Spectral asymptotics of a strong δ interaction supported by a surface

(1)

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Physics Letters A

www.elsevier.com/locate/pla

Spectral asymptotics of a strong δ interaction supported by a surface

Pavel Exner

^a

^,

^b

^,

^∗

, Michal Jex

^a

^,

^c

aDopplerInstituteforMathematicalPhysicsandAppliedMathematics,CzechTechnicalUniversityinPrague,Bˇrehová7,11519Prague,CzechRepublic bNuclearPhysicsInstituteASCR,25068ˇRežnearPrague,CzechRepublic

cDepartmentofPhysics,FacultyofNuclearSciencesandPhysicalEngineering,CzechTechnicalUniversityinPrague,Bˇrehová7,11519Prague,CzechRepublic

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received25February2014 Receivedinrevisedform9June2014 Accepted10June2014

Availableonline12June2014 CommunicatedbyP.R.Holland

Keywords:

δsurfaceinteraction Strongcouplingexpansion

WederiveasymptoticexpansionforthespectrumofHamiltonianswithastrongattractiveδ interaction supportedbyasmoothsurfaceinR³^,êitherînfiniteândasymptoticallyplanar,orcompactand closed.

Its secondtermis foundtobedeterminedbyaSchrödingertypeoperator withaneffectivepotential expressedintermsoftheinteractionsupportcurvatures.

1. Introduction

Quantum mechanics of particles conﬁned to curves, graphs, tubes, surfaces,layers, andother geometrically nontrivial objects isa rich andinspirative subject. On one hand it isuseful physi- cally,inparticular, todescribe various nanostructures,andatthe sametimeit offersnumerousinteresting mathematicalproblems.

Modelsof“leaky”structures[1]inwhichtheconﬁnementisreal- izedby an attractive potentialhave theadvantage that they take quantumtunnelingintoaccount.The potentialisoftentakensin- gular,ofthe

δ

type,becauseitiseasiertohandle[2].

Veryrecently also more singular couplings of the

δ

type at- tractedattention.ThecorrespondingHamiltonianscanbeformally writtenas

H_β

= − Δ − β

⁻¹

δ

( · − Γ ),

(1)

where

Γ

isasmoothsurfacesupportingtheinteraction.Somepre- fer to write the interaction termas

β

⁻¹

(δ

(

·−

Γ ),

·

)δ

(

·−

Γ )

to stressthattheinteractionisinvariantwithrespecttomirrorreﬂec- tion.Whatisimportant,however,isthateitheroftheexpressions ispurelyformal. Aproper deﬁnitionwhichemploys thestandard

δ

concept[3]willbegivenbelow,hereweonlynotethatwewrite

β

⁻¹ tounderline thatastrong

δ

interactioncorresponds tosmall valuesoftheparameter

β

.Wealsonotethatinvestigationofsuch

δ

interactionsisnot justamathematicalexercise. Duetoasem- inalideaof CheonandShigehara[4]made rigorousin[5,6] they

*

Correspondingauthorat:NuclearPhysicsInstituteASCR,25068ˇRežnearPrague, CzechRepublic.

E-mailaddresses:[email protected](P. Exner),jexmicha@fjﬁ.cvut.cz(M. Jex).

canbeapproximatedbyascaled“tripple-layer”potentialcombina- tion.The possibilityofformingsuch systemswithbarrierswhich become moreopaque as theenergy increasesis nodoubt physi- callyattractive.

Thesubjectofthisletteristhe strongcouplingasymptotics of boundstatesofoperators(1)withanattractive

δ

interactionsup- portedbyafiniteorinfinitesurfaceinR³^.^Theânalogous^problem for

δ

interactionsupportedbyinﬁnitesurfacewassolvedin[7].As inthis case,we are going toshow that the asymptoticsis deter- minedbythegeometryof

Γ

.Asabyproduct,wewilldemonstrate the existence of bound states for suﬃciently small

β

for non- planar inﬁnitesurfaceswhichare asymptoticallyplanar, ina way alternativetotheargumentproposedrecentlyin[8].

