4. Straightening of the meridian domain

arXiv:1805.12448v1 [math.SP] 31 May 2018

GENERALIZED PARABOLIC LAYER

PAVEL EXNER AND VLADIMIR LOTOREICHIK

ABSTRACT. We perform quantitative spectral analysis of the self-adjoint Dirichlet LaplacianHon an unbounded, radially symmetric (generalized) parabolic layerP⊂ R³. It was known before thatHhas an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spec- trum asymptotics forHby means of a consecutive reduction to the analogous asymp- totic problem for an effective one-dimensional Schrödinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layerPat infinity.

1. Introduction

1.1. State of the art and motivation

The topic of this paper is the spectral asymptotics for the Dirichlet Laplacian in do- mains of the form of a generalized parabolic layer. By the layer, we understand the Eu- clidean domain consisting of the points, whose distance from a hypersurface, in general unbounded one, is less than some given constant. The self-adjoint Laplace operator on unbounded layers subject to the Dirichlet boundary condition has been considered in numerous papers,e.g. [CEK04, DOR15, DEK01,DLO17, ET10, KL14, LL07, LR12], see also [EK, Chap. 4] and the references therein.

Unbounded layers belong to the class ofquasi-cylindrical domainsin the classification [EE, Dfn. X.6.1 (ii)] introduced by Glazman, because they do not contain arbitrarily large cubes, but they do contain infinitely many disjoint cubes of a fixed size. For this reason, as a rule, the essential spectrum is non-empty and its threshold is strictly positive;cf. [DEK01, KL14] and [EK, Chap. 4]. A challenging question is to prove the existence of bound states below this threshold and to find their properties. Various general results on the bound states can be found in [CEK04, DEK01, LL07, LR12] and in [EK, Chap. 4]. However, there is no result general enough to claim that this problem is completely understood.

In this perspective, more precise analysis of various particular cases deserves attention.

Examples of layers treated so far include mildly curved layers [BEGK01,EKr01], conical layers [DOR15,ET10,OP17], and so-called octant (or Fichera) layers [DLO17].

In the present paper, we initiate an investigation ofgeneralised parabolic layers by per- forming the analysis of radially symmetric layers of this kind. With the indicated layer geometry in mind, the previously known general results yield an explicit formula for the threshold of the essential spectrum and imply the existence of infinitely many bound

1

states below it. Our aspiration here is to understand better the nature of this discrete spectrum and to obtain its quantitative properties.

1.2. Geometric setting

Here and in what follows, we use the shorthand notationsR_{+ := (0,∞)},R_{+ := [0,∞)}, andI_{a := (−a, a)}fora > 0. We also introduce conventional Cartesian coordinates ^x = (x1, x2, x3)in the Euclidean spaceR³.

Given the parametersk > 0,α > 1, andR ≥ 0, we consider the class of real-valued functionsf ∈C^∞₍R₊₎such thatf(0) = ˙f(0) = 0and thatf(x) =kx^αholds for allx≥R. Thegeneralized symmetric paraboloiddetermined by such a functionf is defined by

(1.1) Σ = Σ(f) :=

x, f(|x|)

∈R³_:x∈R² _.

The principal curvatures ofΣwill be denoted byκ1, κ2. Note that the usual paraboloid corresponds to the choiceα= 2andR= 0in the above definition.

The configuration space of our problem is a fixed-width layer built overΣ, in other words, thea-tubular neighborhood ofΣdefined as¹

(1.2) P=P(f, a) :=

x∈R³_{: dist} x,Σ(f)

< a .

The domainP₌P_{(f, a)}will be called generalized radially symmetric parabolic layer, for brevity we will speak of ageneralized parabolic layeras there is no danger of confusion.

In the following,ν(x)stands for the unit normal vector to the surfaceΣat the point (x, f(|x|)). The mappingR²_∋x7→ν(x)is smooth and the orientation ofν(x)is fixed so that its projection on thex3-axis is positive for_|^x_{| ≥}R.

In Subsection 2.3, we show the existence of a constant a⋆ = a⋆(f) > 0 satisfying a⋆kκjk∞<1forj= 1,2such that the restriction of the map

(1.3) L_:R²_×R_→R³_, L₍^x_{, u) = (}^x_{, f(|}^x_|))^⊤_+ν(^x_)u,

ontoR²_×I_a is injective for alla < a⋆. Consequently, the generalized parabolic layer of a half-widtha < a⋆ can alternatively be represented as the image ofR²_×I_a under the mapL. Everywhere in this paper we consider only the generalized parabolic layers the half-width of which satisfies the conditiona < a⋆.

1.3. Definition of the operator

We define a non-negative, symmetric quadratic form on the Hilbert spaceL²(P)by (1.4) Q[ψ] :=k∇ψk²L²(P;C³), domQ:=H₀¹(P).

According to [Dav95, Lem. 6.1.1 and Lem. 6.1.3], the quadratic form Q is closed and densely defined. This allows us to introduce the main object of this paper, namely, the self-adjoint operatorHin L²(P)associated with the formQvia the first representation theorem [Ka, Thm. VI 2.1]. The operator His nothing but theDirichlet Laplacianon the

1dist(x, E) := infy∈E|x−y|Rdis the distance between a pointx∈^Rdand a setE⊂^Rd.

generalized parabolic layerP, the name referring to the fact that the operatorHacts as follows,

Hψ=−∆ψ, domH=

ψ∈H₀¹(P_{) : ∆ψ∈L}²₍P_{) .} (1.5)

1.4. Notation and the main result

Before stating the main result of this paper, we introduce a few further notations. For measurable functions f1,f2: R_{+ →} R₊ satisfying lim_E→0⁺fj(E) = +∞, j = 1,2, and lim_E→0^{+ f}f¹₂^(E)(E) = 1, we agree to write

f1(E) ∼

Eց0f2(E).

