator: The odd sphericalharmonics give the Dirichlet eigenfunctions, the even ones give the Neu-mann eigenfunctions. The eigenvalues are thus of the form (cid:96) ( (cid:96) + 1), Date : September 1, 2020.2010
Mathematics Subject Classification.
Primary 35P20, Secondary 37D50,58J51, 81Q50.
Key words and phrases.
Robin boundary conditions, Robin-Neumann gaps,Laplacian, hemisphere, level spacing distribution.This research was supported by the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation programme (Grantagreement No. 786758) and by the Israel Science Foundation (grant No. 1881/20).We are grateful to Nadav Yesha for discussions on various aspects of this project. a r X i v : . [ m a t h . SP ] A ug ZE´EV RUDNICK AND IGOR WIGMAN
Figure 1.
The hemisphere.where (cid:96) ≥ (cid:96) + 1 for theNeumann case, and (cid:96) for the Dirichlet case.The Robin spectrum is significantly less understood, and it is themain object of our interest. The problem admits separation of vari-ables, and there is a basis of eigenfunctions in the form f ν,m = e imφ P mν (cos θ ), m ∈ Z , where P mν ( x ) is an associated Legendre func-tion. For each m , the admissible ν ’s are determined by the boundarycondition.Both f ν,m and f ν, − m share the same Laplace eigenvalue ν ( ν + 1).Therefore the Robin spectrum admits a systematic double multiplic-ity, and we remove it beforehand by insisting that m ≥
0, resultingin a “desymmetrized spectrum”. Let λ n (0) denote the ordered desym-metrized Neumann eigenvalues (repeated with appropriate multiplic-ity), and for σ > λ n ( σ ) the ordered desymmetrizedRobin eigenvalues, and define the Robin-Neumann (RN) gaps by d n ( σ ) := λ n ( σ ) − λ n (0) . These were recently investigated in [9] in the case of planar domains,and will be the main object of study here.1.2.
Clusters.
We show that the desymmetrized Robin spectrum breaksup into small clusters E (cid:96) ( σ ) of size (cid:98) (cid:96)/ (cid:99) + 1, concentrated around theNeumann eigenvalues (cid:96) ( (cid:96) + 1): For each eigenvalue ν ( ν + 1), there issome m ≥
0, and a corresponding eigenfunction e imφ P mν (cos θ ), so thatthe “degree” ν satisfies a secular equation S m ( ν ) = σ , where S m ( ν ) = 2 tan (cid:18) π ( m + ν )2 (cid:19) Γ (cid:0) ν + m + 1 (cid:1) Γ (cid:0) ν − m + 1 (cid:1) Γ (cid:0) ν + m +12 (cid:1) Γ (cid:0) ν − m +12 (cid:1) . OBIN SPECTRUM FOR THE HEMISPHERE 3
For any integer (cid:96) ≥ m of the same parity ( (cid:96) = m mod 2), there isa unique solution ν (cid:96),m ( σ ) in the open interval ( (cid:96), (cid:96) + 1). Denote byΛ (cid:96),m ( σ ) = ν (cid:96),m ( σ )( ν (cid:96),m ( σ ) + 1) the resulting Laplace eigenvalue. Thenthe desymmetrized spectrum consists of Λ (cid:96),m ( σ ), with 0 ≤ m ≤ (cid:96) , and m = (cid:96) mod 2, and is partitioned into disjoint clusters of size (cid:98) (cid:96)/ (cid:99) + 1: E (cid:96) ( σ ) = { Λ (cid:96),m ( σ ) : 0 ≤ m ≤ (cid:96), m = (cid:96) mod 2 } . We denote by d (cid:96),m ( σ ) the Robin-Neumann (RN) gaps in each cluster: d (cid:96),m ( σ ) = Λ (cid:96),m ( σ ) − (cid:96) ( (cid:96) + 1) . We have an asymptotic formula:
Proposition 1.1.
Fix σ > . Let ≤ m ≤ (cid:96) , m = (cid:96) mod 2 . If (cid:96) − m → ∞ then (1.1) d (cid:96),m ( σ ) ∼ σπ (cid:96) + 1 √ (cid:96) − m . We display a plot of these RN gaps in Figure 2.1.3.
A Szeg˝o type limit theorem.
We show, using (1.1), that theRN gaps from each cluster have a limiting distribution, supported onthe ray [4 σ/π, ∞ ): Corollary 1.2.
Fix f ∈ C ∞ c (0 , ∞ ) . As (cid:96) → ∞ , E (cid:96) ( σ ) (cid:88) λ n ( σ ) ∈E (cid:96) ( σ ) f ( d n ( σ )) = (cid:90) ∞ σ/π f ( y ) 16 σ dyπ y (cid:113) − ( σπy ) . Similarly, we can compute the mean value of the RN gaps withineach cluster ( § (cid:96) →∞ E (cid:96) ( σ ) (cid:88) λ n ( σ ) ∈E (cid:96) ( σ ) d n ( σ ) ∼ σ. Note that 2 = 2 length( ∂ Ω) / area(Ω), and the general theory devel-oped in [9] leads to (1.2) if we average over the entire spectrum. Finerthan that, (1.2) asserts that for the hemisphere the same mean resultholds in each cluster.The cluster structure that we find is similar in nature to that foundfor the spectrum of operator − ∆ + V on the unit sphere S , for asmooth potential V [11, 12]. The eigenvalues of − ∆+ V fall into clusters Strictly speaking, the results of [9] are only for planar domains. Similar results are available for the spectrum of the Laplace Beltrami operatoron Zoll surfaces, which are spheres equipped with a Riemannian metric such thatevery geodesic is closed, and all geodesics have the same length.
