On self-adjoint realizations of sign-indefinite Laplacians

HAL Id: hal-01960406

https://hal.archives-ouvertes.fr/hal-01960406

Preprint submitted on 19 Dec 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On self-adjoint realizations of sign-indefinite Laplacians

Konstantin Pankrashkin

To cite this version:

Konstantin Pankrashkin. On self-adjoint realizations of sign-indefinite Laplacians. 2018. �hal- 01960406�

On self-adjoint realizations of sign-indefinite Laplacians

K

P

Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud, CNRS, Universit´e Paris-Saclay, 91405 Orsay, France

E-mail: konstantin.pankrashkin@math.u-psud.fr Webpage:http://www.math.u-psud.fr/˜pankrashkin/

Abstract

LetΩ⊂R^d be a domain and Σa hypersurface cuttingΩinto two partsΩ±. Forµ>0, consider the functionhwhose value is(−µ)inΩ−and 1 inΩ₊. In the present contribution we discuss the construction and some properties of the self-adjoint realizations of the operatorL=−∇·(h∇)inL²(Ω)with suitable (e.g. Dirichlet) on the exterior boundary. We give first a detailed study for the case when Ω± are two rectangles touching along a side, which is based on operator-valued differential operators, in order to see in an elementary but an abstract level the principal effects such as a loss of regularity and unusual spectral properties. Then we give a review of available approaches and results for more general geometric configurations and formulate some open problems.

Contribution to the proceedings of the conference “Spectral theory and mathematical physics”, Universit´e de Lorraine, Metz (France), May 16–18, 2017, to be published in Revue Roumaine de Math´ematiques Pures et Appliqu´ees in the special issue “Spectral theory and applications in mathematical physics” edited by J. Faupin, M. M˘antoiu and V. Nistor

1 Introduction

The present contribution discusses some approaches to the strict mathematical def- inition of non-elliptic self-adjoint differential operators of a special form. Namely, letΩ⊂R^d be bounded open set andΣbe a hypersurface (calledinterface) splitting Ωinto two partsΩ− andΩ+ (for the moment we do not discuss the precise regu- larity assumptions). Letµ >0 be a parameter andh:Ω→Rtakes the value(−µ) inΩ₋ and is equal to 1 inΩ+. We will be interested in operatorsLacting inL²(Ω) and at least formally given by the differential expressionu7→ −∇·(h∇)uinΩand the Dirichlet boundary conditionu=0 on∂Ω. Such operators appear in the math- ematical theory of negative-index metamaterials arised from the pioneering works by Veselago [43]. At the naive level, if one identifiesL²(Ω)'L²(Ω₋)⊕L²(Ω₊), u'(u₋,u₊)withu_± being the restrictions ofutoΩ_±, then the above operatorLis

expected to act as(u₋,u₊)7→(µ∆u₋,−∆u₊)while the functionsu_±should satisfy the boundary conditions

u_±=0 on∂Ω, u₋ =u₊ onΣ, µ ∂−u₋ =∂+u₊ onΣ,

with∂_± being the outward normal (with respect toΩ_±) derivative onΣ. As will be explained below, understanding the precise regularity properties of the functionsu_± appears to be a non-trivial task depending on a combination of the valueµ and the geometric properties ofΩ±. In some cases, a very low regularity ofu_± is needed in order to have an operator with reasonable properties, and this is an essential feature of the problem which is of relevance for various effects such as the presence of an anomalous localized resonance and the cloaking, see [1, 8, 28, 35, 34, 33], and the critical value of the parameterµ =1 appears to be of a special importance.

Remark that if one has a self-adjoint operatorAin a Hilbert space, then solving the equationAu= f with a given f can be understood in terms of the domain ofA and of its spectral properties. As the above differential expression forLis formally symmetric, it is natural to look for self-adjoint realizations of the differential ex- pression, which then may provide a rigorous reformulation of the above equation.

In the present text we collect some known results and approaches to the study of such self-adjoint differential operators.

It seems that the problem was first addressed in [6] for the case whenΩ− ⊂Ω, both Ω_± are smooth and µ 6= 1, which was then extended to the case of do- mains with corners, see a detailed discussion in Section 5. The question of self- adjointness for the critical caseµ =1 was first addressed in the very recent paper [2], in which the very particular case ofΩ= (−1,1)×(0,1),Ω₋= (−1,0)×(0,1), Ω+= (0,1)×(0,1), h=±1 inΩ_±. was considered. An interesting feature of the model is the fact that the resulting self-adjoint operator appears to have an infinitely degenerate zero eigenvalue, hence, it is not with compact resolvent, although the domainΩis bounded. This comes from the fact that the functions in the operator domain can be very irregular near the interface{0} ×(0,1), and a precise descrip- tion of the resulting operator domain is given as well. On the other hand, the con- struction appears to be very sensible to the symmetries, in particular, the approach had no easy extension to configurations consisting of two non-congruent rectangles touching along a side.

In fact we use the paper [2] as our starting point and use the situation with two rectangles as a toy example illustrating the main features of the problem and the special role of the value µ = 1, which allows for a rather complete spectral analysis and may provide us with some intuition for the general case. Therefore, we prefer to give first a detailed study of this situation, which is done in the following three sections. In Section 2 we recall the basic notions of the theory of self-adjoint extensions. In Section 3 we give some facts related to operator-valued differential equations, which represents an abstract version of the so-called modal analysis, see e.g. [44] and can also be obtained in a much more general setting, see e.g. [18, 19, 20, 32, 39], but we prefer to provide complete “manual” proofs in order to keep the presentation at an elementary level. In Section 4, the preceding constructions are used to study the self-adjointness of indefinite Laplacians on rectangles, which

gives the sought extension of the result of [2]. In the last section 5 we give a review of available approaches and results for more general geometries and state some open questions, in the hope that the present text will attract the attention of the spectral community to this relatively new class of problems.