Furthermore, letTbe the semi-bounded self-adjoint operator associated with a quadratic form T. We denote byσess(T)and σd(T)the essential and the discrete spectrum of T, respectively; byσ(T)we denote the (entire) spectrum ofT,i.e.σ(T) =σess(T)∪σd(T). We setE_∞(T) := infσess(T) and, for a givenj ∈ N, Ej(T) denotes thej^th-eigenvalue ofT in the interval(−∞, E_∞(T)). These eigenvalues are arranged in the ascending order and repeated according to their multiplicities. We define the counting function ofTas

N_E(T) := #

j∈N_{:Ej(T)< E , E≤E∞(T).}

When working with the quadratic formT, we use the notations σess(T), σd(T), σ(T), E_∞(T), Ej(T), and N_E(T) instead. If it is clear from the context, we will not indicate the dependence of the eigenvalues and the threshold of the essential spectrum on the operator,i.e.we will writeEj andE_∞instead ofEj(T)andE_∞(T), respectively.

First, we recall spectral properties of the HamiltonianH, which follow from the previ- ously known general results after checking the appropriate geometric assumptions.

Proposition 1.1. For anya∈ 0, a⋆

, the following claims hold.

(i) σess(H) = [E_∞,∞), whereE_∞= _2a^π2

. (ii) #σd(H) =∞.

To be specific, one has to verify that both the principal curvatures ofΣare bounded, and that the mean and the Gauss curvatures ofΣvanish at infinity, then the item (i) of the above proposition follows from [EK, Prop. 4.2.1]; see also [KL14]. In addition, checking that the total Gauss curvature of Σ does not vanish and making use of the rotational symmetry forPwe derive item (ii) of the above proposition from [EK, Cor. 4.2.2].

In this paper we give an alternative proof of the fact that#σd(H) = ∞and, what is more important, we obtain as the main result the spectral asymptotics ofH.

Theorem 1.2. For anya∈ 0, a⋆

, the counting function of^Hsatisfies N_E∞−E(H) ∼

Eց0

1 2π

αk 2^α

B ³₂,^α₂−¹₂ E^α2−12 , whereB(·,·)is the Euler beta function.

Remark1.3. In the particular case when the layer is ‘genuinely’ parabolic,α= 2, possibly outside a compact, the asymptotic expression on the right-hand side simplifies to ^k

8√ E. We note that for larger values ofαthe accumulation is slower. Moreover, using Stirling formula [AS, 6.1.37] we find that

αlim→∞

1 2π

αk 2^α

B ³₂,^α₂ −¹₂ E^α2−12 = lim

α→∞

√ k

2πα2^αE^α2−12 =∞

holds for any fixedE <1/4. This might formally correspond to the fact that asαgrows, the layer is becoming closer to the one with annular cylindrical end for which the essential spectrum extends belowE_∞; cf. [EK, Prob. 4.10]. On the other hand, for the (possibly locally deformed) conical layer,α= 1, where the discrete spectrum is also infinite [ET10]

one might ask whether the accumulation rate is obtained, upon passing to the limitα→ 1⁺in the above asymptotic expression, but this not the case as it is obvious from the fact that the limit _4π^k is independent of E. The correct spectral asymptotics for the conical layer [DOR15, Thm. 1.4] is given byN_E∞−E(H)∼ _4π^k |lnE|asE →0⁺. Thus, an abrupt transition from the power-type accumulation rate of eigenvalues to an exponential one occurs upon switching fromα >1toα= 1.

In the proof of Theorem1.2we first proceed as in [DOR15, Thm. 1.4]. Namely, we de- compose the HamiltonianHinto an infinite orthogonal sum of the fiber operators, acting on the two-dimensional meridian domain each. In our geometric setting, the meridian domain is the generalized parabolic half-strip. Such a reduction is possible in view of the rotational invariance of the domainPwith respect to thex3-axis. The ‘s-wave’ fiber corresponding to radially symmetric functions turns out to produce an infinite discrete spectrum below the thresholdE_∞, while the remaining fibers in the decomposition ofH altogether produce at most a finite number of eigenvalues belowE_∞, and thus they do not influence the accumulation rate in the vicinity of the threshold. Recall in this respect the conical layer case, where one is able to make a stronger claim, namely thatnoneof the fibers except the ‘s-wave’ one contributes to the discrete spectrum.

The spectral analysis of the ‘s-wave’ fiber is reduced to the well-known properties of the one-dimensional Schrödinger operators corresponding to the differential expression

−d^ds²²+qon the positive semi-axis, where the bounded potentialq:R_+→Rasymptotically behaves as

q(s)∼ −1 4

k s

_α²

as s→+∞.

Note that for the conventional paraboloid (α= 2), the potentialq behaves at infinity as the Coulomb potential. This reduction is the core of the paper. As an intermediate step, it contains straightening of the underlying generalized parabolic half-strip in suitably chosen curvilinear coordinates, in the spirit of [EŠ89,DE95]; see also [EK, Chap. 1]. This step is new compared to the conical layer, where the meridian domain is straight right from the beginning.

2. Preliminaries

In this preparatory section we collect the auxiliary material, which is needed in the proof of the main result. First, in Subsection2.1, we provide the standard decomposi- tion of the self-adjoint operatorHinto fibers. Furthermore, in Subsection2.2 we derive the arc-length parametrization for the graph off(·), define its signed curvature and in- troduce curvilinear coordinates in the associated generalized parabolic strip. Then, in Subsection2.3, we check that the generalized paraboloidΣsatisfies some of the assump- tions in [EK, Chap. 4] obtaining in this way a proof of Proposition1.1. Finally, we recall in Subsection2.4several standard spectral results on a class of one-dimensional Schrödinger operators we will need in the following.

2.1. The fiber decomposition ofH

In view of the rotational symmetry, it is convenient to express the HamiltonianHin the cylindrical coordinates. LetR²

+ be the positive half-planeR_+×R. We consider the cylindrical coordinates(r, z, θ)∈R²

+×S¹linked with the Cartesian coordinates(x1, x2, x3) via the following conventional relations

(2.1) x1=rcosθ, x2=rsinθ, x3=z.