ZE´EV RUDNICK AND IGOR WIGMAN
10 20 30 40 50 60 700.51.01.52.02.53.03.5
Figure 2.
The RN differences d (cid:96),m ( σ ) in the cluster E (cid:96) ( σ ) for (cid:96) = 150 and σ = 1. The horizontal line (red)is their mean value 2. The solid curve (green) is thetheoretical formula (1.1). C (cid:96) of diameter O (1) around the eigenvalues (cid:96) ( (cid:96) + 1) of the sphere (inour case, the clusters are bigger, of diameter ≈ √ (cid:96) ), and moreover theeigenvalues in each cluster C (cid:96) become equidistributed with respect toa suitable measure.1.4. RN gaps.
We next examine the totality of the Robin-Neumanngaps d n ( σ ) := λ n ( σ ) − λ n (0). Theorem 1.3.
There are constants < c < C so that for each σ > ,(a) For all n , λ n ( σ ) − λ n (0) ≤ Cλ n (0) / · σ. (b) There are arbitrarily large n so that λ n ( σ ) − λ n (0) ≥ cλ n (0) / · σ. In particular the Robin-Neumann gaps for the hemisphere are un-bounded. We note that at this point, we do not know of any planardomain where the RN gaps are provably unbounded [9]. The upperbound is better than what is known for general smooth planar domains[9], which is d n ( σ ) ≤ Cλ n (0) / · σ .As a corollary to Theorem 1.3 we establish the limit level spacingdistribution for the Robin spectrum, which is the distribution P ( s ) (as-suming it exists) of the nearest-neighbour gaps λ n +1 ( σ ) − λ n ( σ ), normal-ized to have mean unity (cf § λ n +1 (0) − λ n (0)are zero and P ( s ) is the delta function at the origin. We show that theRobin spectrum has the same level spacing distribution; OBIN SPECTRUM FOR THE HEMISPHERE 5
Corollary 1.4.
For every σ > , the level spacing distribution for thedesymmetrized Robin spectrum on the hemisphere is a delta-functionat the origin. However, unlike in the Neumann or Dirichlet case, the delta functionis not a result of multiplicities, as there are none here:
Theorem 1.5.
Fix σ > . Then the desymmetrized Robin spectrum issimple: λ m ( σ ) (cid:54) = λ n ( σ ) for all n (cid:54) = m . We note that there are few deterministic simplicity results available,unlike generic simplicity which is more common, e.g. the Dirichletspectrum of generic triangles is simple [5]. For instance, simplicity ofthe desymmetrized Dirichlet spectrum on the disk was proved by Siegelin 1929 (Bourget’s hypothesis) [10], and the same result holds for theNeumann spectrum [1]. However, there are arbitrarily small σ > σ sufficientlysmall, but for rectangles with irrational squared aspect ratio, it failsfor arbitrarily small σ [8].Finally, we note that the theory developed here for the hemisphere isquite singular when compared to what we expect to hold for all otherspherical caps. In that case we do not expect a cluster structure andmoreover, we believe that the level spacing distribution will be Pois-sonian ( P ( s ) = exp( − s )), as is expected for most integrable systems[3, 7], compare Figure 5.2. The Robin problem
Basics.
Denote by Ω the upper hemisphere on the unit sphere,given in spherical coordinates asΩ = (cid:110) (sin θ cos ϕ, sin θ sin ϕ, cos θ ) : 0 ≤ φ < π, ≤ θ ≤ π/ (cid:111) so that the north pole is at θ = 0, and the equator, which is theboundary ∂ Ω, is at θ = π/ F + ν ( ν + 1) F = 0 , ∂F∂n + σF = 0with ν >
0, where ∂/∂n is the derivative in the direction of the out-ward pointing normal to the equator, and σ >
0. We will call ν the“degree”, in keeping with the case of Dirichlet or Neumann boundaryconditions, when the eigenfunctions are spherical harmonics of degree (cid:96) , with eigenvalue (cid:96) ( (cid:96) + 1). ZE´EV RUDNICK AND IGOR WIGMAN
For σ >
0, all eigenvalues λ = ν ( ν + 1) are positive, hence ν is realand ν > ν < −
1. Since the two solutions of λ = ν ( ν + 1) are ν and − − ν , we may assume that ν > { R φ } around the north-south pole, which defines “sectors” consisting of func-tions transforming as F ( R φ x ) = e imφ F ( x ) (here m ∈ Z ). We write sucha Robin eigenfunction as F ( φ, θ ) = e imφ f ν,m (cos θ )where f ( x ) is a solution of ( x := cos θ )(2.1) (1 − x ) f (cid:48)(cid:48) − xf (cid:48) + (cid:18) ν ( ν + 1) − m − x (cid:19) f = 0 . The Robin boundary condition σF + ∂F∂n = 0 is then translated to(2.2) σf (0) − f (cid:48) (0) = 0 . Indeed, the equator is θ = π/
2, or x = 0; and the normal derivative(outward normal) is ∂∂n (cid:12)(cid:12)(cid:12) θ = π/ = − ddx (cid:12)(cid:12)(cid:12) x =0 . Desymmetrization.