2. TheHamiltonian

The ﬁrst thing todo is to deﬁne properlythe operator (1).It acts,ofcourse,asLaplacianoutsideofthesurface

Γ

(

H_β

ψ )(

x

) = − (Δψ )(

x

)

forx∈R³\

Γ

andtheinteractionwillbeexpressedthroughsuit- ableboundaryconditionsonthesurfacewhich,inaccordwith[3], would include continuity of the normal derivative together with a jumpofthefunctionvalue.Speciﬁcally,thedomainoftheoper- atorwillbe

D

(

H_β

) = ψ ∈

^H²

R

³

\ Γ ∂

nΓ

ψ (

x

) = ∂

₋nΓ

ψ (

x

) =: ψ

(

x

)

Γ

,

− βψ

(

x

)

Γ

= ψ (

x

)

∂₊Γ

− ψ (

x

)

∂₋Γ

,

(2)

where n_Γ is the normal to

Γ

and

ψ(

x

)

|∂_±Γ are the appropri- ate traces of the function

ψ

; all these quantities exist in view of the Sobolev embedding theorem. Being interested in the attractive

δ

interactions, we choose the above form of boundary conditions with

β >

0. Another way to deﬁne the operator H_β is by the means of the associated quadratic form as discussed in [2]. Its domain is H¹

(R

³\

Γ )

and the formvalue fora function

ψ

∈^H¹

(R

³\

Γ )

isgivenby

h_β

[ ψ ] = ∇ ψ

²

− β

⁻¹

ψ (

x

)

∂₊Γ

− ψ (

x

)

∂₋Γ

²

L²(Γ )

.

(2)

As indicated we are interested in the spectrum of Hβ inthe strong-couplingregime,

β

→⁰+,fortwokindsofsurfaces

Γ

.The ﬁrstisaninﬁnitesurfaceofwhichweassumethat:

(a1)

Γ

is C⁴ smooth andallows a globalnormal parametrization withuniformlyboundedelliptictensor,

(a2)

Γ

has no “near self-intersections”, i.e. there exists its sym- metric layer neighborhood of a ﬁnite thickness which does notintersectwithitself,

(a3)

Γ

is asymptotically planar in the sense that its curvatures vanish asthe geodetic distancefrom a ﬁxed point tends to inﬁnity,

andﬁnally

(a4) trivialcaseisexcluded,

Γ

isnotaplane.

Infact,theassumption(a1)can beweakenedinawaysimilarto [9],however,forthesakeofsimplicitywesticktotheexistenceof aglobalnormalparametrization. Thesecond classtoconsiderare ﬁnite surfaces. The compactness makes the assumptions simpler in this case, on the other hand, we have to require additionally absenceofaboundary:

(b)

Γ

isaclosedC⁴ smoothsurfaceofaﬁnitegenus.

In this case no global parametrization exists, of course, but the geometryof

Γ

canbedescribedby anatlasofmapsrepresenting normalparameterizations withauniformlyboundedelliptictensor.

3. Geometricpreliminaries

Letuscollectnowsomeneededfactsaboutthegeometryofthe surface and its neighborhoods;for a more complete information werefer,e.g.,to[10].Weconsiderinﬁnitesurfacesﬁrstandwein- troducenormalcoordinateson

Γ

startingfromalocalexponential map

γ

:^To

Γ

→^Uowiththeorigino∈

Γ

totheneighborhoodU_o ofthepointo;thecoordinatessaregivenby

s

= (

s₁

,

s₂

) →

^expo i

s_ie_i

(

o

)

(3) where{^e1

(

o

),

e₂

(

o

)}

^is ^anorthonormal basis of T_o

Γ

.Byassump- tion(a1)onecanﬁndapointo∈

Γ

suchthatthemap(3)canbe extendedtoaglobaldiffeomorphismfromTo

Γ

R² ^to

Γ

.