2 Basic constructions for self-adjoint extensions

2.1 Notation

The scalar product in a Hilbert spaceH will be denoted ash·,·i_H or simply ash·,·i if there is no ambiguity. The scalar produce will be assumed anti-linear with respect to the first argument. IfBis a linear operator in a Hilbert space, then by domB, kerB, ranB,σ(B)andρ(B)we denote its domain, kernel, range, spectrum and resolvent set, and Band B^∗ stand for the closure of B and for its adjoint, respectively. IfB is self-adjoint, thenσ_ess(B)andσ_p(B)stand respectively for its essential spectrum and point spectrum (defined as the set of eigenvalues). ByB(G,H )we mean the Banach space of the bounded linear operators from a Hilbert spaceG to a Hilbert spaceH , and we will denoteB(H ):=B(H,H ).

2.2 Boundary triples

Let us briefly recall the key points of the boundary triple approach to self-adjoint extensions following the first sections of [9]. A detailed discussion can also be found in [17, 20].

LetSbe a closed densely defined symmetric operator in a Hilbert spaceH . A triple(Ξ,Γ,Γ⁰), whereΞis a Hilbert space andΓ,Γ⁰: domS^∗→Ξare linear maps, is called aboundary tripleforSif the following three conditions are satisfied:

(a)hf,S^∗gi_H − hS^∗f,gi_H =hΓf,Γ⁰gi_Ξ− hΓ⁰f,Γgi_Ξfor f,g∈domS^∗, (b) the map domS^∗3 f 7→(Γf,Γ⁰f)∈Ξ×Ξis surjective,

(c) kerΓ∩kerΓ⁰=domS.

A boundary triple forSexists if and only ifSadmits self-adjoint extensions, i.e. if its deficiency indices are equal, dim ker(S^∗−i) =dim ker(S^∗+i) =:n(S). A boundary triple is not unique, but for any choice of a boundary triple(Ξ,Γ,Γ⁰)forSone has dimΞ=n(S), and the mapsΓ,Γ⁰are bounded with respect to the graph norm ofS^∗. Assume from now on that the deficiency indices ofSare equal and pick a bound- ary triple(Ξ,Γ,Γ⁰). LetΠ:Ξ→Ξ_Π:=ranΠ⊆Ξbe an orthogonal projector inΞ andΘbe a linear operator in the Hilbert spaceΞ_Πcarrying the induced scalar prod- uct. Denote byA_Π,Θ the restriction ofS^∗onto

domA_Π,Θ=

f ∈domS^∗:Γf ∈domΘ,ΠΓ⁰f =ΘΓf ,

then the closedness, the symmetry and the self-adjointness ofA_Π,ΘinH are equiv- alent to the closedness, the symmetry and the self-adjointness of the operatorΘin

Ξ_Π, and for the closures one has A_Π,Θ=A_Π,Θ. Finally, any self-adjoint extension ofSwrites asA_Π,Θ.

The spectral analysis of the self-adjoint extensions can be carried out using the associated Weyl function. Namely, let Abe the restriction of S^∗ to kerΓ, i.e. cor- responds to(Π,Θ) = (0,0)in the above notation, which is therefore a self-adjoint operator. For z∈ρ(A) the restriction Γ: ker(S^∗−z)→Ξ is a bijection, and we denote its inverse by G(z). In other word, for ξ ∈Ξ, one has G(z)ξ := f with f ∈domS^∗ uniquely determined by the conditions(S^∗−z)f =0 andΓf =ξ. The mapz7→G(z), called the associatedγ-field, is then a holomorphic map fromρ(A) toB(Ξ,H), and one has the identity

G(z)−G(w) = (z−w)(A−z)⁻¹G(w)for anyz,w∈ρ(A). (1) The associatedWeyl function is then the holomorphic map z7→M(z):=Γ⁰G(z)∈ B(Ξ)defined forz∈ρ(A). The above mapsMandGsatisfy a number of identities, in particular,

M⁰(z) =G(z)^∗G(z)>0, 0∈σ M⁰(z)

for anyz∈R∩ρ(A). (2) To describe the spectral properties of the self-adjoint operatorsA_Π,Θlet us con- sider first the caseΠ=1, thenΘis a self-adjoint operator inH , and the following holds, see Theorems 1.29 and Theorem 3.3 in [9]:

Proposition 1. For any z∈ρ(A)∩ρ(A_1,Θ)one has0∈ρ Θ−M(z) and (A_1,Θ−z)⁻¹= (A−z)⁻¹+G(z) Θ−M(z)−1

G(z)^∗. (3)

For any z∈ρ(A)one has

(1) z∈σ(A_1,Θ)if and only if0∈σ Θ−M(z) , (2) z∈σess(A_1,Θ)if and only0∈σess Θ−M(z)

, (3) G(z): ker Θ−M(z)

→ker(A_1,Θ−z)is an isomorphism, hence, z∈σp(A_1,Θ)if and only if0∈σp Θ−M(z)

.

Now consider a self-adjoint extensionA_Π,Θfor an arbitrary orthogonal projector Π. Denote by S_Π the restriction of S^∗ to domS_Π ={u∈domS^∗:Γu=ΠΓ⁰u= 0}, which is a closed densely defined symmetric operator whose adjointS^∗

Π is the restriction of S^∗ to domS^∗_Π ={u∈domS^∗ :Γu∈ranΠ}, then (Ξ_Π,Γ_Π,Γ_Π) with Ξ_Π =ranΠ, Γ_Π :=ΠΓ and Γ⁰_Π :=ΠΓ⁰ , is a boundary triple for S_Π, while the restriction ofS^∗

Π to kerΓ_Π is still the original operatorA=S^∗|_kerΓ. The associated γ-field G_Π and Weyl function M_Π take the form z7→G_Π(z):=G(z)Π^∗ and z7→

M_Π(z):=ΠM(z)Π^∗, and domA_Π,Θ:={u∈domS^∗_Π:Γ⁰_Πu=ΘΓ_Πu}, see [9, Remark 1.30]. A direct application of Proposition 1 gives the following assertions:

Corollary 2. For any z∈ρ(A)∩ρ(A_Π,Θ)one has0∈ρ Θ−M_Π(z) and (A_Π,Θ−z)⁻¹= (A−z)⁻¹+G_Π(z) Θ−M_Π(z)−1

G_Π(z)∗

. (4)

For any z∈ρ(A)one has:

(1) z∈σ(A_Π,Θ)if and only if0∈σ Θ−M_Π(z) , (2) z∈σess(A_Π,Θ)if and only if0∈σess Θ−M_Π(z)

, (3) G_Π(z): ker Θ−M_Π(z)

→ ker(A_Π,Θ−z) is an isomorphism, hence, z ∈ σ_p(A_Π,Θ)if and only if0∈σ_p Θ−M_Π(z)

.