Since the equation (2.1) is independent ofthe sign of m , we see that the Robin spectrum has a systematic dou-ble multiplicity. We will remove it (desymmetrization) by insisting that m ≥
0. Note that this is equivalent to taking only eigenfunctions whichare symmetric with respect to the reflection ( x, y, z ) (cid:55)→ ( x, − y, z ). Weorder the desymmetrized Neumann eigenvalues (including multiplici-ties) by λ = 0 < λ = 2 < λ = λ = 6 < . . . The eigenfunctions.
The solutions of the differential equation(2.1) which are nonsingular in 0 ≤ x ≤ mν [6, 14.3.4]P mν ( x ) = ( − m Γ( ν + m + 1)2 m Γ( ν − m + 1) (1 − x ) m/ F (cid:18) ν + m + 1 , m − ν ; m + 1 , − x (cid:19) = ( − m Γ( ν + m + 1)2 m Γ( ν − m + 1) (cid:18) − x x (cid:19) m/ F (cid:18) ν + 1 , − ν ; m + 1; 1 − x (cid:19) . (2.3) OBIN SPECTRUM FOR THE HEMISPHERE 7
Here F ( a, b ; c ; z ) is Olver’s hypergeometric seriesF ( a, b ; c ; z ) = ∞ (cid:88) s =0 ( a ) s ( b ) s Γ( c + s ) s ! z s , | z | < a ) s = Γ( a + s ) / Γ( a ), so that (2.3) converges absolutely if x ∈ ( − , x = cos θ ∈ [0 ,
1] which is relevantfor the hemisphere. 3.
The secular equation
For integer m ≥
0, we set(3.1) S m ( ν ) = 2 tan (cid:18) π ( m + ν )2 (cid:19) Γ (cid:0) ν + m + 1 (cid:1) Γ (cid:0) ν − m + 1 (cid:1) Γ (cid:0) ν + m +12 (cid:1) Γ (cid:0) ν − m +12 (cid:1) . Plots of S ( ν ) and S ( ν ) are displayed in figure 3. - m = = Figure 3. S ( ν ) (dashed) and S ( ν ) (solid). Theorem 3.1.
Let σ > .(a) For each m ≥ , the degree ν > for which the boundaryvalue problem (2.1) and (2.2) admits nonzero regular solutionssatisfies the secular equation S m ( ν ) = σ. (b) The secular equation has no solutions in < ν < m .Proof. We saw that for all ν , there is a one-dimensional space of solu-tions of the ODE (2.1) which are regular for x ∈ [ − , ZE´EV RUDNICK AND IGOR WIGMAN the associated Legendre function P mν ( x ). The boundary condition (2.2)gives the secular equation f (cid:48) ν,m (0) f ν,m (0) = (cid:16) d P mν dx (cid:17) (0)P mν (0) = σ. The values at x = 0 of P mν and its derivative are [6, § mν (0) = 2 m √ π Γ (cid:0) ν − m + 1 (cid:1) Γ (cid:0) − ν − m (cid:1) = 2 m √ π cos (cid:18) π ( ν + m )2 (cid:19) Γ (cid:0) ν + m +12 (cid:1) Γ (cid:0) ν − m + 1 (cid:1) and (cid:18) d P mν dx (cid:19) (0) = − m +1 √ π Γ (cid:0) ν − m +12 (cid:1) Γ (cid:0) − ν + m (cid:1) = 2 m +1 √ π sin (cid:18) π ( m + ν )2 (cid:19) Γ (cid:0) ν + m + 1 (cid:1) Γ (cid:0) ν − m +12 (cid:1) and therefore (cid:16) d P mν dx (cid:17) (0)P mν (0) = 2 tan (cid:18) π ( m + ν )2 (cid:19) Γ (cid:0) ν + m + 1 (cid:1) Γ (cid:0) ν − m + 1 (cid:1) Γ (cid:0) ν + m +12 (cid:1) Γ (cid:0) ν − m +12 (cid:1) . Hence we obtain the secular equation in the form S m ( ν ) = σ with S m as in (3.1).We transform S m ( ν ) by using Euler’s reflection formula Γ( s )Γ(1 − s ) = π/ sin( πs ) to convertΓ( ν − m + 1)Γ( ν − m +12 ) = (cid:32) π sin( π ( m − ν )2 )Γ( m − ν ) (cid:33) / (cid:32) π sin π ( ν − m +1)2 Γ(1 − ν − m +12 ) (cid:33) = Γ( m − ν +12 )Γ( m − ν ) · cos( π m − ν )sin( π m − ν ) = Γ( m − ν +12 )Γ( m − ν ) cot (cid:18) π m − ν (cid:19) . Moreover, for integer m ,tan (cid:18) π ( m + ν )2 (cid:19) · cot (cid:18) π ( m − ν )2 (cid:19) = − . Thus we obtain(3.2) S m ( ν ) = − ν + m + 1)Γ( m − ν +12 )Γ( m + ν +12 )Γ( m − ν ) . The expression (3.2) allows us check that if m ≥
1, there is no solutionfor the secular equation if 0 < ν < m (recall σ > < ν < m , hence so are the Gamma functions. Therefore S m ( ν ) is negative for ν < m . Thus for 0 < ν < m there is no solution of thesecular equation if σ > (cid:3) Proposition 3.2.