Usingthesecoordinates,weexpresscomponentsofthesurface metric tensor gμν as gμν=

γ

,μ·

γ

,ν anddenote gμν=

(

gμν

)

⁻¹. The invariant surface element is denoted as d

Γ

=^g¹²^d²^s ^where g:=^det^gμν.Theunitnormaln

(

s

)

isdeﬁnedasthecrossproduct ofthelinearlyindependenttangentvectors

γ

,μ,i.e.n

(

s

)

=_|^γ_γ^,1_,1^×_×^γ_γ^,2_,2_|^. TheGausscurvatureK andmeancurvatureMcanbecalculatedby meansoftheWeingartentensorhν

μ:= −ⁿ,μ·

γ

,σgσ ν, K

=

^det^h^ν_μ

=

^k1k₂

,

M

=

¹

2Trh^ν_μ

=

¹

2

(

k₁

+

^k2

).

We recall thatthe eigenvalues ofhν

μ are theprincipal curvatures k₁_,₂ andthattheidentity K−^M²= −¹₄

(

k₁−^k2

)

² holds.

We also need neighborhoodsof the surface

Γ

. Alayer

Ω

d of halfwidthd

>

0 will bedeﬁnedastheimageof Dd:= {

(

s

,

u

)

:^s∈ R²

,

u∈

(−

d

,

d

)}

bythemap

L

:

Dd q

≡ (

s

,

u

) → γ (

s

) +

un

(

s

)

(4) Thisdeﬁnitionprovidesatthesametimeaparametrizationof

Ω

d, andtheassumption(a2)canberephrasedas

(a2) there is a d0

>

0 such that the map (4)is injective forany d

<

d₀.

Moreover, in view of(a1) such an Lis a diffeomorphism,which willbecrucialfortheconsiderationstofollow.Thelayer

Ω

d canbe regardedasamanifoldwithaboundaryandcharacterized bythe metric tensorwhich can be expressed inthe parametrization(4) as

G_{i j}

=

(

G_μν

)

0

0 1

,

whereGμν=

(δ

_μ^σ−^uh^σμ

)(δ

_ρ^σ−^uh^σρ

)

gρν.Weuseheretheconven- tioninwhichtheLatinindicesrunthrough1

,

2

,

3,numberingthe coordinates

(

s₁

,

s₂

,

u

)

in

Ω

d,andtheGreekonesthrough1

,

2.The volume element ofthe manifold

Ω

_d can be written in the form d

Ω

d:=√

Gd²sduwith G

:=

detGi j

=

g

(

1

−

uk1

)(

1

−

uk2

)

2

=

g

1

−

2Mu

+

K u²

2

;

withthefuturepurposeinmindweintroduceashorthandforthe last factor,

ξ(

s

,

u

)

:=¹−^2M

(

s

)

u+^K

(

s

)

u².Thecurvaturesalsoal- lowustoexpressmoreexplicitlythenextassumption:

(a3) K

,

M→^{0 as}|^s|:=

s²₁+^s²₂→ ∞^.

Recall next a few useful estimates made possible by the assumption(a3),cf.[11].Incombinationwith(a1)and(a2)itimplies thattheprincipalcurvaturesk1andk2 areuniformlybounded.We set

ρ _:=

max

^k1

∞

,

^k2

∞

₋1

;

note that

ρ >

d0 holds for the critical halfwidth of assumption (a2). It can be checked easily that for a givend

< ρ

thefollow- inginequalitiesaresatisﬁedinthelayerneighborhood

Ω

dof

Γ

,

C₋

(

d

) ≤ ξ ≤

^C+

(

d

),

(5)

whereC_±:=

(

1±^d

ρ

⁻¹

)

²,andthisinturnimplies

C₋

(

d

)

g_μν

≤

^Gμν

≤

^C+

(

d

)

g_μν

.

