Some classes of generalized boundary triplets, Weyl functions, and - - PowerPoint PPT Presentation

Some classes of generalized boundary triplets, Weyl functions, and local point interactions

Seppo Hassi (University of Vaasa) Joint work with M. Malamud and V. Derkach Vienna, Dec. 20, 2016

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 1 / 23

• 1. Ordinary boundary triples and Weyl functions

Background

• H a separable Hilbert space with inner product (·, ·)
• S a closed symmetric linear relation in H with equal defect numbers
• Goal: descriptions of selfadjoint (dissipative etc.) extensions of S

In the beginning of thirties J. von Neumann created the extension theory of symmetric operators in Hilbert spaces. His approach relies on two fundamental formulas allowing one to describe all selfadjoint (m-dissipative) extensions parameterizing them by means of isometric (contractive)

• perators between the defect subspaces

V : Ni → N−i, where Nz is defined by Nz := ker (S∗ − z), z ∈ C \ R and n±(S) := dim N±i.

• J. von Neumann approach to extension theory was further developed e.g. by M.G. Kre˘

ın, M.A. Krasnosel’skii, M.A. Naimark, M. I. Viˇ sik, M. Sh. Birman, A.V. ˇ Strauss, H. Langer. Another approach to extension theory based on the notion of abstract boundary condition was proposed by J.W. Calkin [3] under the name of reduction operators. In a more specific situation this approach was later independently developed under the name of boundary value spaces or boundary triplet in V.M. Bruk [2] and A.N. Kochubei [11] obviously being motivated by M. I. Viˇ sik [15], F.S. Rofe-Beketov -69, M. L. Gorbachuk [9]. This approach relies on concepts of abstract boundary mappings and abstract Green’s identity. In the terminology of Calkin a boundary triplet is a reduction operator that is bounded (w.r.t. the graph norms).

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 2 / 23

• 1. Ordinary boundary triples and Weyl functions

Ordinary boundary triples

Definition

A collection Π = {H, Γ0, Γ1} consisting of a Hilbert space H and two linear mappings Γ0 and Γ1 from S∗ to H, is said to be an ordinary boundary triple for S∗ if: (i) the abstract Green’s identity (f ′, g) − (f , g′) = (Γ1 f , Γ0 g)H − (Γ0 f , Γ1 g)H (1.1) holds for all f = f f ′

• ,

g = g g′

• ∈ S∗;

(ii) the mapping Γ := Γ0 Γ1

• : S∗ → H2 is surjective.

In ODE setting formula (1.1) turns into the classical Lagrange identity, a key tool in treatment of BVP’s. In applications to BVP’s for elliptic equations formula (1.1) becomes a second Green’s

• formula. However, in this case the assumption (ii) is violated and this circumstance has been
• vercome in the classical papers by M. Visik [15], G. Grubb [10] and book of K. Moren (1965)

[14]. Relying on the Lions-Magenes trace theory they regularized the classical Dirichlet and Neumann trace mappings to get a correct version of Definition 1. Definition 1 yields a parametrization of all selfadjoint extensions A of S via

• A = AΘ := {

f ∈ S∗ : Γ f ∈ Θ} (1.2) where Θ ranges over the set of all selfadjoint relations in H. This correspondence is bijective and equivalently Θ := Γ( A). (1.3)

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 3 / 23

• 1. Ordinary boundary triples and Weyl functions

Weyl functions

The main analytical tool in description of spectral properties of selfadjoint extensions of S is the Weyl function introduced by Derkach and Malamud, which is the boundary triple analog of the Krein-Langer Q-function. Fix the notation for the following selfadjoint extensions of S: A0 := ker Γ0 = AΘ∞ and A1 := ker Γ1 = AΘ1 (1.4) where Θ∞ = {0} × H and Θ1 = O. Let Nλ := ker (S∗ − λ), λ ∈ ρ(A0), be the defect subspace of S and let

• Nλ =
• fλ =

fλ λfλ

• : fλ ∈ Nλ
• .