2.3 Construction of boundary triples using trace maps

In various situations one deals with a symmetric operator obtained by restricting a self-adjoint operator with well-known properties to the kernel of an explicitly given linear map. This observation may be used to construct a boundary triple and to study other self-adjoint extensions of the symmetric operator. We present here very briefly the respective construction, which is described in detail e.g. in [38] or in [9, Section 1.4.2].

Let A be a self-adjoint operator in a Hilbert space H , then one denotes by H (A) the Hilbert space given by the linear space domA equipped with the scalar product hu,vi_H(A)=hu,vi_H +hAu,Avi_H. Let a Hilbert space Ξ and a bounded surjective linear operator τ :H (A)→Ξ be such that the kernel kerτ is a dense linear subspace of H. Such a map τ will be referred to as a trace map forA. It follows that the restriction S of A to kerτ (i.e. the restriction to the vectors with zero traces) is a closed densely defined symmetric operator inH . To avoid some technicalities we additionally assume thatρ(A)∩R6= /0 and pick an arbitrary value λ ∈ρ(A)∩R. Forz∈ρ(A)consider the maps

G(z):= τ(A−z)⁻¹∗

∈B(Ξ,H), M(z):=τ G(z)−G(λ)

∈B(Ξ),

then the adjointS^∗is given by domS^∗:=n

u=u_λ+G(λ)f_u:u_λ ∈domAand f_u∈Ξ o

, (S^∗−λ)u= (A−λ)u_λ.

Furthermore, the triple(Ξ,Γ,Γ⁰)withΓu:= f_uandΓ⁰u:=τu_λ is a boundary triple for S, and z7→G(z) and z7→M(z) are the associated γ-field and Weyl function, respectively.

2.4 Trace maps for direct sums

In what follows we will be interested in boundary triples for infinite direct sums of operators. In view of the preceding considerations it is useful to understand how to

construct a trace map for a direct sum of operators. The construction below follows [29, 37].

LetA_nbenon-negativeself-adjoint operators in Hilbert spacesHn, n∈N. For each n we consider a trace map τ_n:H (A_n)→Ξ_n for A_n, where Ξ_n is a Hilbert space. Due to the constructions of the preceding subsection this gives rise to closed densely defined symmetric operatorsS_n:=A_n|_kerτn and the associatedγ-fields and Weyl functions z7→G_n(z) and z7→M_n(z). To construct a trace map for the self- adjoint operatorA:=^L_n∈NA_nwe use [37, Theorem 2.5], which gives the following result: Forn∈NsetR_n:=p

M_n⁰(−1)∈B(Ξ_n), then the linear operator τ :H(A)3(u_n)_n∈N7→(R⁻¹_n τ_nu_n)_n∈N∈^M

n∈N

Ξ_n, (5) is bounded and surjective. This construction can be adjusted as follows:

Proposition 3. Let K_n∈B(Ξ_n) be strictly positive and such that for some a≥1 one has

a⁻¹K_n²≤M_n⁰(−1)≤aK_n²for any n∈N. (6) then the linear operator

τ:H (A)3(u_n)7→(K_n⁻¹τ_nu_n)∈^M

n∈N

Ξ_n

is bounded and surjective.

Proof. RepresentingK_n⁻¹= (K_n⁻¹R_n)R⁻¹_n and comparing with (5) one sees that it is sufficient to show that for someb>0 there holds

kK_n⁻¹R_nk ≤band

(K_n⁻¹R_n)⁻¹

≤bfor alln∈N, which is due to the self-adjointness ofR_nandB_nequivalent to

kR_nK_n⁻¹k ≤band

(R_nK_n⁻¹)⁻¹

≤bfor alln∈N, (7) Remark that the condition (6) takes the forma⁻¹K_n²≤R²_n≤aK_n². Therefore, for any ξ ∈Ξnone has

kR_nK_n⁻¹ξk²_Ξn =hK_n⁻¹ξ,R²_nK_n⁻¹ξi_Ξn ≤ahK_n⁻¹,K_n²K_n⁻¹ξi_Ξn =akξk²_Ξn, kR_nK_n⁻¹ξk²_Ξn =hK_n⁻¹ξ,R²_nK_n⁻¹ξi_Ξn ≥1

ahK_n⁻¹,K_n²K_n⁻¹ξi_Ξn =1

akξk²_Ξn, which gives the estimates (7) withb:=√

a.

3 Sign-changing operator-valued Sturm-Liouville operators

3.1 Sturm-Liouville operator on a bounded interval

In order to illustrate the constructions of the preceding section let us consider first a simple one-dimensional situation. Let(a,b)⊂Rbe a non-empty bounded interval.

In the Hilbert spaceH =L²(a,b)consider the self-adjoint Dirichlet Laplacian A: f 7→ −f⁰⁰, domA=

f ∈H²(a,b): f(a) = f(b) =0 ,

whose spectrum consists of the simple eigenvaluesπ²n²/(b−a)², n∈N. For the subsequent computations we chooseλ =0∈ρ(A)∩R. The resolvent(A−z)⁻¹is an integral operator,

(A−z)⁻¹f (t) =

Z _b

a

K_z(t,s)f(s)ds,

where the integral kernelK_z is given by K_z(t,s) = 1

√−zsinh √

−z(b−a)

×

(sinh √

−z(t−a)

sinh √

−z(b−s)

, t<s, sinh √

−z(s−a)

sinh √

−z(b−t)

, t>s.