(6) Since themetrictensor gμν uniformlyellipticbyassumption,we alsohave

c₋

δ

_μν

≤

^gμν

≤

^c+

(

d

)δ

_μν (7)

asamatrixinequalityforsomepositiveconstantsc_±.

Letusbrieflydescribemodificationsneededifwepasstoclosed surfaces.Aswehaveindicatedaglobalparametrizationisreplaced now by a finiteatlas A ôf^maps; ⁱⁿêach ^part Mi we introduce normal coordinates anddefine layer neighborhoodsby the maps MîonD_i_,_d:= {(^s

,

u

)

:^s∈^domMi

,

u∈

(−

^d

,

d

)}

^with^a^given^d

>

0, M

ˆ

i

:

^Di,d q

≡ (

s

,

u

) → γ

i

(

s

) +

^un

(

s

)

(8) In view ofassumption(b)there isa criticald₀

>

0 suchthat ev- erymap Mˆi:^Di,d→

Ω

_d fromA^is^injective^provided^d

<

d0 and

(3)

a diffeomorphism. Furthermore, Mˆi

(

s_i

,

u_i

)

= ˆMj

(

s_j

,

u_j

)

implies Mi

(

si

)

=Mj

(

sj

)

. The above estimates of the metric tensor re- mainsvalidalsoforcompact

Γ

.

4. Theresults

As in the caseof a

δ

interaction supported by a surface, the asymptoticsisdeterminedbythegeometryof

Γ

.Tostatethere- sults,weintroducethefollowingcomparisonoperator,

S

= −Δ

Γ

−

¹

4

(

k1

−

k2

)

²

= −Δ

Γ

+

K

−

M²

,

(9) where

Δ

Γ istheLaplace–Bertrami operatoronthesurface

Γ

and k1,2 aretheprincipalcurvaturesof

Γ

.ThespectrumofSispurely discrete if

Γ

is compact. In the noncompact case the potential vanishes at inﬁnity andhas negative valuesunless

Γ

is a plane which is, however, excluded by assumption (a4). Consequently,

σ

ess

(

S

)

= [⁰

,

∞

)

andthe discrete spectrum isnonempty. We de- note the eigenvalues of S, arranged in the ascending orderwith themultiplicitytakenintoaccount,as

μ

j.

Firstwe inspectthe essentialspectrum inthe strong-coupling regime:

Theorem1.Letaninﬁnitesurface

Γ

satisfyassumptions(a1)–(a4),then

σ

ess

(

H_β

)

⊆ [

(β),

∞)^,^where

(β)

= −_β⁴2 +O(^e⁻^c^/β

)

holdsas

β

→ 0₊forsomeconstantc

>

0.

Wenotethatincaseofacompact

Γ

wehave

σ

ess

(

Hβ

)

= [⁰

,

∞

)

; aproof can befound in[8].Thenext two theoremsdescribe the asymptoticsofthenegativepointspectrumofHβ.

Theorem2.Letaninﬁnite surface

Γ

satisfyassumptions(a1)–(a4), then Hβhasatleastoneisolatedeigenvaluebelowthethresholdofthe essentialspectrumforallsuﬃcientlysmall

β >

0,andthe j-theigen- valuebehavesinthelimit

β

→⁰+as

λ

j

= −

⁴

β

²

+ μ

j

+

O

(−β

^ln

β).

Theorem3.Letacompactsurface

Γ

satisfyassumption(b),thenH_β hasatleastoneisolatedeigenvaluebelowthethresholdoftheessential spectrumforall

β >

0,andthej-theigenvaluebehavesinthelimit

β

→ 0₊as

λ

j

= −

⁴

β

²

+ μ

j

+

O

(−β

^ln

β).