We denote the graph of the functionx7→f(x)by

(2.2) Γ = Γ(f) :=

(x, f(x)) :x >0 ⊂R²

+.

In order to describe in cylindrical coordinates the generalized parabolic layerP₌P_{(f, a)} of the half-widtha∈(0, a⋆), we first define its meridian domainG₌G_{(f, a)⊂}R²₊by

G₌G_{(f, a) :=}

x∈R²_{+: dist} x,Γ(f)

< a .

Note that the domainGcan be viewed as a generalized parabolic half-strip. The general- ized parabolic layerPin (1.2) is defined in the cylindrical coordinates (2.1) by

(2.3) P₌P_{(f, a) :=}G_{(f, a)×}S¹_.

The layerPcan be seen as a sub-domain of R³constructed via rotation of the meridian domainGaround thex3-axis. For later purposes, we split the boundary∂GofGinto two disjoint parts,

∂0G_{:={0} ×}I_a and ∂1G_:=∂G_\∂0G_.

Next, we introduce the usual cylindricalL²-space and the first order cylindrical Sobolev space onPas

L²_cyl(P_{) :=L}²₍G_×S¹_{;rdrdzdθ), H}¹

cyl(P_{) :=}

ψ:ψ, ∂rψ, ∂zψ, r⁻¹∂θψ∈L²_cyl(P_{) .} The spaceH_cyl¹ (P₎is endowed with the norm_{k · k}H_cyl¹ (P), defined for allψ∈H_cyl¹ (P₎by

kψk²H_cyl¹ (P):=kψk²L²_cyl(P)+ Z

P

|∂rψ|²+|∂zψ|²+|∂θψ|² r²

rdrdzdθ.

The expression for the quadratic formQassociated withHcan be rewritten in the cylin- drical coordinates as follows

Qcyl[ψ] = Z

S¹

Z

G

|∂rψ|²+|∂zψ|²+|∂θψ|² r²

rdrdzdθ.

Following the strategy of [DOR15,KLO17,ET10], we consider an orthonormal basis of the Hilbert spaceL²(S¹₎given by

(2.4) vm(θ) = (2π)^−1/2e^imθ, m∈Z_. For anym∈Z, we define the mapping

(2.5) Πm:L²_cyl(P_)→L²₍G_{;rdrdz), (Πm}ψ)(r, z, θ) = ψ(r, z,·), vm

L²(S¹).

Performing a ‘partial wave’ decomposition [RS-II, App. to X.1], see also [DOR15,KLO17, LO16], with respect to the basis in (2.4), we obtain

(2.6) H∼=M

m∈Z

Fm,

where the symbol^∼=stands for the unitary equivalence relation and, for allm∈Z, the op- eratorsFmacting onL²(G_;rdrdz)are called thefibersofH. They are associated through the first representation theorem with closed, densely defined, symmetric, and non-negative quadratic forms in the Hilbert spaceL²(G_;rdrdz)

Fm[ψ] :=Qcyl[ψ⊗vm] = Z

G

|∂rψ|²+|∂zψ|²+m² r²|ψ|²

rdrdz, domFm:= Πm H₀¹(P₎

. (2.7)

The domain of the operatorFmcan be deduced from the quadratic formFmin the stan- dard way via the first representation theorem.

Next, we introduce the unitary operator

U:L²(G_;rdrdz)→L²₍G_{), Uψ:=}^√_rψ.

This unitary operator allows us to transform the quadratic formsFminto equivalent ones expressed in a flat metric. Indeed, the quadratic formFmis unitarily equivalent viaUto the form in the Hilbert spaceL²(G₎defined as

fm[ψ] :=Fm[U⁻¹ψ] = Z

G

|∂rψ|²+|∂zψ|²+m²−¹₄ r² |ψ|²

drdz, domfm:=U(domFm).

(2.8)

In fact, it is not difficult to check thatC^∞_{0 (}G₎is a core for the formfmand that for allm6= 0 its form domain satisfies

(2.9) domfm=U(domFm) =H₀¹(G_).

Finally, for the sake of convenience, we introduce form ∈ Z the following shorthand notation for frequently used integrands:

(2.10) E_{m[ψ] :=|∂rψ|}²_+|∂zψ|²₊^m

2

r²|ψ|² and F_{m[ψ] :=|∂rψ|}²_+|∂zψ|²₊^m

2−¹4

r² |ψ|².

2.2. The arc-length parametrization ofΓand associated coordinates onG The arc-length parametrization for the curveΓ = Γ(f)in (2.2) is given by

R_{+∋s7→ φ(s), f(φ(s))}

∈R²_,

whereφ:R_{+ →} R₊ is a monotonously increasing function such that φ(0) = 0and that lim_s→∞φ(s) =∞, and which satisfies the ordinary differential equation

(2.11) φ^˙2 1 + ˙f²(φ)

= 1.

On the interval[R,∞), the above equation reduces toφ^˙2 1 +α²k²φ^2α−2

= 1. The equa- tion (2.11) combined withφ(0) = 0andf^˙(0) = 0immediately implies

(2.12) φ(0) = 1^˙ and 0<φ˙≤1.

The signed curvature ofΓis given by

(2.13) γ= f(φ) ¨˙ φ+ ¨f(φ) ˙φ²_˙

φ−f˙(φ) ˙φφ¨= ¨f(φ) ˙φ³.

On the interval[R,∞), the above expression simplifies toγ=α(α−1)kφ˙³φ^α−2. Asymp- totic properties ofφandγ that will be needed in the following are obtained in Appen- dixA.

The unit tangential vector toΓat the point(φ(s), f(φ(s)))is given by( ˙φ(s), f(φ(s)) ˙˙ φ(s))^⊤. Consequently, the unit normal vector toΓat the same point can be expressed as

(2.14) νΓ(s) = −f˙(φ(s)) ˙φ(s), φ(s)˙ ⊤

.