Fix σ > . Then OBIN SPECTRUM FOR THE HEMISPHERE 9 (a) S m vanishes at the points m + 2 k , with k ≥ integer, tends toinfinity as ν (cid:37) m +2 k +1 , and S m ( ν ) is negative for m +2 k − <ν < m + 2 k , positive in m + 2 k < ν < m + 2 k + 1 and increasingfor m + 2 k − < ν < m + 2 k + 1 .(b) Let (cid:96) = m + 2 k with integer k = 0 , , , . . . . Then there is aunique solution ν (cid:96),m ( σ ) ∈ ( (cid:96), (cid:96) + 1) of the secular equation.(c) Write ν (cid:96),m ( σ ) = (cid:96) + δ (cid:96),m ( σ ) , with δ = δ (cid:96),m ( σ ) ∈ (0 , . Then (3.3) δ < (cid:113) π σ √ ν . Proof.
We use S m in the form S m ( ν ) = 2 tan (cid:18) π ( m + ν )2 (cid:19) G ( ν + m ) G ( ν − m )where G ( s ) := Γ( s + 1)Γ( s +12 ) . Note that G ( s ) is positive for s >
0. We have for s > G (cid:48) ( s ) = 12 G ( s ) (cid:18) ψ (cid:16) s (cid:17) − ψ (cid:18) s (cid:19)(cid:19) with ψ the digamma function [6, 5.9.16] ψ ( s ) := Γ (cid:48) ( s )Γ( s ) = − γ + (cid:90) − t s − − t dt, (cid:60) ( s ) > G (cid:48) ( s ) G ( s ) = 12 (cid:90) (1 − t s/ ) − (1 − t ( s − / )1 − t dt = 12 (cid:90) t ( s − / √ t dt is clearly positive for s >
0. Since G ( s ) > G (cid:48) ( s ) > s >
0, so that G ( s ) is increasing, and(3.4) 0 < G (cid:48) ( s ) G ( s ) < . The function S m ( ν ) is positive for m + 2 k < ν < m + 2 k + 1 becauseboth G ( ν ± m ) are positive for ν > m , and writing ν = m + 2 k + δ givestan π ( m + ν ) = tan π δ which is positive for δ ∈ (0 , δ ∈ ( − , S m is(3.5) S (cid:48) m S m ( ν ) = π sin πδ + G (cid:48) G ( ν − m ) + G (cid:48) G ( ν + m ) . Since G (cid:48) /G >
0, we find that if δ ∈ (0 ,
1) then S (cid:48) m /S m ( ν ) > S m ( ν ) > v > m we obtain that S (cid:48) m ( ν ) > ν ∈ ( m + 2 k, m + 2 k + 1), so that S m is increasing there. Otherwise, if ν ∈ ( m + 2 k − , m + 2 k ), then δ ∈ ( − , S m ( ν ) <
0. Then, since in this range π sin πδ < − π , the inequality(3.4) shows, with the use of the triangle inequality, that the r.h.s. of(3.5) is π sin πδ + G (cid:48) G ( ν − m ) + G (cid:48) G ( ν + m ) < − π + 12 + 12 < , and so is the l.h.s. of (3.5), and then S (cid:48) m ( ν ) > G ( s ) is positive and increasing, for ν > m we get G ( ν − m ) ≥ G (0) = 1 √ π . By Stirling’s formula G ( s ) ∼ (cid:112) s + O (1 / √ s ) as s → ∞ , in fact [6,5.6.4](3.6) (cid:114) s < G ( s ) < (cid:114) s , s > . Also notetan (cid:18) π ( m + ν )2 (cid:19) = tan π (cid:18) m + k + δ (cid:19) = tan πδ ≥ πδ . We obtain σ = 2 tan( π ( m + ν )2 ) G ( ν + m ) G ( ν − m ) > πδ (cid:114) ν + m G (0) ≥ δ (cid:114) π √ ν so that δ < (cid:113) π σ/ √ ν . (cid:3) Corollary 3.3.
Fix σ > . For (cid:96) ≥ m ≥ , (cid:96) = m mod 2 , let ν = ν (cid:96),m ( σ ) be the unique solution of the secular equation S m ( ν ) = σ with ν ∈ ( (cid:96), (cid:96) + 1) . Write ν = (cid:96) + δ , with δ ∈ (0 , . Thena. As σ → , δ → ,b. As σ → ∞ , we have δ → . Consequently, as σ → ν → (cid:96) , while as σ → ∞ , ν → (cid:96) + 1.Thus, as σ varies between 0 and + ∞ , Λ (cid:96),m ( σ ) := ν (cid:96),m ( σ ) · ( ν (cid:96),m ( σ ) + 1)interpolates between a Neumann eigenvalue (cid:96) ( (cid:96) + 1) with (cid:96) of the sameparity as m , and a Dirichlet eigenvalue ( (cid:96) + 1)( (cid:96) + 2) with same m andopposite parity between (cid:96) and m . OBIN SPECTRUM FOR THE HEMISPHERE 11
Proof.