The map

τ :H(A)3 f 7→

f⁰(a)

−f⁰(b)

∈C²

is surjective and bounded due to the Sobolev embedding theorem, and the restriction ofAto kerτ is the closed symmetric operator

S: f 7→ −f⁰⁰, domS=H₀²(a,b).

A direct computation shows that the operators G(z):= τ(A−z)¯ ⁻¹∗

:C² →H are given by

G(z) ξ_a

ξ_b

(s) =∂_tK_z¯(a,s)ξ_a−∂_tK_z¯(b,s)ξ_b

= sinh √

−z(b−s) sinh √

−z(b−a)ξ_a+sinh √

−z(s−a) sinh √

−z(b−a)ξ_b, and the associated mapM(z) =τ G(z)−G(0)

:C²→C²is M(z) =m(z,b−a)−m(0,b−a)

with m(z, `):=

√−z sinh(`√

−z)

−cosh(`√

−z) 1

1 −cosh(`√

−z)

. (8)

In order to construct a boundary triple forSwe remark first that the adjointS^∗acts as f 7→ −f⁰⁰ on the domain domS^∗=H²(a,b). Each f ∈domS^∗ can be uniquely represented as f = f₀+G(0)(ξ_a,ξ_b)with f₀∈domAand(ξ_a,ξ_b)∈C², i.e.

f(s) = f₀(s) + b−s

b−aξ_a+s−a

b−aξ_b, s∈(a,b),

and as a boundary triple(C²,Γ,Γ⁰)forSone can take Γf =

ξ_a ξ_b

= f(a)

f(b)

,

Γ⁰f =τf₀=

f₀⁰(a)

−f₀⁰(b)

=

f⁰(a)

−f⁰(b)

−m(0,b−a) f(a)

f(b)

.

For a later use we remark that, by a direct computation, M⁰(z) = 1

2√

−z· 1 sinh²ζ

sinhζcoshζ−ζ ζcoshζ−sinhζ ζcoshζ−sinhζ sinhζcoshζ−ζ

, (9)

ζ := (b−a)√

−z.

3.2 Direct sums of Sturm-Liouville operators on a bounded interval

Consider an infinite number sequence (λ_n)_n∈N such that λn ≥0 for alln∈N and that

n→+∞lim λ_n= +∞.

Let(a,b)⊂Rbe a non-empty bounded interval. For eachn∈N, in the Hilbert spaceHn:=L²(a,b)consider the closed densely defined symmetric operators

S_n: f_n7→ −f_n⁰⁰+λnf_n, domS_n=H₀²(a,b).

For eachS_none can construct a boundary triple as in the preceding subsection, and the associated γ-fields G_n and the Weyl functions M_n are then given by G_n(z) = G(z−λn)andM_n(z) =m(z−λn,b−a)−m(−λ_n,b−a)withmas in (8). We would like to construct a boundary triple for the direct sum S=^L_nS_n. Remark that we clearly haveS^∗=^L_nS^∗_n. ByA_nwe denote the self-adjoint extension ofS_ndefined by the Dirichlet boundary condition f_n(a) = f_n(b) =0, andB_n will stand for the self-adjoint extension of S_n defined by the Neumann boundary condition f_n⁰(a) =

f_n⁰(b) =0. In addition, denote A:=^M

n∈N

A_n, B:=^M

n∈N

B_n,

then bothAandBare self-adjoint with compact resolvents.

Lemma 4. The linear map

τ:H (A)→`²(N)⊗C², τ(f_n) =

1+λ_n)¹⁴

f_n⁰(a)

−f_n⁰(b)

n∈N

, is bounded and surjective.

Proof. An elementary computation with the help of (9) shows that (λ_n+1)¹²M_n⁰(−1)≡(λ_n+1)¹²M⁰ −(λ_n+1)

→ 1

2Id forn→+∞, and it is sufficient to use Proposition 3 withK_n= (λ_n+1)⁻¹⁴.

Using the map τ from Lemma 4 and the constructions of Subsection 2.3 one then easily computes a boundary triple(G,Γ,Γ⁰)for the operatorS≡A|_kerτ,

G :=`²(N)⊗C², Γ(f_n) =

(1+λ_n)⁻¹⁴

f_n(a) f_n(b)

n∈N

,

Γ⁰(f_n) =

(1+λn)¹⁴

f_n⁰(a)

−f_n⁰(b)

−m(−λ_n,b−a)

f_n(a) f_n(b)

n∈N

.

In particular, in view of the asymptotic behaviorm(−λ)' −√

λId forλ →+∞it follows that the linear maps

f 7→

(1+λn)⁻¹⁴

f_n(a) f_n(b)

n∈N

∈`²(N)⊗C², f 7→

(1+λ_n)⁻³⁴

f_n⁰(a)

−f_n⁰(b)

n∈N

∈`²(N)⊗C², are bounded with respect to the graph norm ofS^∗.

It is instructive to look at the self-adjoint operatorBfrom the point of view of the above boundary triple:

Lemma 5. The linear map

τ:H (B)→`²(N)⊗C², τ(f_n) =

1+λ_n)³⁴

f_n(a) f_n(b)

n∈N

,

is bounded and surjective.

Proof. In terms of the above boundary triple, the boundary condition forBtake the formΓ⁰f =LΓf withΓf ∈domL, where

L(ξ_n) =−

(1+λ_n)¹⁴m(−λ_n,b−a)ξ_n

n∈N,

i.e. the mapH(B)3 f 7→Γf ∈H (L)is bounded and surjective. Using the asymp- totics lim_λ→+∞λ⁻

1

2m(−λ,b−a) =−Id we obtain the result.

For a later use we give two additional estimates:

Proposition 6. On domA and domB, the graph norm of S^∗ is equivalent to the norm

kfk_H2 :=

∑

n∈N

kf_n⁰⁰k²_L2(a,b)+ (1+λn)²kf_nk²_L2(a,b)

<∞.