Next, on the half-stripΩ :=R_+×I_a, we introduce the mapτ: Ω→Gby (2.15) τ(s, u) = τ1(s, u)

τ2(s, u)

!

:= φ(s) f(φ(s))

!

+uνΓ(s) = φ(s)−uf˙(φ(s)) ˙φ(s) f(φ(s)) +uφ(s)˙

! .

This map defines convenient curvilinear coordinates(s, u)on the generalized parabolic stripG. Note also that the JacobianJ₌J_{(s, u)}of the mapτis given by

J₌

∂sτ1 ∂uτ1

∂sτ2 ∂uτ2

=

φ˙−uf¨(φ) ˙φ²−uf˙(φ) ¨φ −f˙(φ) ˙φ f˙(φ) ˙φ+uφ¨ φ˙

= ˙φ²−uf¨(φ) ˙φ³+ ˙f²(φ) ˙φ²= 1−uγ, (2.16)

where (2.11) was used in the last step. We also set

(2.17) g(s, u) := (J_{(s, u))}²_{= (1−uγ(s))}²_. 2.3. Parametrization and curvatures ofΣ

In this subsection we provide the natural parametrization ofΣas a surface of revolu- tion in the sense [dC, §3.3, Example 4]. Using this parametrization we obtain formulæ for

the curvatures ofΣand a parametrization for the generalized parabolic layerP. Finally, we check the assumptions on layers [EK, Chap. 4] and eventually prove Proposition1.1.

The generalized paraboloidΣcan be alternatively parametrized as (2.18) R_+×S¹_{∋(s, θ)7→σ(s, θ) := φ(s) cos}θ, φ(s) sinθ, f(φ(s))⊤

,

where the functionsf and φ are as in Subsection 1.2 and Subsection2.2, respectively.

According to [dC, §3.3, Example 4], see also [EK, §4.2.2], the principal curvatures ofΣat the point^x=σ(s, θ)are explicitly given by

κ1(s, θ) =−f˙(φ(s)) ˙φ(s)

φ(s) =− f˙(φ(s))

φ(s)(1 + ˙f²(φ(s)))^1/2, κ2(s, θ) =−γ(s) =−f(φ(s)) ˙¨ φ³(s) =− f¨(φ(s))

1 + ˙f²(φ(s)^3/2. (2.19)

Hence, the meanM =¹₂(κ1+κ2)and the GaussK=κ1κ2curvatures ofΣcan be evaluated as

M(s, θ) =−f˙(φ(s)) + ˙f³(φ(s)) + ¨f(φ(s))φ(s) 2φ(s)(1 + ˙f²(φ(s)))^3/2 , K(s, θ) = f˙(φ(s)) ¨f(φ(s))

φ(s)(1 + ˙f²(φ(s)))². (2.20)

It follows from [EK, Sec. 4.2.2], usinglim_s→∞φ(s) = 0˙ (cf.Proposition A.2(ii)) that the total Gauss curvature ofΣis given byK_:= R

ΣK = 2π. Furthermore, the normal vector νΣ(s, θ)toΣat^x=σ(s, θ)with(s, θ)∈R_+×S¹can be expressed as

(2.21) νΣ(s, θ) = −f˙(φ(s)) ˙φ(s) cosθ,−f˙(φ(s)) ˙φ(s) sinθ,φ(s)˙ ⊤

, thus the layerPis the image ofR_+×S¹_×I_aunder the map

(2.22) R_+×S¹_×R_{∋(s, θ, u)7→}π(s, θ, u) := φ(s) cosθ, φ(s) sinθ, f(φ(s))⊤

+uνΣ(s, θ).

Now, we have all the tools to prove Proposition1.1.

Proof of Proposition1.1. First, recall thatΣis a surface of revolution with a non-zero total Gauss curvatureK _{= 2π}. Furthermore, we immediately notice that both the principal curvaturesκj(s, θ),j= 1,2, ofΣcomputed in (2.19) areC^∞-smooth onR_+×S¹, pointwise positive, and vanish in the limits → ∞; cf.LemmaA.1and PropositionA.3(i). Conse- quently, the minimal normal curvature radius ofΣdefined in [EK, Chap. 4, Assumption (ii)] and given by

ρm:= max{kκ1k∞,kκ2k∞}−1

is positive and finite. In addition, the layerPis asymptotically planar in the sense of [EK,

§4.1.2, Assumption (iii)], because bothM(s, θ)andK(s, θ)vanish in the limits→ ∞. The remaining step is more technical: we have to check that there existsa⋆ ∈ (0, ρm) such that the generalized parabolic layerP ₌ P_{(f, a)}in (1.2) is not self-intersecting for

alla ∈ (0, a⋆)in the sense of [EK, Chap. 4, Assumption (i)]. The latter is equivalent to showing that the restriction of the mapπ(·)in (2.22) ontoR_+×S¹_×I_ais injective for all a∈(0, a⋆). Upon checking this last assumption, [EK, Thm. 4.4] yieldsσess(H) = [E_∞,∞) and#σd(H) =∞.

We will proceed byreductio ad absurdum. If such ana⋆>0did not exist, then it would be possible to find(sj, θj, uj)∈R_+×S¹_×I_ρ

m,j = 1,2, such that(s1, θ1, u1)6= (s2, θ2, u2), s1 ≤ s2, and at the same timeπ(s1, θ1, u1) = π(s2, θ2, u2)withu1, u2 ∈ (−ρm, ρm)having arbitrarily small absolute values. Without loss of generality we can assume that both u1andu2are positive, because the surfaceΣsplitsR³into its hypograph and epigraph, respectively. Using the explicit expression of the normal vector in (2.21) together with the fact thatπ(s1, θ1, u1)−π(s2, θ2, u2)has trivial projections onto all the axes, we get

(2.23)











f˙(φ(s1)) cosθ1c1−f˙(φ(s2)) cosθ2c2=φ(s1) cosθ1−φ(s2) cosθ2, f˙(φ(s1)) sinθ1c1−f˙(φ(s2)) sinθ2c2=φ(s1) sinθ1−φ(s2) sinθ2, c1−c2=f(φ(s2))−f(φ(s1)),

wherecj :=ujφ(s˙ j)forj = 1,2. The above system of linear equations with respect toc1

andc2has a non-trivial solution if, and only if

∆ :=

1 −1 f(φ(s2))−f(φ(s1))

f˙(φ(s1)) cosθ1 −f˙(φ(s2)) cosθ2 φ(s1) cosθ1−φ(s2) cosθ2

f˙(φ(s1)) sinθ1 −f˙(φ(s2)) sinθ2 φ(s1) sinθ1−φ(s2) sinθ2

= 0.