5. Bracketingestimates

Thebasic ideais analogoustothe one usedin [7], namelyto estimatetheoperator H_β fromaboveandbelow,inatightenough manner, by suitable operators for which we are able to calcu- late the spectrum directly. The starting point for such estimates isthebracketing trick,that is,imposing additionalDirichlet/Neu- mannconditionsattheboundary oftheneighborhood

Ω

d ofthe surface

Γ

.Weintroducequadraticformsh⁺_β andh⁻_β,bothofthem givenbytheformula

∇ψ

²_L2(Ωd)

− β

⁻¹

Γ

ψ (

s

,

0₊

) − ψ (

s

,

0₋

)

²d

Γ

withthedomainsD(^h⁺_β

)

= ˜^H¹₀

(Ω

d\

Γ )

andD(^h⁻_β

)

=^H¹

(Ω

d\

Γ )

, respectively, the former being understood as a set of functions whicharelocallyH¹andvanishattheboundaryof

Ω

a.Wedenote theself-adjoinedoperatorsassociatedwiththeseformsasH^±_β.By thestandardbracketingargumentweget

−Δ

_R^N3\Ωd

⊕

H_β⁻

≤

Hβ

≤ −Δ

_R^D3\Ωd

⊕

H_β⁺

,

(10) where−

Δ

^D_R^,3^N\Ωd istheDirichlet–Laplacian andNeumann–Laplacian respectivelyon theset R³\

Ω

_d. Theoperators −

Δ

_R^D₃^,^N_\Ω

d arepos- itive, hence all the information about the negative spectrum is encodedintheoperators H_β^±.

Thenextstepistotransformtheoperators H_β^±intothecurvi- linear coordinates

(

s

,

u

)

. This is done by means of the unitary transformation

U

ψ = ψ ◦

L

:

^L²

(Ω

_d

) →

^L²

(

D_d

,

d

Ω).

By

(·,

·)G we denotethescalarproductin L²

(

D_d

,

d

Ω)

.The opera- torsU H_β^±U⁻¹ actingonthisspaceareassociatedwiththeforms h^±_β

U⁻¹

ψ

=

∂

i

ψ,

G^{i j}

∂

j

ψ

G

− β

⁻¹

Γ

ψ (

s

,

0₊

) − ψ (

s

,

0₋

)

²d

Γ

havingthedomains H˜₀¹

(

Dd\

Γ,

d

Ω)

and H˜¹

(

Dd\

Γ,

d

Ω)

,respectively. Next we employ another unitary transformation, inspired by [11], withthe aimto getridof thetransverse coordinatede- pendence,i.e.switchfromthemetricd

Ω

tod

Γ

duby

U

˜ ψ = ξ

¹²

ψ :

L²

(

Dd

,

d

Ω) →

L²

(

Dd

,

d

Γ

du

).

Similarlyasbefore,wedenotethescalarproductin L²

(

D_d

,

d

Γ

du

)

as

(

·

,

·

)

g andconsidertheoperators

F_β^±

:= ˜

^{U U H}^±_β^U⁻¹U

˜

⁻¹

which act in L²

(

D_d

,

d

Γ

du

)

. The quadratic forms

ζ

_β^± associated withF_β^±canbecalculatedash^±_β

(

U˜⁻¹U⁻¹

ψ)

withtheresult

ζ

_β⁺

[ψ] =

∂

μ

ψ,

G^μν

∂

ν

ψ

g

+

ψ, (

V₁

+

^V2

)ψ

g

+ ∂

3

ψ

g

− β

⁻¹

Γ

ψ (

s

,

0₊

) − ψ (

s

,

0₋

)

²d

Γ

−

Γ

M

ψ (

s

,

0₊

)

²

− ψ (

s

,

0₋

)

²

d

Γ

ζ

_β⁻

[ψ] = ζ

_β⁺

[ψ] +

Γ

ς (

s

,

d

) ψ (

s

,

d

)

²_d

Γ

−

Γ

ς (

s

, −

^d

)ψ (

s

, −

^d

)

²d

Γ,

where

ς

₌ ^M⁻_ξ^{K u},thetwocurvature-inducedpotentialsare

V₁

=

^g⁻¹²

g¹²G^μνJ_,_μ

,ν

+

^J,μG^μνJ_,_ν

,

V₂

=

^K

−

^M²

ξ

²

with J= ^ln₂^ξ^. ^The corresponding form domains are H˜¹₀

(

D_d\

Γ,

d

Γ

du

)

andH˜¹

(

D_d\

Γ,

d

Γ

du

)

,respectively.