Proof. The graph norm ofS^∗for f = (f_n)∈domS^∗is given by

n∈

∑

kf_n⁰⁰+λ_nf_nk²_L2(a,b)+kf_nk²_L2(a,b)

,

and kf_n⁰⁰+λ_nf_nk²_L2(a,b) =kf_n⁰⁰k²_L2(a,b)+λ_n²kf_nk²_L2(a,b)+2λ_nℜhf_n⁰⁰,f_ni_L2(a,b). Using the integration by parts and the Cauchy-Schwarz inequality, for f ∈domAor f ∈ domBwe obtain

0≤2λ_nkf_n⁰k²_L2(a,b)=2λ_nℜhf_n⁰⁰,f_ni_L2(a,b)≤ kf_n⁰⁰k²_L2(a,b)+λ_n²kf_nk²_L2(a,b), which gives the result.

3.3 Sturm-Liouville operator with an operator-valued potential

LetG be a Hilbert space and T be a non-negative self-adjoint operator in G with a compact resolvent. Let us pick an orthonormal basis (e_n) of G consisting of eigenfunctions ofT:

Te_n=λ_ne_n, λ_n∈R, n∈N, he_k,e_ni=δ_k,n.

Fors≥0 we will consider the Hilbert spacesGs≡Gs(T) =dom(T+1)^2s equipped with the scalar producthu,vi_s=

(T+1)^s2u,(T+1)^2sv

G, then, in particular,G0= G, andG2coincides withH(T)algebraically and topologically. ByG−s≡G−s(T) we denote then the dual of Gs realized as the completion ofG with respect to the scalar product hu,vi_−s =

(T +1)^−s2u,(T +1)^−s2

G and introduce the subspace G∞:=^T_t>0Gt which is then dense in any Gs, s∈R. With these definitions, the operatorT :G∞→G∞extends uniquely to a bounded linear mapT :Gs→Gs−2for anys∈R, while the operator 1+T :Gs→Gs−2becomes unitary.

Furthermore, let I = (a,b)⊂R be a non-empty bounded interval and H = L²(I,G). Each function f ∈H can then be uniquely represented as

f(t) =

∑

n∈N

f_n(t)e_n, f_n(·):=

f(·),e_n

G ∈L²(a,b).

Denote byS₀the operator f 7→ −f⁰⁰+T f defined on the domain domS₀=n

f : f(t) =

N n=1

∑

f_n(t)e_n: N∈N, f_n∈H₀²(a,b)o

and letSbe its closure, which will be called theminimal operatorgenerated by the differential expression−d²/dt²+T on(a,b), and its adjoint S^∗will be called the associated maximal operator. Using the unitary transformU :H → ⊕_nL²(a,b), f 7→(f_n), one easily sees that S is unitarily equivalent to ⊕S_n with S_n defined as in the preceding section 3.2, which gives a choice of a boundary triple. For what follows is will be more instructive to not to use directly the eigenbasis(e_n)and the unitary transformU associated withT and to reformulate the preceding construc- tions using the above spacesGsas follows:

Lemma 7. The domain of the adjoint S^∗consists of the functions f ∈L²(I,G)such that(−f⁰⁰+T f)∈L²(I,G), where f⁰⁰is computed inG−2, and the operator S^∗acts by f 7→ −f⁰⁰+T f . For any f ∈domS^∗, there exist boundary values

f(a),f(b)∈G₋¹

2, f⁰(a),f⁰(b)∈G₋³

2,

which are bounded with respect to the graph norm of S^∗, and(G ⊗C²,Γ,Γ⁰)with Γf =D⁻¹

f(a) f(b)

, Γ⁰f =D

f⁰(a)

−f⁰(b)

−m(−T,b−a) f(a)

f(b)

,

is a boundary triple for S, and the respective Weyl function is M(z) =D

m(z−T,b−a)−m(−T,b−a) D,

D:= (T+1)¹⁴ 0 0 (T+1)¹⁴

! .

The minimal operator S is exactly the restriction of S^∗to the functions f satisfying f(a) = f(b) = f⁰(a) = f⁰(b) =0.

Let A be the extension of S corresponding to the boundary condition f(a) = f(b) =0, which will be called theDirichlet realization of−d²/dt²+T on (a,b), andBbe the extension corresponding to the boundary condition f⁰(a) = f⁰(b) =0, called then the Neumann realization. By construction of the previous subsection, the both operators are self-adjoint and with compact resolvents.

We introduce the operator Sobolev spaceH_T²(I):=L²(I,G2)∩H²(I,G). In other words,H_T²(I)consists of the functions f ∈L²(I,G)satisfying

kfk²

H_T²(I):=

∑

n∈N

kf_n⁰⁰k²_L2(I)+ (1+λ_n)²kf_nk²_L2(I)

<+∞,

and one has the obvious inclusionH_T²(I)⊂domS^∗. Lemma 8. The linear maps

H_T²(I)3 f 7→ f(a),f(b)

∈G³

2 ×G³

2, H_T²(I)3 f 7→ f⁰(a),−f⁰(b)

∈G¹

2×G¹

2

are surjective. Furthermore, if f∈domS^∗, then f ∈H_T²(I)if and only if f(a),f(b)∈ G³

2

.

Proof. By Proposition 6, one has domA⊂H_T²(I)and domB⊂H_T²(I), and the sur- jectivity follows from Lemmas 4 and 5. Now let f ∈domS^∗with f(a),f(b)∈G³

2, then due to the surjectivity there exists g∈H_T²(I) with g(a) = f(a) and g(b) = f(b). The function f₀:= f −g satisfies f₀(a) = f₀(b) =0, hence, f₀∈domA⊂ H_T²(I).

As a direct consequence we obtain:

Corollary 9. domS=H_T²_,0(I):=

f ∈H_T²(I): f(a) = f(b) =0, f⁰(a) = f⁰(b) = 0 .