The determinant∆can be explicitly computed, giving

∆ =

−f˙(φ(s2)) cosθ2 φ(s1) cosθ1−φ(s2) cosθ2

−f˙(φ(s2)) sinθ2 φ(s1) sinθ1−φ(s2) sinθ2

+

f˙(φ(s1)) cosθ1 φ(s1) cosθ1−φ(s2) cosθ2

f˙(φ(s1)) sinθ1 φ(s1) sinθ1−φ(s2) sinθ2

+ f(φ(s2))−f(φ(s1))

f˙(φ(s1)) cosθ1 −f˙(φ(s2)) cosθ2

f˙(φ(s1)) sinθ1 −f˙(φ(s2)) sinθ2

= sin(θ2−θ1)

f(φ(s˙ 2))φ(s1)−f˙(φ(s1))φ(s2) + ˙f(φ(s1)) ˙f(φ(s2))

f(φ(s1))−f(φ(s2) . Hence, the requirement∆ = 0can only be satisfied in two disjoint cases:

(i) θ:=θ1=θ2ands1< s2;

(ii) θ16=θ2andf^˙(φ(s2))φ(s1)−f˙(φ(s1))φ(s2) + ˙f(φ(s1)) ˙f(φ(s2))

f(φ(s1))−f(φ(s2)

= 0. These cases require separate analysis.

Case (i). PickR1 > Rsuch thatαkφ(R1)^α−1 >1. There are two sub-cases to distinguish.

Ifs1 < R1, then for simple geometric reasons there is a constant R2 = R2(f, R1) > R1

such that necessarilys2 < R2 holds. Hence, [BEHL16, Prop. B.2], applied for the image of(0, R2)×S¹under the mapping in (2.18), implies that there existsa1∈(0, ρm)such that the conditionu1, u2∈(0, a1)is incompatible withs1< s2.

In the second sub-case (s1≥R1), we reduce the linear system (2.23) to (_˙

f(φ(s1))c1−f˙(φ(s2))c2=φ(s1)−φ(s2), c1−c2=f(φ(s2))−f(φ(s1)).

Solving it and returning to the initial variablesuj,j= 1,2, we find u1=φ(s1)−φ(s2) + ˙f(φ(s2))[f(φ(s2)−f(φ(s1)]

φ(s˙ 1)[ ˙f(φ(s2)−f˙(φ(s1)] , u2=φ(s1)−φ(s2) + ˙f(φ(s1))[f(φ(s2)−f(φ(s1)]

φ(s˙ 2)[ ˙f(φ(s2)−f˙(φ(s1)] .

For the sake of brevity, setx1:=φ(R1). We estimateu1usingφ^˙ ≤1and applying Cauchy’s mean-value theorem

u1≥ inf

x∈[x1,∞)

f˙(x1) ˙f(x)−1

f¨(x) = inf

x∈[x1,∞)

α²k²(xx1)^α−1−1 α(α−1)kx^α−2

≥ inf

x∈[x1,∞)

x(α²k²x^α−1₁ −x^1−α)

α(α−1)k = x1(α²k²x^α−1₁ −x^1−α₁ ) α(α−1)k >0.

Hence, there existsa2∈(0, ρm)such thatu1, u2∈(0, a2)is incompatible withs1< s2. Case (ii). The linear system (2.23) reduces to

(2.24)

( _˙

f(φ(s1)) cosθ1c1−f˙(φ(s2)) cosθ2c2=φ(s1) cosθ1−φ(s2) cosθ2, f˙(φ(s1)) sinθ1c1−f˙(φ(s2)) sinθ2c2=φ(s1) sinθ1−φ(s2) sinθ2. Solving it and returning to the initial variablesuj,j= 1,2, we find

uj = φ(sj)

φ(s˙ j) ˙f(φ(sj)), j= 1,2.

In view of LemmaA.1, we conclude that there existsa3 ∈(0, ρm)such that the condition u1, u2∈(0, a3)can not be satisfied together withθ16=θ2.

Merging the outcome of the analysis for the Cases (i) and (ii), we get the claim for

a⋆= min{a1, a2, a3}.

Note that the principal curvaturesκ1, κ2ofΣcomputed in (2.19) do not depend onθ. Using this observation we introduce the functions

ξp:=a sup

s∈[p,∞)

max

|κ1(s)|,|κ2(s)| =a sup

s∈[p,∞)

max

|γ(s)|,|f˙(φ(s))|φ(s)˙ φ(s)

,

ζp:=

1−ξp

1 +ξp

2

. (2.25)

In view of the conditionakκjk∞<1forj= 1,2, LemmaA.1and PropositionA.3(i) yield that the functionsR_{+∋p7→ξp, ζp}satisfy0< ξp<1,lim_p→∞ξp= 0, andlim_p→∞ζp= 1.

2.4. Spectral properties of auxiliary one-dimensional Schrödinger operators Consider the one-dimensional Schrödinger operator

(2.26) hqψ=−ψ^′′+qψ, domhq =H²(R_+)∩H¹

0(R_+),

acting in the Hilbert spaceL²(R₊₎with a real-valued potentialq∈L^∞(R₊₎. The operator hq is obviously self-adjoint, and the corresponding quadratic form will be denoted by hq. The next proposition describes relations between the spectral properties ofhq and the behavior of the potentialq(s)ass→ ∞. It covers only the classes of potentials required for the proof of our main result.