6. ProofofTheorem 1

Intheexcludedcasewhen

Γ

isaplane,thespectrumiseasily found byseparationofvariables. Sincea

δ

interactioninone di- mensionhasasingle eigenvalueequalto−_β⁴2,cf.[3,Sec. I.4],we get

σ (

Hβ

)

=

σ

ess

(

Hβ

)

= [−_β⁴2

,

∞

)

.We want to show that under the assumption (a3) the essential spectrum does not change, at least asymptotically.We employ an estimate which follows from Lemma 4thatwewillprovebelow,namely

(4)

d

−d

^d_du^f

²^du

− β

⁻¹

f

(

0₊

) −

^f

(

0₋

)

²

≥

−

⁴

β

²

−

¹⁶

β

²^exp

−

^4d

β

^f

L²(−^d,d)

.

(11) As we shall seethe inequality holds for suﬃciently small

β

and

d

β

>

2.Theinclusion

σ

ess

(

Hβ

)

⊆ [

(β),

∞

)

isequivalentto inf

σ

ess

(

Hβ

) ≥ (β)

which will be satisﬁed if inf

σ

ess

(

H⁻_β

)

≥

(β)

for H⁻_β acting in L²

(Ω

d

)

for d

<

g₀

< ρ

. This is obvious from inequalities (10) and the fact that the operator −

Δ

^N_R₃_\_Ω

d is positive and cannot thuscontribute to thenegative partofthe spectrum.In thenext stepwe dividethe surface

Γ

into twoparts,namely

Γ

_τ^int:= {^s∈

Γ

|^r

(

s

) < τ

_} and

Γ

_τ^ext:=

Γ

\

Γ

_τ^int. The layer neighborhoods cor- respondingto

Γ

_τ^int and

Γ

_τ^ext are D^intτ = {

(

s

,

u

)

∈^Dd|^s∈

Γ

_τînt} ând Dêxtτ =^Dd\^Dîntτ .WeintroducetheNeumannoperatorsonrespec- tiveneighborhoods,H^−,_β,τ^z forz=înt

,

ext associatedwiththeforms

∂

i

ψ,

G^{i j}

∂

j

ψ

G

− β

⁻¹

Γτ^z

ψ (

s

,

0₊

) − ψ (

s

,

0₋

)

²d

Γ

deﬁnedonH˜¹

(

D^zτ\

Γ,

d

Ω)

.UsingoncemoreNeumannbracketing wegetH⁻_β ≥^H^−,β,τînt⊕^H^−,β,τêxt.Theinnerpartiscompact,hencethe spectrumof H_β,⁻^,τînt ispurelydiscrete.Consequently, themin–max principleimplies

inf

σ

ess

H⁻_β

≥

^inf

σ

ess

H⁻_β,^,_τ^ext

,

and it is suﬃcient to check that the right-hand side cannot be smaller than

(β)

. The quantities m⁺τ :=^sup_Γ_τ^ext

ξ

and m⁻τ :=

inf_Γext

τ

ξ

tend to one as

τ

→ ∞ⁱⁿ ^view ^of^assumption ^(a3). ^We havethefollowingestimate,

ψ,

H_β,⁻^,_τ^ext

ψ

G

≥

D^extτ

∂

3

ψ (

q

)

²d

Ω

− β

⁻¹

Γτ^ext

ψ (

s

,

0₊

) − ψ (

s

,

0₋

)

²d

Γ

≥

^m⁻_τ

D^extτ

∂

3

ψ (

q

)

²d

Γ

du

− β

⁻¹

Γ_τ^ext

ψ (

s

,

0₊

) − ψ (

s

,

0₋

)