That δ → σ → G ( s ) we obtain σ = S m ( ν ) ≤ πδ G (2 m + 2 k + 1) G (2 k + 1) (cid:28) m,k tan πδ σ → ∞ , we have δ → (cid:3) Multiplicity one
We have seen (Theorem 3.1) that the desymmetrized Robin spectrumof the hemisphere is given by the energies(4.1) Λ (cid:96),m ( σ ) = ν (cid:96),m ( σ ) · ( ν (cid:96),m ( σ ) + 1)with (cid:96) ≥
0, and 0 ≤ m ≤ (cid:96) satisfying m ≡ (cid:96) mod 2, satisfying thesecular equation S m ( ν ) = σ , with S m given by (3.1): S m ( ν ) = 2 tan (cid:18) π ( m + ν )2 (cid:19) Γ( ν + m + 1)Γ( ν − m + 1)Γ( ν + m +12 )Γ( ν − m +12 ) . To show that there are no degeneracies in the desymmetrized spec-trum (Theorem 1.5), it therefore suffices to prove:
Proposition 4.1.
Fix σ > . For all (cid:96) ≥ and ≤ m ≤ (cid:96) − with m ≡ (cid:96) mod 2 , ν (cid:96),m +2 ( σ ) > ν (cid:96),m ( σ ) . Figure 4.
Plots of ν ,m , m = 0 , , , , ,
10 on [0 , m . The picture emerging for ν ,m ( σ ) on [0 , ≤ m ≤ m ≡ (cid:96) mod 2, is displayed within figures 4. This clearlysupport the statement of Proposition 4.1. Proof.
Recall that ν (cid:96),m ( σ ) ∈ ( (cid:96), (cid:96) + 1), and that (cid:96) = m + 2 k , k ≥
0. ByProposition 3.2, both S m ( ν ) and S m +2 ( ν ) are increasing and positivein ( (cid:96), (cid:96) + 1). Using the recurrence Γ( s + 1) = s Γ( s ) we find S m +2 ( ν ) S m ( ν ) = ν + m + 1 · ν − m − ν + m +12 · ν − m = 1 − m + 1)( ν − m )( ν + m + 1) < . Hence for ν ∈ ( (cid:96), (cid:96) + 1), where both S m ( ν ) and S m +2 ( ν ) are positive,we must have S m +2 ( ν ) < S m ( ν ). Therefore S m +2 ( ν (cid:96),m ( σ )) < S m ( ν (cid:96),m ( σ )) = σ = S m +2 ( ν (cid:96),m +2 ( σ )) . Since S m +2 is increasing in ( (cid:96), (cid:96) +1), we deduce that ν (cid:96),m ( σ ) < ν (cid:96),m +2 ( σ )as claimed. (cid:3) Clusters and a Szeg˝o type limit theorem
Cluster structure.
Denote the cluster (a multiset) of desym-metrized multiple Neumann eigenvalues sharing a common value of (cid:96) ( (cid:96) + 1) by E (cid:96) (0) = (cid:110) (cid:96) ( (cid:96) + 1) : 0 ≤ m ≤ (cid:96), m = (cid:96) mod 2 (cid:111) . This cluster has size E (cid:96) (0) = (cid:98) (cid:96)/ (cid:99) + 1. We label the eigenvaluesthere by E (cid:96) (0) = (cid:8) λ L , λ L +1 , . . . , λ L + (cid:98) (cid:96)/ (cid:99) (cid:9) where L = L (cid:96) is given by L = E (0) ∪ E (0) ∪ · · · ∪ E (cid:96) − (0)) = (cid:96) − (cid:88) (cid:96) (cid:48) =0 (cid:22) (cid:96) (cid:48) (cid:23) + 1 = (cid:96) O ( (cid:96) ) . The distance of the Neumann eigenvalue cluster E (cid:96) (0) to the closestother Neumann eigenvalue cluster, which for (cid:96) ≥ E (cid:96) − (0) (in otherwords, the distance between distinct nearby Neumann eigenvalues), is(5.1) min (cid:96) (cid:48) : (cid:96) (cid:48) (cid:54) = (cid:96) dist (cid:16) E (cid:96) (0) , E (cid:96) (cid:48) (0) (cid:17) = (cid:96) ( (cid:96) + 1) − ( (cid:96) − (cid:96) = 2 (cid:96). We saw that the Robin eigenvalues are ν ( ν + 1) where ν = ν (cid:96),m ( σ ) ∈ ( (cid:96), (cid:96) + 1), (cid:96) = m mod 2, is a solution of the secular equation S m ( ν ) = σ .Denote by(5.2) E (cid:96) ( σ ) = { Λ (cid:96),m ( σ ) : (cid:96) ≥ m ≥ , (cid:96) = m mod 2 } OBIN SPECTRUM FOR THE HEMISPHERE 13 which is the evolution of the Neumann eigenvalue cluster E (cid:96) (0). Since (cid:96) < ν (cid:96),m ( σ ) < (cid:96) + 1, the spectral cluster E (cid:96) ( σ ) is contained in theopen interval (cid:16) (cid:96) ( (cid:96) + 1) , ( (cid:96) + 1)( (cid:96) + 2) (cid:17) , and in particular the evolvedeigenvalue clusters E (cid:96) ( σ ) do not mix with each other.5.2. Asymptotics of the Robin-Neumann gaps.