Proposition 2.1. Let the self-adjoint operatorhq be as in(2.26)with the potentialq: R_{+ →}R satisfying

(2.27) _|q(s)| ≤ C

(1 +s)^β and lim

s→∞s^βq(s) =−c, withβ∈(0,2],C >0, andc≥0. Then the following claims hold.

(i) σess(hq) = [0,∞).

(ii) Forβ = 2andc= 0, it holdsN_0(hq)<∞. (iii) Forβ ∈(0,2)andc >0, it holdsN₀(hq) =∞and

N_{−E(hq) ∼}

Eց0

c^β1B

3

2,¹_β−¹₂ πβE^β1−12 , whereB(·,·)is the Euler beta-function.

Proof. The claim (i) follows from [Te, Lem. 9.35], because the potential q satisfies the condition

Z n+1

n |q(s)|ds≤ Z n+1

n

Cds

(1 +s)^β ≤ C

(1 +n)^β →0, n→ ∞.

The claims (ii) and (iii) follow from [BKRS09, Lem 4.9], see also [RS-IV, Thm. XIII.9 (a)]

and [RS-IV, Thm. XIII.82]. We remark that the original claim of [BKRS09, Lem 4.9] is formulated for operators on the full line, but the result for half-line operators follows easily with an additional factor ¹₂ in the asymptotic formula for the eigenvalue counting

function.

3. Bracketing lemma

Pick an arbitraryp >0and define the line segmentS_⊂Gby S_{:=τ({p} ×}I_a),

where the mappingτ is as in (2.15). For the sake of brevity, we mostly avoid indicating the dependence onpas long as there is no danger of misunderstanding. We define the

open setsΩ0 := (0, p)×I_a andΩ1 := (p,∞)×I_a. Clearly, the line segmentSsplits the parabolic stripGinto two disjoint sub-domainsG_{j := τ(Ωj)},j = 0,1. In particular, we have the orthogonal decompositions

L²(G_{) =L}²₍G_0)⊕L²₍G₁₎ and L²(G_{;rdrdz) =L}²₍G_0;rdrdz)⊕L²₍G_1;rdrdz).

Next, we introduce form∈Zthe following symmetric, densely defined quadratic forms inL²(G₁₎,

f^N_m1[ψ] :=

Z

G1

F_{m[ψ]drdz, dom}f^N_m1:={Ψ|^G1: Ψ∈H₀¹(G_)}, (3.1a)

f^D_m1[ψ] :=

Z

G1

F_{m[ψ]drdz, dom}f^D_m1:=H₀¹(G_1), (3.1b)

where the integrandF_m[ψ]is as in (2.10). It is easy to see that both the formsf^N_m1andf^D_m1 are closed and semi-bounded. Now we are in position to formulate the bracketing lemma which we employ in the following.

Lemma 3.1. For anyp >0,m∈Z, and allE >0the following claims hold.

(i) N_E∞−E(Fm)≥N_E∞−E(f^D_m1).

(ii) N_E∞−E(Fm)≤N_E∞−E(f^N_m1) +C_m^N for some constantC_m^N=C_m^N(p)>0.

Proof. (i) First, we define the restriction of the quadratic form Fm,m ∈ Z, to functions vanishing onS,

(3.2) F^S_m[ψ] :=Fm[ψ], domF^S_m:=

ψ∈domFm:ψ|^S= 0 .

The quadratic form F^S_m can be naturally decomposed into the orthogonal sum F^S_m = F^S_m0⊕F^S_m1, where the quadratic forms F^S_m0 and F^S_m1 are defined in the Hilbert spaces L²(G₀;rdrdz)andL²(G₁;rdrdz), respectively. Using the orderingFm ≺ F^S_m of the forms and the min-max principle we obtain

N_E∞−E(Fm)≥N_E∞−E(F^S_m) =N_E∞−E(F^S_m0) +N_E∞−E(F^S_m1), ∀E >0.

It remains to notice that the formF^S_m1is unitarily equivalent via (3.3) U1:L²(G_1;rdrdz)→L²₍G_1), U1ψ:=√

rψ, to the formf^D_m1in (3.1b), and therefore the desired inequality in (i) holds.

(ii) Letχ0:R_{+ → [0,1]}be aC^∞-smooth function satisfyingχ0(s) = 1for alls < ¹₂ and χ0(s) = 0for alls >1. Define the functionχ1:R_+→[0,1]through the identityχ²₀+χ²₁≡1 onR₊. Employing the curvilinear coordinates(s, u)on the parabolic stripGwe introduce the cut-off functionsX_j,p: G_→[0,1],j= 0,1, by

X_{0,p(s, u) :=χ0(p}⁻¹_s) and X_{1,p(s, u) :=χ1(p}⁻¹_{s), p >0.}

These cut-off functions can also be viewed as functions of the arguments(r, z). Define the expressionXp: G_→R₊by

(3.4) Xp:=|∇X_0,p|²_+|∇X_1,p|²_.

SetΩ^′₀ := (0, p)×I_a andΩ^′₁ := (^p₂,∞)×I_a. The parabolic stripGcan be represented as the union of the domainsG^′_{0 := τ(Ω}^′₀₎and G^′_{1 := τ(Ω}^′₁₎, having a non-empty bounded intersectionG^′_{01 :=}G^′_0∩G^′₁. The expressionXpin (3.4) can be pointwise estimated from above asXp ≤ CXp⁻²¹G^′₀₁ with some constantCX > 0, here¹G^′₀₁ is the characteristic function ofG^′₀₁. Next we define a family of quadratic forms parametrized bym∈Zand j= 0,1through

E^D_mj[ψ] :=

Z

G^′_j

E_m[ψ]rdrdz−CXp⁻² Z

G^′₀₁

|ψ|²rdrdz,

domE^D_mj:=

Ψ|^G′_j: Ψ∈domFm,supp Ψ⊂G^′_{j ,}

where the integrandE_m[ψ]is as in (2.10). Further, applying the IMS formula [CFKS, §3.1]

to the quadratic formFm, we obtain that(X_j,pψ)|G^′

j ∈domE^D_mj,j = 0,1and with a slight abuse of notation we have

Fm[ψ]≥E^D_m0[X_{0,pψ] +}E^D_m1[X_{1,pψ], ∀ψ∈dom}Fm. Hence, by [DLR12, Lem. 5.2] we get

N_E∞−E(Fm)≤N_E∞−E(E^D_m0) +N_E∞−E(E^D_m1), ∀E >0.