(1.5)

Definition

The abstract Weyl function and γ-field of S, corresponding to the ordinary boundary triple Π = {H, Γ0, Γ1} are defined by M(λ)Γ0 fλ = Γ1 fλ, γ(λ)Γ0 fλ = fλ,

• fλ ∈

Nλ, λ ∈ ρ(A0), (1.6) where fλ is given by (1.5). The γ-field γ(λ) and the Weyl function M(·) are holomorphic on ρ(A0) and satisfy the identities γ(λ) = [I + (λ − µ)(A0 − λ)−1]γ(µ), λ, µ ∈ ρ(A0), (1.7) M(λ) − M(µ)∗ = (λ − ¯ µ)γ(µ)∗γ(λ), λ, µ ∈ ρ(A0), (1.8) meaning that M(·) is a Q-function of S in the sense of Krein and Langer.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 4 / 23

• 1. Ordinary boundary triples and Weyl functions

Characterization of Weyl functions of ordinary boundary triples

It follows from (1.8) that M belongs to the Herglotz-Nevanlinna class R[H], i.e. M(λ) = M(¯ λ)∗ and Im M(λ) ≥ 0 for all λ ∈ C \ R. (1.9) Furthermore, γ(λ) maps H onto Nλ and (1.8) ensures that Im M(λ) is positive definite, i.e. M(·) belongs to the subclass Ru[H] of uniformly strict Herglotz-Nevanlinna functions: M(·) ∈ Ru[H] ⇐ ⇒ M ∈ R[H] and 0 ∈ ρ(Im M(λ)) for all λ ∈ C \ R. The converse is also true: every uniformly strict R[H]-function can be realized as the Weyl function of a symmetric operator A.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 5 / 23

• 2. BG-boundary triples and their Weyl functions

BG-boundary triples and their Weyl functions

The following notion of BG-boundary triple was introduced in Derkach-Malamud [7].

Definition

The collection Π = {H, Γ0, Γ1}, where H is a Hilbert space and Γ = {Γ0, Γ1} is a single-valued linear mapping from S∗ into H2, is said to be a B-generalized boundary triple, shortly, a BG-boundary triple for S∗, if S∗ is dense in S∗ and: (B1) the abstract Green’s identity (1.1) holds for all f = f f ′

• ,

g = g g′

• ∈ S∗;

(B2) ran Γ0 = H; (B3) A0 := ker Γ0 is a selfadjoint relation in H. The Weyl function corresponding to a BG-boundary triple is defined by the same equality (1.6) where fλ runs through Nλ ∩ S∗, a dense part of Nλ. The Weyl function M(·) of a B-generalized boundary triple, satisfies still the properties (1.7)–(1.9). However, instead of the property 0 ∈ ρ(Im M(i)) one has a weaker condition 0 ∈ σp(Im M(i)). We denote by Rs[H] the class of strict Nevanlinna functions, that is F(·) ∈ Rs[H] ⇐ ⇒ F(·) ∈ R[H] and 0 ∈ σp(Im F(i)) The class Rs[H] characterizes BG-boundary triples, in particular, every function from the class Rs[H] is a Weyl function of some BG-boundary triple.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 6 / 23

• 3. Unitary boundary triples and their Weyl families

Boundary relations and unitary boundary triples

The following definitions and facts are taken from Derkach-H-Malamud-de Snoo [4] (2006):

Definition

With H a Hilbert space a linear relation Γ : H2 → H2 is a unitary boundary relation for S∗, if: (G1) dom Γ is dense in S∗ and (f ′, g)H − (f , g′)H = (h′, k)H − (h, k′)H, (3.1) holds for every { f , h}, { g, k} ∈ Γ; (G2) Γ is maximal in the sense that if { g, k} ∈ H2 × H2 satisfies (3.1) for every { f , h} ∈ Γ, then { g, k} ∈ Γ. The condition (G1) can be interpreted as an abstract Green’s identity. Associate with Γ the following linear relations which are not necessarily closed: Γ0 =

• {

f , h} : { f , h} ∈ Γ, h = {h, h′}

• ,

Γ1 =

• {

f , h′} : { f , h} ∈ Γ, h = {h, h′}

• .

(3.2)

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 7 / 23

• 3. Unitary boundary triples and their Weyl families

Boundary relations as unitary mappings between Krein spaces

Consider (H2, JH) as a Kre˘ ın space with scalar product f f ′

• ,

g g′

• J

:= i(f ′, g) − i(f , g′) determined on H2 = H × H by JH := −iIH iIH

• .