Recall that wewrite ν (cid:96),m ( σ ) = (cid:96) + δ (cid:96),m ( σ ). Lemma 5.1. As (cid:96) → ∞ , with ≤ m < (cid:96) , (cid:96) = m mod 2 , (5.3) δ (cid:96),m ( σ ) = 2 σπ √ (cid:96) − m (cid:18) O (cid:18) (cid:96) − m (cid:19)(cid:19) . For m = (cid:96) , we have (5.4) δ (cid:96),(cid:96) ( σ ) ∼ σ √ π(cid:96) . Proof.
For 0 < (cid:96) − m = O (1), (5.3) is just the upper bound (3.3), soassume (cid:96) − m → ∞ . The cluster E (cid:96) ( σ ) consists of (cid:98) (cid:96)/ (cid:99) + 1 eigenvaluesΛ (cid:96),m ( σ ) = ν (cid:96),m ( ν (cid:96),m + 1) with m + 2 k = (cid:96) , m, k ≥
0, and where ν (cid:96),m ( σ )is the unique solution of the secular equation S m ( ν ) = σ in the interval( (cid:96), (cid:96) + 1). We write ν = ν (cid:96),m ( σ ) = (cid:96) + δ = m + 2 k + δ, δ = δ (cid:96),m ( σ ) . Recall that the S m of the secular equation S m ( ν ) = σ is given by(5.5) S m ( ν ) = 2 tan (cid:18) π ( m + ν )2 (cid:19) G ( ν + m ) G ( ν − m )where G ( s ) = Γ( s + 1) / Γ( s +12 ) satisfies (cf. (3.1)) G ( s ) = (cid:114) s (cid:18) O (cid:18) s (cid:19)(cid:19) , s → ∞ . Since we assume that (cid:96) − m = 2 k → ∞ , both arguments of G in(5.5) tend to infinity, because ν + m = 2 k + 2 m + δ = (cid:96) + m + δ and ν − m = 2 k + δ = (cid:96) − m + δ . Moreover,tan π ν + m ) = tan π δ and we know (Proposition 3.2) that(5.6) δ (cid:28) σ/ √ (cid:96) → π ν + m ) = tan π δ = π δ + O (cid:18) (cid:96) / (cid:19) . Therefore we can write S m ( ν ) = 2 π δ (cid:18) O (cid:18) (cid:96) (cid:19)(cid:19) · (cid:114) ν − m (cid:18) O (cid:18) (cid:96) − m (cid:19)(cid:19) · (cid:114) ν + m (cid:18) O (cid:18) (cid:96) + m (cid:19)(cid:19) = πδ · (cid:114) k + δ (cid:114) k + m + δ (cid:18) O (cid:18) (cid:96) − m (cid:19)(cid:19) . (5.7)Furthermore, since 2 k = (cid:96) − m , (cid:114) k + δ √ k (cid:18) O (cid:18) δ(cid:96) − m (cid:19)(cid:19) = (cid:114) (cid:96) − m (cid:18) O (cid:18) √ (cid:96) ( (cid:96) − m ) (cid:19)(cid:19) and likewise since k + m = ( (cid:96) + m ) / (cid:114) k + m + δ (cid:114) (cid:96) + m (cid:18) O (cid:18) (cid:96) / (cid:19)(cid:19) . Inserting (5.7) into the secular equation S m ( ν ) = σ gives, when (cid:96) − m → ∞ , that δ (cid:96),m ( σ ) = 2 σπ √ (cid:96) − m (cid:18) O (cid:18) (cid:96) − m (cid:19)(cid:19) . When m = (cid:96) , we use δ = δ (cid:96),(cid:96) ( σ ) (cid:28) / √ (cid:96) → G (0) = 1 / √ π toobtain σ = S (cid:96) ( σ ) = 2 tan (cid:16) π δ (cid:17) G (2 (cid:96) + δ ) G ( δ ) ∼ πδG (2 (cid:96) ) G (0) ∼ πδ √ (cid:96) √ π as (cid:96) → ∞ , which gives (5.4). (cid:3) We derive an asymptotic for the RN gaps d (cid:96),m ( σ ) = Λ (cid:96),m ( σ ) − (cid:96) ( (cid:96) (+1)in each cluster: Corollary 5.2. As (cid:96) → ∞ , for all ≤ m < (cid:96) with m = (cid:96) mod 2 , theRobin-Neumann gaps satisfy (5.8) d (cid:96),m ( σ ) = 2 σπ · (cid:96) + 1 √ (cid:96) − m + O (cid:32) √ (cid:96) ( (cid:96) − m ) / (cid:33) . For m = (cid:96) we have (5.9) d (cid:96),(cid:96) ( σ ) ∼ σ √ π √ (cid:96). OBIN SPECTRUM FOR THE HEMISPHERE 15
Proof.