3.4 Sign-changing operator on an interval

LetT be as in the preceding subsection 3.3. Consider three parametersa>0,b>0, µ 6=0, and the intervals

I₋:= (−a,0), I₊:= (0,b), I:= (−a,b).

We would like to construct and study self-adjoint realizations of the operator Lin L²(I,G)formally given by the differential expression

(L f)(t) =

−µ

−f⁰⁰(t) +T f(t)

, t ∈I₋,

−f⁰⁰(t) +T f(t) , t ∈I₊,

the transmission conditions f(0⁻) = f(0⁺) and −µf⁰(0⁻) = f⁰(0+) at zero and the Dirichlet boundary condition f(−a) = f(b) =0 at the endpoints. To be more precise, one uses the natural identification L²(I,G)' L²(I₋,G)⊕L²(I+,G) and considers the following linear operatorLinL²(I₋,G)⊕L²(I₊,G):

L f₋

f₊

=

−µ(−f₋⁰⁰+T f₋)

−f₊⁰⁰+T f₊

, domL=

f_± ∈H_T²(I_±): f₋(−a) = f₋(0)−f₊(0) = f₊(b) =0,

−µf₋⁰(0) = f₊⁰(0) .

We will consider the operatorLas an extension of another closed densely defined symmetric operator. Namely, letA_±be the Dirichlet realizations of−d²/dt²+T in I_±, which are both self-adjoint inL²(I_±,G)and with compact resolvents. Consider the operatorA:= (−µA₋)⊕A₊ and the linear map

τ:H (A)→G⁴, τ f₋

f₊

=

D 0

0 D









−µ 0 0 0

0 −µ 0 0

0 0 1 0

0 0 0 1

















f₋⁰(−a)

−f₋⁰(0) f₊⁰(0)

−f₊⁰(b)







 ,

where

D:= (T+1)¹⁴ 0 0 (T+1)¹⁴

!

which is bounded and surjective due to the above constructions, and consider the restrictionSofAto kerτ. Indeed, the operatorSdecomposes asS= (−µS₋)⊕S₊, whereS_± are closed densely defined symmetric operators in L²(I_±,G)covered by the constructions of Lemma 7, and the adjoint is similarly decomposed as S^∗ =

(−µS^∗₋)⊕S^∗₊. As a result, we obtain a boundary triple(G⁴,Γ,Γ⁰)forSwith

Γf =











(T+1)⁻¹⁴f₋(−a) (T+1)⁻¹⁴f₋(0) (T+1)⁻¹⁴f₊(0) (T+1)⁻¹⁴f₊(b)









 ,

Γ⁰f =











−µD

"

f₋⁰(−a)

−f₋⁰(0)

−m(−T,a) f₋(−a) f₋(0)

!#

D

f₊⁰(0)

−f₊⁰(b)

−m(−T,b)

f₊(0) f₊(b)









 ,

the associated Weyl function is M(z) =

D 0

0 D

N(z)

D 0

0 D

,

N(z):= −µh m(−^z

µ−T,a)−m(−T,a)i

0

0 m(z−T,b)−m(−T,b)

! ,

and the expression of the respectiveγ-fieldsz7→G(z)will have no importance, we just remark that due to (1) the only possible singularities ofz7→G(z)at the points ofσ(A)are poles with finite-dimensional residues.

In view of Lemma 8, the domain ofLcan be rewritten as domL=

n

f = (f₋,f₊)∈domS^∗: f_±(0)∈G³

2, −µf₋⁰(0) = f₊⁰(0), f₋(−a) = f₋(0)−f₊(0) = f₊(b) =0o

. (10)

The last boundary condition in (10) can be rewritten asΓf =ΠΓf, or, equivalently, asΓf ∈ranΠ, whereΠ:G⁴→G⁴is the orthogonal projector given by

Π(f₁,f₂,f₃,f₄) = 1

2(0,f₂+f₃,f₂+f₃,0 .

In the subsequent constructions it will be convenient to use the unitary mapU : G →ranΠgiven byU g= ^√1

2(0,g,g,0), thenU^∗(0,g,g,0) =√

2g, and (10) takes the form

domL=

f = (f₋,f₊)∈domS^∗:Γf ∈U(domΘ),

U^∗ΠΓ⁰f =ΘU^∗Γf . (11) whereΘis a linear operator inG given by

Θ= 1

2(T+1)¹⁴√ T

coth(b√

T)−µcoth(a√ T)

(T+1)¹⁴, (12) domΘ=G2.

Therefore, in terms of boundary triples one represents L=A_Π,UΘU∗ (see Subsec- tion 2.2), and we remark that

M(z):=U^∗ΠM(z)Π^∗U

=1

2(T+1)¹⁴√ T

coth(b√

T)−µcoth(a√ T)

(T+1)¹⁴

−1

2(T+1)¹⁴ √

T−zcoth(b√ T−z)

−µ r

T+ z µ coth

a

r T+ z

µ

(T+1)¹⁴. Proposition 10. Forµ 6=1, the operator L is self-adjoint and has a compact resol- vent.

Proof. According to the discussion of Subsection 2.2, the self-adjointness of Lis equivalent to the self-adjointness ofΘinG. One easily sees that on domΘone has Θ= ^1−µ₂ T+Cwith a bounded self-adjoint operatorC. AsT defined onG2≡domT is self-adjoint, the operatorsΘand thenLare self-adjoint too. Furthermore, for non- realzthe operatorΘ−M(z)has a compact inverse. AsAis with compact resolvent as well, it follows from the resolvent formula (4) that the resolvent ofLis a compact operator.