Notice that the quadratic formE^D_m0 corresponds to a self-adjoint operator with a com- pact resolvent, while the formE^D_m1is unitarily equivalent viaU^′₁:L²(G^′_1;rdrdz)→L²₍G^′₁₎, U^′₁ψ:=√rψ, to the following form in the Hilbert spaceL²(G^′₁),

H₀¹(G^′_1)∋ψ7→

Z

G^′₁

F_{m[ψ]drdz−CXp}⁻² Z

G^′₀₁

|ψ|²drdz,

which is larger in the sense of ordering than the orthogonal sumf_⊕f^N_m1of the formf^N_m1in the Hilbert spaceL²(G₁)defined in (3.1a) and the form in the Hilbert spaceL²(G^′₀₁),

f[ψ] :=

Z

G^′₀₁

F_m[ψ]−CXp⁻²_|ψ|²

drdz, domf:=

Ψ|^G′01: Ψ∈H₀¹(G^′_{1) .}

It only remains to notice that the last form also corresponds to a self-adjoint operator in the Hilbert spaceL²(G^′₀₁₎with a compact resolvent. Thus, the desired inequality holds with the constantC_m^N =N_E∞(E^D_m0) +N_E∞(f).

4. Straightening of the meridian domain

Recall that the curveΓin (2.2) is parametrized via the mappingR_+∋s7→(φ(s), f(φ(s))) withφsatisfying the differential equation (2.11). Furthermore, recall that the quadratic

formFmin (2.7) is unitarily equivalent through the unitary transformationUto the form fmin (2.8) expressed in the flat metric.

Let us introduce auxiliary potentials V_m^C, Vm: Ω → R, m ∈ Z, on the half-strip Ω = R_+×I_aby the formulæ

V_m^C(s, u) := m²−¹4

φ(s)−uf˙(φ(s)) ˙φ(s)^2, Vm(s, u) := u¨γ(s)

2g^3/2(s, u)− γ²(s) 4g(s, u)−5

4

u²γ˙²(s)

g²(s, u)+V_m^C(s, u).

(4.1)

The first three terms in the definition ofVmcorrespond to the curvature-induced potential arising while straightening the curved two-dimensional half-stripG;cf. [EK, Sec. 1.1]. The fourth termV_m^Ccorresponds to the centrifugal potential written in the(s, u)-coordinates.

For the sake of brevity, we introduce shorthand notations for the integrands,

(4.2)

T_m[ψ](s, u) := |∂sψ|²

g(s, u)+|∂uψ|²+Vm|ψ|², m∈Z_, B_{p[ψ](u) :=} ^γ

′(p)u|ψ(p, u)|²

2g^3/2(p, u) , p≥0.

Next, we define the unitary operatorV:L²(G)→L²(Ω)by (Vψ)(s, u) :=g(s, u)^1/4ψ τ(s, u)

,

where the mapτis as in (2.15), its JacobianJis given by (2.16) andg=J²as in (2.17). The quadratic formfmis unitarily equivalent via the mappingVto the quadratic form (4.3) tm[ψ] :=fm[V⁻¹ψ], domtm:=V(domfm),

on the Hilbert spaceL²(Ω). Furthermore, recall thatΩ1 = (p,∞)×I_{a ⊂Ω}and introduce the unitary operatorV1:L²(G_1)→L²_(Ω1)by

(V1ψ)(s, u) :=g(s, u)^1/4ψ τ(s, u) .

The quadratic formsf^D_m1 and f^N_m1 in (3.1) can be further unitarily transformed into the forms

t^D_m1[ψ] :=f^D_m1[V⁻₁¹ψ], domt^D_m1:=V1(domf^D_m1), (4.4)

t^N_m1[ψ] :=f^N_m1[V⁻¹₁ ψ], domt^N_m1:=V1(domf^N_m1), (4.5)

on the Hilbert spaceL²(Ω1).

In the remaining part of this section we will get more explicit expressions for the forms tm,t^D_m1andt^N_m1. LetΨ∈H¹(G₎andψ∈H¹(Ω)be connected through the relationΨ◦τ = J^−1/2_ψ. Let also(t1, t2)^⊤ and (n1, n2)^⊤ be, respectively, the unit tangential and normal vectors toΓ. Using the Frenet formula we find

∂s(J^−1/2_{ψ) = (∂rΨ◦τ)∂sτ1+ (∂zΨ◦τ)∂sτ2}

= (∂rΨ◦τ)t1(1−uγ) + (∂zΨ◦τ)t2(1−uγ) =J_{((∂rΨ◦τ)t1+ (∂zΨ◦τ)t2),}

∂u(J^−1/2_{ψ) = (∂rΨ◦τ)∂uτ1+ (∂zΨ◦τ)∂uτ2= (∂rΨ◦τ)n1+ (∂zΨ◦τ)n2,}

Hence, we obtain

(4.6) _|(∇Ψ)◦τ|²=|∂s(J^−1/2_ψ)|²

J^{2 +|∂u(}J^−1/2_ψ)|²_.