We have d (cid:96),m ( σ ) = Λ (cid:96),m ( σ ) − Λ (cid:96),m (0) = ( ν − (cid:96) )( ν + (cid:96) + 1) = δ (2 (cid:96) + 1 + δ )= (2 (cid:96) + 1) δ (cid:96),m + δ (cid:96),m = (2 (cid:96) + 1) δ (cid:96),m + O (cid:18) (cid:96) (cid:19) where we have used (5.6). Moreover, for m < (cid:96) we have the asymptoticformula (5.3) for δ (cid:96),m , and hence d (cid:96),m ( σ ) = 2(2 (cid:96) + 1) σπ √ (cid:96) − m (cid:18) O (cid:18) (cid:96) − m (cid:19)(cid:19) + O (cid:18) (cid:96) (cid:19) = 2(2 (cid:96) + 1) σπ √ (cid:96) − m + O (cid:32) √ (cid:96) ( (cid:96) − m ) / + 1 (cid:96) (cid:33) = 2(2 (cid:96) + 1) σπ √ (cid:96) − m + O (cid:32) √ (cid:96) ( (cid:96) − m ) / (cid:33) . For the case m = (cid:96) , (5.9) similarly follows from (5.4). (cid:3) Equidistribution of gaps in the cluster.
We can now deducethe equidistribution of gaps in each cluster (Corollary 1.2) and computethe average gap in a cluster as asserted in (1.2). Since the argumentsare similar, we do the latter:
Corollary 5.3. As (cid:96) → ∞ , E (cid:96) ( σ ) (cid:88) λ n ( σ ) ∈E (cid:96) ( σ ) d n ( σ ) ∼ σ. Proof.
Using d (cid:96),m = (2 (cid:96) + 1) δ (cid:96),m + δ (cid:96),m (cid:28) √ (cid:96) by (3.3), we see that wemay restrict the average to m ≤ (cid:96) − O ( (cid:96) − / ):1 E (cid:96) ( σ ) (cid:88) λ n ( σ ) ∈E (cid:96) ( σ ) d n ( σ ) = 1 (cid:96)/ O (1) (cid:88) ≤ m ≤ (cid:96) − m = (cid:96) mod 2 d (cid:96),m + O ( (cid:96) − / ) . Then we use (5.8) to obtain1 (cid:96)/ O (1) (cid:88) ≤ m ≤ (cid:96) − m = (cid:96) mod 2 d (cid:96),m = 1 (cid:96)/ (cid:88) ≤ m ≤ (cid:96) − m = (cid:96) mod 2 (cid:96) + 1) σπ √ (cid:96) − m + O (cid:18) (cid:96) / (cid:19) . Moreover, using standard bounds for the rate of convergence of Rie-mann sums gives1 (cid:96)/ (cid:88) ≤ m ≤ (cid:96) − m = (cid:96) mod 2 (cid:96) + 1) σπ √ (cid:96) − m = (cid:18) σπ + O ( 1 (cid:96) ) (cid:19) (cid:18)(cid:90) dx √ − x + O (cid:18) (cid:96) / (cid:19)(cid:19) = 2 σ + O (cid:18) (cid:96) / (cid:19) . Altogether, we obtain1 E (cid:96) ( σ ) (cid:88) λ n ( σ ) ∈E (cid:96) ( σ ) d n ( σ ) = 2 σ + O (cid:18) (cid:96) / (cid:19) ∼ σ as claimed. (cid:3) Bounds for the RN gaps: Proof of Theorem 1.3
Proof.
Using (3.3) shows that for (cid:96) (cid:29) ≤ m ≤ (cid:96) with m ≡ (cid:96) mod 2,Λ (cid:96),m ( σ ) − (cid:96) ( (cid:96) + 1) = ( ν l,m ( σ ) − (cid:96) )( ν (cid:96),m ( σ ) + (cid:96) + 1) (cid:28) σ √ (cid:96) so that max (cid:110) | λ − (cid:96) ( (cid:96) + 1) | : λ ∈ E (cid:96) ( σ ) (cid:111) (cid:28) σ √ (cid:96). Therefore, for all n , we have(6.1) λ n ( σ ) − λ n (0) (cid:28) σλ n (0) / . This proves Theorem 1.3(a).To show that we can actually attain the upper bound in (6.1),note that Proposition 4.1 demonstrates that to get the largest pos-sible Robin-Neumann gaps, it is worth, given (cid:96) ≥
0, to take m = (cid:96) . Wethen use (5.9) to obtain d (cid:96),(cid:96) ( σ ) ∼ σ √ π √ (cid:96) ∼ √ π Λ (cid:96),(cid:96) (0) / σ, which proves Theorem 1.3(b). (cid:3) We note that Λ (cid:96),(cid:96) (0) ∈ E (cid:96) (0), and therefore for each (cid:96) (cid:29)
1, we havefound n = (cid:96) / O ( (cid:96) ) for which λ n ( σ ) − λ n (0) (cid:29) λ n (0) / · σ, and in particular that the Robin-Neumann gaps are unbounded. OBIN SPECTRUM FOR THE HEMISPHERE 17 Level spacings
In this section, we show that the level spacing distribution of thedesymmetrized Robin spectrum on the hemisphere is a delta functionat the origin, as is the case with Neumann or Dirichlet boundary condi-tions. We note that for other spherical caps (cf [4] for background), weexpect that the level spacing distribution is Poissonian. A numericalplot for the desymmetrized Dirichlet spectrum on the cap with openingangle θ = π/ θ = π/
2) is displayed in Figure 5.