Proposition 11. For µ =1, the operator L is not closed, but it is essentially self- adjoint. Its closureL is the restriction of S^∗to the domain

domL =

f = (f₋,f₊)∈domS^∗: f₋(−a) = f(b) =0,

f₋(0) = f₊(0), −f₋⁰(0) = f₊⁰(0) . (13) and the essential spectrum ofL is {0}. If a=b, the zero is an isolated infinitely degenerate eigenvalue of L. If a6=b, then 0is not an eigenvalue of L. In this case, there existε>0and N >0such that there exist a bijection E between the set {n:n≥N}and the set of the eigenvalues of L in (−ε,ε) such that for n→+∞

there holds

E(n)∼ −2λ_ne^−2a

√

λn if a<b, E(n)∼2λne^−2b

√

λn if a>b. (14) Proof. One easily sees that Θ=Φ(T)|_G2 with a bounded function Φ:R+ →R satisfyingΦ(+∞) =0. Therefore,Θis a compact operator inG. As its domainG2is dense inG, it has a unique self-adjoint extension, which is just the closureΘdefined on the the whole space. According to the constructions of subsection 2.2 it implies thatLis essentially self-adjoint, and the domain of the closureL =L=A_Π,UΘU∗

is given as in (11) withΘreplaced byΘ, and by using the explicit expressions for ΓandΓ⁰one arrives at (13).

In order to study the essential spectrum ofL let us remark first that one has 0∈ σess(Θ)due to the compactness ofΘ, hence, by Corollary 2 one has 0∈σess(L).

Therefore, it remains to show that L has no essential spectrum in R\ {0}. To

see this, remark first thatM(z)−₂^zId is a compact operator for z∈/σ(A). Denote Σ:=σ(A)∪{0}, then forz∈C\Σone can representM(z)−Θ=¹₂z Id+K (z)

, whereK (z)are compact operators meromorphically depending onz∈C\ {0}and having at most simple poles with finite-dimensional residues at the points ofσ(A).

Due to the meromorphic Fredholm alternative, see e.g. [40, Theorem XIII.13], only two situations are possible:

(a) 0∈σ(M(z)−Θ)for allz∈C\Σ,

(b) there exists a subset B⊂C\ {0}, without accumulation points in C\ {0}, such that the inverse(M(z)−Θ)⁻¹exists and is bounded forz∈C\ {0} ∪ B∪σ(A)

and extends to a meromorphic function inC\ {0} ∪B

such that the coefficients in the Laurent series of(M(z)−Θ)⁻¹at the points ofBare finite-dimensional operators.

The case (a) is impossible, in fact, by Corollary 2 this would imply the presence of a non-empty non-real spectrum for the self-adjoint operatorL. Therefore, the case (b) is realized. It follows then from the resolvent formula (4) that the only possible singularities of the resolvent of L ≡A_Π,UΘU∗ in C\ {0} are poles with finite-dimensional residues, which shows thatL has no essential spectrum inC\ {0}.

By Corollary 2 one has dim kerL =dim kerΘ. Fora=bone has simplyΘ=0, which gives dim kerΘ=∞. Forzclose to 0 one can representM(z)−Θ≡M(z) = zM⁰(0) +z²B(z) and the norms ofB(z) are uniformly bounded, and M⁰(0) has a bounded inverse by (2). It follows that there existsε>0 such that 0∈ρ M(z)−Θ for 0<|z|<ε, and in view of Corollary 2 this shows that 0 is an isolated point in the spectrum ofL.

Fora6=bwe represent Θ=

√

Tsinh (b−a)√ T sinh(a√

T)sinh(b√ T),

which shows that dim kerΘ= 0. Now it remains to show the asymptotics (14).

Let us take a smallρ >0, then by Corollary 2 the eigenvalues of L are exactly the values E ∈(−ρ,ρ) for which 0 is an eigenvalue of M(E)−Θ (with same multiplicities), i.e. iff for somen∈None has

p

λ_n−Ecoth(bp

λ_n−E)−p

λ_n+Ecoth

ap λ_n+E

.

Remark for each fixed n this equation admits at most finitely many solutions in (−ρ,ρ), hence, in order to study the accumulation it is sufficient to look at large values ofn. Then one can assume without loss of generality thatλn+E 6=0, and the equation rewrites asF_λn(E) =0, where

F_λ(z) = r

λ−z

λ+zcoth(bp

λ−z)−coth(ap λ+z).

An elementary analysis shows that there existsλ₀such that forλ >λ₀>0 one has (F_λ)⁰<0 in (−ρ,ρ) with F_λ(−ρ)>0 and F_λ(−ρ)<0. Therefore, for λ >λ₀ there is a uniquez_λ ∈(−ρ,ρ)withF_λ(z_λ) =0. Then one has

s λ−z_λ

λ+z_λ =coth(ap

λ+z_λ) coth(bp

λ−z_λ)

and taking Taylor expansions with respect toz_λ/λ for largeλ one obtains 1−z_λ

λ +Oz_λ λ

2

= 1+2e^−2a

√

λ+o(e^−2a

√ λ) 1+2e^−2b

√

λ+o(e^−2b

√ λ), which shows thatz_λ =2λ(e^−2b

√

λ −e^−2a

√

λ) +o(e^−2a

√

λ+e^−2b

√

λ)as λ →+∞.

UsingE(n) =z_λn we arrive at the result.

4 Indefinite Laplacians with separated variables

4.1 Indefinite Laplacian on a cylinder

Let us see how the preceding constructions apply to a simple two-dimensional ex- ample. Leta,b, `be strictly positive constants. LetC be a circle of length 2` >0, i.e. C =R/(2`Z), and Ω:= (−a,b)×C. We define a function h:Ω→ R by h(t,s) =−µ fort <0 andh(t,s) =1 fort>0. Our objective is to find self-adjoint realizations of the operatoru7→ −∇·(h∇)uinΩwith the Dirichlet boundary con- ditionu=0 on∂Ω.

We denoteI₋= (−a,0),I₊:= (0,b),I:= (−a,b)andΩ_±:=I_±×C. By setting G :=L²(C), we obtain the identificationsL²(Ω_±)'L²(I_±,G). Furthermore, we will identifyL²(Ω)'L²(Ω₋)×L²(Ω₊), u'(u₋,u₊), whereu_± is the restriction of u to Ω_±. With these conventions, let us consider the following operator L in L²(Ω_±)'L²(I_±,G)acting asL(u₋,u₊) = (µ∆u₋,−∆u+)on the domain

domL=n

(u₋,u₊): u_±∈H²(Ω_±),

u₋(−a,·) =u₊(b,·) =0, u₋(0,·) =u₊(0,·),

−µ∂u₋

∂t (0,·) =∂u₊

∂t (0,·)o . In order to make a link with the constructions of the preceding section we denote by T the self-adjoint Laplacian inL²(C)acting as f 7→ −f⁰⁰ on the domain domT = H²(C). The associated spaces Gs(T) are then the usual Sobolev spaces H^s(C).