²d

Γ

≥

¹

β

²m⁺_τm⁻_τ

−

4

−

16 exp

−

^4d

β

×

D^extτ

ψ (

q

)

²d

Ω,

andsince

τ

isarbitrary,weobtain

(β)

≥ −_β⁴2−_β¹⁶2exp

(

−^4d_β

)

. 7. ProofofTheorem 2

Toprovethesecondtheorem,wewillneedseveralauxiliaryre- sults.Theoperators F_β^± arestill notsuitable toworkwithandso wereplacethem witha slightlycruder bounds.Firstwe estimate thevaluesofthepotentialsV₁ andV₂.Withthehelpofinequali- ties(5)–(7)weareabletocheckthat

dv⁻

≤

V1

≤

dv⁺

holdsforsuitablenumbersv^±andd

<

d0

< ρ

.Ontheotherhand, V2canbeestimatedas

C⁻₋²

K

−

M²

≤

V2

≤

C⁻₊²

K

−

M²

,

where C_± are thesame asin (5).This allows usto replace (10) withtheestimatesusingoperators D^±_β,

D⁻_d_,β

:=

^U_d⁻

⊗

^I

+

⊕ Γ

T_d⁻_,β

(

s

)

d

Γ ≤

^F_β⁻

≤

^Hβ

H_β

≤

^F_β⁺

≤

^U⁺_d

⊗

^I

+

⊕

Γ

T_d⁺_,β

(

s

)

d

Γ =:

^D⁺_d_,β ⁽¹²⁾

where

U_d^±

= −

^C±

Δ

_Γ

+

^C⁻_±²

K

−

^M²

+

^v^±^d

withthedomainD

(

U^±_d

)

=^L²

(

R²

,

d

Γ )

andthetransversepartacts as

T_d^±_,β

(

s

)ψ = −Δψ

withthedomains D

T_a⁺_,β

(

s

)

=

f

∈

^H²

(−

^a

,

a

) \ {

⁰

}

^f

(

a

) =

^f

(−

^a

) =

⁰

,

f

(

0₋

) =

^f

(

0₊

) = −β

⁻¹

f

(

0₊

) −

^f

(

0₋

) +

^M

f

(

0₊

) +

^f

(

0₋

)

and D

T_a⁻_,β

(

s

)

=

f

∈

^H²

( −

^a

,

a

) \ {

⁰

} ∓

^M

∞

+

^d

^K

∞ C₋ f

( ±

^a

)

=

^f

( ±

^a

),

f

(

0₋

) =

^f

(

0₊

) = −β

⁻¹

f

(

0₊

) −

^f

(

0₋

) +

^M

f

(

0₊

) +

^f

(

0₋

) ,

respectively. The negative spectrumis describedby the following resulttheproofofwhichcanbefoundin[12].

Lemma4.EachoftheoperatorsT_d^±_,β

(

s

)

hasexactlyonenegativeeigen- value t_±

(

d

, β)

, respectively,whichis independentof s providedthat

d

β

>

2and

β(

^M∞+^d^K∞

) <

1.Forall

β >

0suﬃcientlysmall theseeigenvaluessatisfytheinequalities

−

⁴

β

²

−

¹⁶

β

²^exp

−

^4d

β

≤

^t−

(

d

, β) ≤ −

⁴

β

²

≤

^t+

(

d

, β)

≤ −

⁴

β

²

+

¹⁶

β

²^exp

−

^4d

β

.

On the other hand, the spectrum of the operators U_d^± has the asymptotic expansion governed by the operator S which we can adoptfrom[7]:

Lemma5.TheeigenvaluesofU_d^±satisfytherelations

μ

^±_j

(

d

) = μ

j

+

C^±_jd

+

O

d²

for d

→

0

,

where

μ

jisthej-theigenvalueoftheoperatorS andtheconstantsC^±_j areindependentond.