Figure 5.
The level spacing distribution P ( s ) for all1258 desymmetrized Dirichlet eigenvalues ν ( ν + 1) with ν <
100 for the spherical cap with opening angle θ = π/
3. The solid curve is the Poisson result exp( − s ). Proof of Corollary 1.4.
The statement of Corollary 1.4 is equivalent tothe fact that for every y > N →∞ N { n ≤ N : λ σn +1 − λ σn > y } = 0 . Recall that we divided the ordered desymmetrized Robin eigenvalues { λ σn } n ≥ into disjoint clusters E (cid:96) ( σ ) (see (5.2)), each at distance O ( √ (cid:96) )from the Neumann eigenvalues (cid:96) ( (cid:96) + 1), so diam E (cid:96) ( σ ) (cid:28) √ (cid:96) (Theo-rem 1.3(a)), and hence of distance 2 (cid:96) + O ( √ (cid:96) ) from the closest othercluster, and of size E (cid:96) ( σ ) = (cid:98) (cid:96)/ (cid:99) + 1 = (cid:96)/ O (1).For N (cid:29)
1, denote by L the index of the cluster to which λ N ( σ )belongs, so that (cid:91) (cid:96) ≤ L − E (cid:96) ( σ ) ⊂ { λ n ( σ ) : n ≤ N } ⊆ (cid:91) (cid:96) ≤ L E (cid:96) ( σ )and therefore N = (cid:88) (cid:96) ≤ L − E (cid:96) ( σ ) + O ( L ) = L O ( L ) so that L = O ( √ N ). Then(7.2) { n ≤ N : λ σn +1 − λ σn > y } ≤ L (cid:88) (cid:96) =0 (cid:88) λ σn +1 − λ σn >yλ σn ∈E (cid:96) ( σ ) . Denote by n + the maximal index of an eigenvalue in E (cid:96) ( σ ), and by n − the minimal index. Then the gaps corresponding to the cluster E (cid:96) ( σ ) are firstly those with λ n +1 ( σ ) − λ n ( σ ) with n − ≤ n ≤ n + − λ n + +1 ( σ ) − λ n + ( σ ). The number of those gaps ofthe second kind is at most L + 1 = O ( √ N ).For the gaps > y of the first kind, we have in each cluster (cid:88) λ σn +1 − λ σn >yλ σn ∈E (cid:96) ( σ ) n
References [1] M. Ashu, Some properties of Bessel functions with applications to Neumanneigenvalues in the unit disc, Bachelor’s thesis 2013:K1 (E. Wahl´en advisor),Lund University. http://lup.lub.lu.se/student-papers/record/7370
OBIN SPECTRUM FOR THE HEMISPHERE 19 [2] P. B´erard and G. Besson. Spectres et groupes cristallographiques. II. Domainessph´eriques. Ann. Inst. Fourier (Grenoble) 30 (1980), no. 3, 237–248.[3] M.V.Berry and M. Tabor. Level clustering in the regular spectrum. Proc. R.Soc. London A356, 375–394 (1977).[4] G. V. Haines. Spherical cap harmonic analysis. J. Geophys. Res. 90 (B3):2583–2591.[5] L. Hillairet and C. Judge. Spectral simplicity and asymptotic separation ofvariables. Comm. Math. Phys. 302 (2011), no. 2, 291–344. Erratum: Comm.Math. Phys. 311 (2012), no. 3, 839–842.[6] NIST Digital Library of Mathematical Functions (DLMF). Available online athttps://dlmf.nist.gov/[7] Z. Rudnick. What is quantum chaos? Notices Amer. Math. Soc. 55 (2008), no.1, 32–34.[8] Z. Rudnick and I. Wigman, in preparation.[9] Z. Rudnick, I. Wigman and N. Yesha. Differences between Robin and Neumanneigenvalues, available online https://arxiv.org/abs/2008.07400 .[10] C. L. Siegel, ¨Uber einige Anwendungen diophantischer Approximationen, Abh.Preuss. Akad. Wiss., Phys.-math. Kl., (1929).[11] A. Weinstein. Asymptotics of eigenvalue clusters for the Laplacian plus a po-tential. Duke Math. J. 44 (1977), no. 4, 883–892.[12] H. Widom. Eigenvalue distribution theorems for certain homogeneous spaces.J. Functional Analysis 32 (1979), no. 2, 139–147.[13] N. Yesha, private communication.
School of Mathematical Sciences, Tel Aviv University, Tel Aviv69978, Israel
E-mail address : [email protected] Department of Mathematics, King’s College London, UK
E-mail address ::