Furthermore, we introduce the minimal operatorsS_±generated by −d²/dt²+T in L²(I_±,G), defined onH_T,0² (I_±,G), and their adjoints, i.e. the maximal operators, S^∗_±.

Proposition 12. There holds H_T,0² (I_±) = H₀²(Ω_±), and the operators S_± act by u_±7→ −∆u_±.

Proof. AsΩ_±have smooth boundaries, the spaceH₀²(Ω_±)is the closure ofC₀^∞(Ω_±) in the norm

kuk²_H2(Ω_±)= Z

Ω±

|∆u|²+|u|² dx.

On the other hand, the spaceH_T,0² (I_±)is the closure, with respect to the graph norm k · k_±ofS_±, of the set

D±,0:=n

u(t,s) =

N n=−N

∑

u^±_n(t)e^2πins/`:

N∈N, u^±_n ∈C₀^∞(I_±)o

⊂C₀^∞(Ω_±), and foru∈D±,0one haskuk²_± =kuk²

H²(Ω±). The classical theory of Fourier series shows that each function from C₀^∞(Ω_±) can be approximated by functions from D±,0 in anyC^k-norm, hence, also in the k · k_± norms, which shows the equality between the spaces.

In view of Proposition 12, the maximal operatorsS^∗_± are defined as in the clas- sical PDE theory, i.e. they act as

domS^∗_±=

u_± ∈L²(Ω_±): ∆u_±∈L²(Ω_±) , S^∗_±u_±=−∆u_±, and the functionsu_±∈domS^∗_±admit boundary values

u_±|_∂Ω±∈H⁻¹²(∂Ω)'H⁻¹²(C)×H⁻¹²(C),

∂±u_±|_∂Ω±∈H⁻³²(∂Ω±)'H⁻³²(C)×H⁻³²(C),

(15) where∂_±is the outward normal derivative on the boundary ofΩ_±, and these bound- ary values are bounded with respect to the graph norm of S^∗_±, see [31, Chapter 2, Section 6.5].

Lemma 13. For u_±∈domS^∗_±, the values of u_± and∂u_±/∂t at the endpoints of I_± defined as in Lemma 7 coincide with the Sobolev boundary values in(15).

Proof. Recall that by construction the sets D±:=

n

u_±(t,s) =

N n=−N

∑

u^±_n(t)e^2πins/`:

N∈N, u^±_n ∈C^∞(I_±) o

⊂C^∞(Ω_±).

are dense in domS^∗_± in the respective graph norms. For u_± ∈D±, the two trace versions coincide. As the traces are bounded with respect to the graph norm ofS_±^∗, the result follows.

Finally we arrive at the following identification:

Lemma 14. H_T²(I_±) =H²(Ω_±).

Proof. As the boundary of∂Ω± is smooth, it is a standard elliptic regularity result that, ifu_±∈domS_±^∗, thenu∈H²(Ω_±)iffu_±|_∂Ω±∈H³²(∂Ω_±). Now it is sufficient to substitute the result of Lemma 13 into the second assertion of Lemma 8.

With the above Lemmas at hand, the study of the operatorL is reduced to the constructions of the subsection 3.4 by considering it as an extension of the operator S:= (−µS₋)⊕S₊, and by using Propositions 10 and 11 one arrives at the following results:

Proposition 15. Ifµ 6=1, then the above operator L is self-adjoint and has a com- pact resolvent. Forµ =1, the above operator L is essentially self-adjoint, and its closureL acts as(u₋,u₊)7→(µ∆u₋,−∆u₊)on the domain

domL =n

(u₋,u₊):u_±∈L²(Ω_±), ∆u_±∈L²(Ω_±),

u₋(−a,·) =u₊(b,·) =0,u₋(0,·) =u₊(0,·),

−∂u₋

∂t (0,·) = ∂u₊

∂t (0,·)o ,

where the boundary values are understood as the Sobolev traces. One has σ_ess(L) ={0}. If a =b, then 0 is an isolated infinitely degenerate eigenvalue ofL. If a6=b, then0is not an eigenvalue of L, and the eigenvalues accumulate to the zero from below (respectively, from above) if a<b (respectively, a>b).

4.2 Indefinite Laplacian on a rectangle

Let us modify the example of the preceding section. Leta,b, ` be strictly positive constants andR:= (−a,b)×(0, `). We define a functionh:R→Rbyh(t,s) =−µ fort <0 andh(t,s) =1 fort>0. Our objective is to find self-adjoint realizations of the operatoru7→ −∇·(h∇)uinRwith the Dirichlet boundary conditionu=0 on

∂R.

We denote I₋ = (−a,0), I₊ := (0,b), I := (−a,b), R_± :=I_±×(0, `) and use the natural identificationL²(R)'L²(R₋)×L²(R₊),u'(u₋,u₊), whereu_± is the restriction ofutoR_±. Let us consider the following operatorL₀inL²(R₋)×L²(R+) acting asL₀(u₋,u₊) = (µ∆u₋,−∆u₊)on the domain

domL₀=n

(u₋,u₊): u_±∈H²(R_±),

u₋(−a,·) =u₊(b,·) =0, u_±(·,0) =u_±(·, `) =0, u₋(0,·) =u₊(0,·), −µ∂u₋

∂t (0,·) =∂u₊

∂t (0,·)o . In order to simplify the construction, we can reduce the study to the case of a cylin- der. Namely, letC :=R/(2`Z)andΩ:=I×C. InL²(Ω)consider the orthogonal