(5)

Withthese prerequisites we are ready to prove the second theorem. We put d

(β)

= −

β

ln

β

. Using the fact that each of the operators T_d^±_,β

(

s

)

hasexactly one negative eigenvaluet_±

(

d

(β), β)

togetherwiththeexplicitformof D^±_d_,β wecanwritetheirspectra ast_±

(

d

(β), β)

+

μ

^±_j

(

d

(β))

, where

μ

^±_j are theeigenvalues ofthe operatorsU_d^±.UsingnowLemmata 4 and5weareabletorewrite thisas

t_±

d

(β), β

+ μ

^±_j

d

(β)

= −

⁴

β

²

+ μ

j

+

O

β |

^ln

β |

,

hencethemin–maxprincipleincombinationwithinequalities(12) concludetheargument.

8. ProofofTheorem 3

Theexistenceofisolatedeigenvaluescanbecheckedvariation- ally as in [8]. For a test function

ξ

one chooses characteristic functionofthe volumeenclosed by thesurface

Γ

; thisyields an estimateofthegroundstateenergyfromabove,

λ

0

≤

^h^β

(ξ )

ξ

²

= β

⁻¹^S

V (13)

wherehβ isquadratic(2),S isthearea ofthesurface

Γ

andV is thevolumeenclosedby

Γ

.Theproofoftheasymptoticexpansion proceed with minimum modiﬁcations as for the inﬁnite surface, henceweomitthedetails.

Acknowledgements

The research was supported by theCzech Science Foundation

within the project 14-06818S andby Grant Agency of theCzech TechnicalUniversityinPrague,grantNo.SGS13/217/OHK4/3T/14.

References

[1]P.Exner,Leakyquantumgraphs:areview,in:ProceedingsoftheIsaacNewton InstituteProgramme“AnalysisonGraphsandApplications”,in:AMSProceed- ings ofSymposiainPure MathematicsSeries, vol. 77, AMS, Providence,RI, 2008,pp. 523–564.

[2]J. Behrndt, M. Langer, V. Lotoreichik, Schrödinger operators with δ and δ-potentials supported on hypersurfaces, Ann. Henri Poincaré 14 (2013) 385–423.

[3]S.Albeverio,F.Gesztesy,R.Høegh-Krohn,H.Holden,SolvableModelsinQuan- tumMechanics,2ndeditionwithanappendixbyP. Exner,AMSChelseaPub- lishing,Providence,RI,2005.

[4]T.Cheon,T.Shigehara,Realizingdiscontinuouswavefunctionswithrenormal- izedshort-rangepotentials,Phys.Lett.A243(1998)111–116.

[5]S.Albeverio,L.Nizhnik,Approximationofgeneralzero-rangepotentials,Ukr.

Math.J.52(2000)582–589.

[6]P.Exner,H.Neidhardt,V.A.Zagrebnov,Potentialapproximationstoδ:anin- verseKlauderphenomenonwithnorm-resolventconvergence,Commun.Math.

Phys.224(2001)593–612.

[7]P.Exner,S.Kondej,Boundstateduetoastrongδinteractionsupportedbya curvedsurface,J.Phys.A,Math.Gen.36(2003)443–457.

[8]J. Behrndt, P. Exner, V. Lotoreichik, Schrödinger operators with δ- and δ-interactionsonLipschitzsurfacesandchromaticnumbersofassociatedpar- titions,arXiv:1307.0074[math-ph].

[9]G. Carron,P.Exner,D. Krejˇciˇrík,Topologicallynon-trivialquantumlayers, J.

Math.Phys.45(2004)774–784.

[10]M.Fecko,DifferentialGeometryandLieGroupsforPhysicists,CambridgeUni- versityPress,2006.

[11]P.Duclos,P.Exner,D.Krejˇciˇrík,Boundstatesincurvedquantumlayers,Com- mun.Math.Phys.223 (223)(2001)13–28.

[12]P.Exner,M.Jex, Spectralasymptoticsofastrongδ interactiononaplanar loop,J.Phys.A,Math.Theor.46(2013)345201.