In the spirit of [EK, Sec. 1.1], we find that domtm = H₀¹(Ω)for allm ∈ Z_{\ {0}}and by performing elementary computations relying on (4.6) we get

tm[ψ] = Z

Ω

|∂s(J^−1/2_ψ)|²

J ^+|∂u(J^−1/2ψ)|²J+V_m^C|J^−1/2ψ|²J

= Z

Ω

|∂sψ|² J^{2 −}

uγRe (ψ∂˙ sψ) J^{3 +}

u²γ˙²|ψ|²

4J^{4 +|∂uψ|}

2−γRe (ψ∂uψ)

J ⁺

γ²|ψ|² 4J^{2 +V}

mC|ψ|²

= Z

Ω

|∂sψ|² J^{2 −}

uγ∂˙ s(|ψ|²) 2J^{3 +}

u²γ˙²|ψ|²

4J^{4 +|∂uψ|}

2−γ∂u(|ψ|²)

2J ⁺

γ²|ψ|² 4J^{2 +V}

C m|ψ|². Integrating by parts in the above formula, we obtain

tm[ψ] = Z

Ω

|∂sψ|²

J^{2 +|∂uψ|}

2+ u²γ˙²

4J^{4 +∂s} uγ˙

2J³

+∂u

γ 2J

+ γ²

4J^{2 +V}

mC

|ψ|²

= Z

Ω

|∂sψ|²

J^{2 +|∂uψ|}

2+ u²γ˙²

4J^{4 +} u¨γ 2J^{3 −}

3u²γ˙² 2J^{4 −}

γ² 2J²⁺

γ² 4J^{2 +V}

C m

|ψ|²

= Z

Ω

|∂sψ|²

J^{2 +|∂uψ|}

2+ u¨γ

2J³⁻ 5u²γ˙²

4J^{4 −} γ² 4J^{2 +V}

mC

|ψ|²

= Z

Ω

T_m[ψ], (4.7)

the boundary terms vanished thanks to the Dirichlet boundary condition.

Analogously, we get for allm∈Z t^D_m1[ψ] :=

Z

Ω1

T_m[ψ]dsdu, domt^D_m1:=H₀¹(Ω1),

Mimicking the above computation for the form t^N_m1, m ∈ Z, we arrive at domt^N_m1 = {ψ|Ω1:ψ∈H₀¹(Ω)}and obtain that

t^N_m1[ψ] :=

Z

Ω1

|∂sψ|² J^{2 −}

uγ∂˙ s(|ψ|²) 2J^{3 +}

u²γ˙²|ψ|²

4J^{4 +|∂uψ|}

2−γ∂u(|ψ|²)

2J ⁺

γ²|ψ|² 4J^{2 +V}

C m|ψ|². Again integrating by parts, we finally get

t^N_m1[ψ] = Z

Ω1

|∂sψ|²

J^{2 +|∂uψ|}

2+Vm|ψ|²

dsdu+ Z

I_a

˙

γ(p)u|ψ(p, u)|² 2J³_{(p, u)} ^du

= Z

Ω1

T_m[ψ]dsdu+ Z

I_a

B_p[ψ]du.

While the Dirichlet boundary condition on∂G_1\ S is preserved upon straightening of G₁, the Neumann boundary condition on the line segmentStransforms into the Robin boundary condition with the coupling functionI_{a ∋u7→} ^γ(p)u˙

2J³(p,u). This peculiarity mani- fests in the presence of the boundary in the expression fort^N_m1.

5. Properties of N

∞−E(F^m)

for

Z

In this section, we investigate the spectral counting function of the fiber operatorsFm

for allm ∈ Z_{\ {0}}. First, we show that for allm 6= 0the discrete spectrum of the fiber operators below the thresholdE_∞is at most finite, and secondly, that this discrete spec- trum is empty for all but finitely many such fiber operators. This informal explanation is precisely formulated in the proposition below, the proof of which relies on the min-max principle and on the finiteness of the discrete spectrum for a class of one-dimensional Schrödinger operators stated in Proposition2.1(ii).

Proposition 5.1. Let^Fm, m ∈ Z_{\ {0}}, be the self-adjoint fiber operators in the Hilbert space L²(G_;rdrdz)associated with the quadratic forms in(2.7). Then there exists anM =M(f, a)∈N₀ such that:

(i) 1≤N_E∞(Fm)<∞for allm= 1,2, . . . , M; (ii) N_E∞(Fm) = 0for allm > M.

Proof. First of all, recall that the formtm in (4.3) is unitarily equivalent to the form Fm

which is represented by the operatorFm. We organize the argument into three steps.

Step 1: bracketing. We infer that the orderingsm≺tmholds, where the formsmis given by (5.1) sm[ψ] =

Z

Ω

|∂sψ|²

(1 +ξ0)² +|∂uψ|²+Um|ψ|²

duds, domsm=H₀¹(Ω), withξ0as in (2.25) forp= 0and the potentialUmdefined by

(5.2) Um(s) = 1 φ²(s)

_m2−¹₄

(1 +ξ0)² −φ²(s)γ²(s)

4(1−ξ0)² −aφ²(s)|¨γ(s)| 2(1−ξ0)³ −5

4

a²φ²(s) ˙γ²(s) (1−ξ0)⁴

.

Step 2: large|m| ∈N.In view of PropositionsA.2(i) andA.3(i)-(iii), the functions s7→φ²(s)γ²(s), s7→φ²(s)|γ(s)¨ | and s7→φ²(s) ˙γ²(s)

are all bounded onR₊. Hence, for any_|m| ≥M withM ∈Nlarge enough, the potential Umis pointwise positive. Fur suchm’s we obtain

N_E∞(Fm) =N_E∞(fm) =N_E∞(tm)≤N_E∞(sm) = 0.

Step 3: small|m| ∈N.In view of the asymptotics ofγ(s),γ(s)˙ andγ(s)¨ ass→ ∞shown in PropositionA.3, the potentialq:R_+→Rdefined by

(5.3) q(s) :=−1

ζ0

_γ2(s)

4 + a|γ(s)¨ | 2(1−ξ0)+5

4

a²γ˙²(s) (1−ξ0)²

,

whereξ0 andζ0 are as in (2.25) withp= 0, satisfies the condition (2.27) withβ = 2and c = 0and therefore by Proposition2.1the negative discrete spectrum of the self-adjoint operatorhqin (2.26) is finite.