Now the condition (G1) can be interpreted as follows: Γ is an isometric multivalued mapping from the Kre˘ ın space (H2, JH) to the Kre˘ ın space (H2, JH): (JH f , g)H2 = (JH h, k)H2, { f , h}, { g, k} ∈ Γ. The maximality condition (G2) guarantees that a boundary relation Γ is a unitary relation from (H2, JH) to (H2, JH): Γ−1 = Γ[∗]. In particular, Γ is closed and linear. Converse is also true:

Proposition

Let Γ be a unitary relation from the Kre˘ ın space (H2, JH) to the Kre˘ ın space (H2, JH). Then: Γ boundary relation for S∗ ⇐ ⇒ ker Γ = S.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 8 / 23

• 3. Unitary boundary triples and their Weyl families

Weyl families and γ-fields

Definition

The Weyl family M(·) of S corresponding to the boundary relation Γ : H2 → H2 is defined by M(λ) := Γ( Nλ(T)), λ ∈ C \ R, i.e., M(λ) =

• h ∈ H2 : {

fλ, h} ∈ Γ, fλ = {f , λf } ∈ H2 , If M(·) is operator-valued, then it is called the Weyl function of S corresponding to Γ.

Definition

The γ-field γ(·) of S corresponding to the boundary relation Γ : H2 → H2 is defined by γ(λ) :=

• {h, f } ∈ H × H : {

fλ, h} ∈ Γ, fλ ∈ H2 , where fλ = {f , λf }, h = {h, h′}, and λ ∈ C \ R. Moreover, γ(λ) stands for

• γ(λ) :=
• {h,

fλ} ∈ H × H2 : {h, f } ∈ γ(λ)

• .

γ-field is a single-valued mapping from Γ0( Nλ(T)) = dom M(λ) onto Nλ(T), T = dom Γ (T dense in S∗).

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 9 / 23

• 3. Unitary boundary triples and their Weyl families

Nevanlinna families

A family of linear relations M(λ), λ ∈ C \ R, in a Hilbert space H is called a Nevanlinna family if: (i) M(λ) is maximal dissipative for every λ ∈ C+ (resp. max. accumulative for every λ ∈ C−); (ii) M(λ)∗ = M(¯ λ), λ ∈ C \ R; (iii) for some, and hence for all, µ ∈ C+(C−) the operator family (M(λ) + µ)−1(∈ [H]) is holomorphic for all λ ∈ C+(C−). By the maximality condition, M(λ), λ ∈ C \ R, is closed. The class of all Nevanlinna families in a Hilbert space is denoted by R(H). Nevanlinna families M(λ) ∈ R(H) admit the following decomposition to the operator part Ms(λ), λ ∈ C \ R, and constant multi-valued part M∞: M(λ) = Ms(λ) ⊕ M∞, M∞ = {0} × mul M(λ). Here Ms(λ) is a Nevanlinna family of densely defined operators in H ⊖ mul M(λ).

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 10 / 23

• 3. Unitary boundary triples and their Weyl families

Realization theorem for Nevanlinna families

Two boundary relations Γ(j) : (H(j))2 → H2, j = 1, 2, are said to be unitarily equivalent if there is a unitary operator U : H(1) → H(2) such that Γ(2) = Uf Uf ′

• ,

h h′

• :

f f ′

• ,

h h′

• ∈ Γ(1)
• .

(3.3) If the boundary relations Γ(1) and Γ(2) are connected by (3.3) and Sj = ker Γ(j), Tj = dom Γ(j), j = 1, 2, then S2 = US1U−1, T2 = UT1U−1. The boundary relation Γ : H2 → H2 is minimal, if H = Hmin := span { Nλ(T) : λ ∈ C+ ∪ C− }.

Theorem (DHMS2006)

Let Γ : H2 → H2 be a boundary relation for S∗. Then the corresponding Weyl family M(·) belongs to the class R(H). Conversely, if M(·) belongs to the class R(H) then there exists (up to unitary equivalence) a unique minimal boundary relation whose Weyl function coincides with M(·).

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 11 / 23

• 4. SG-boundary triples and their Weyl functions

SG-boundary triples and their Weyl functions

Ordinary and BG-boundary triplets are examples of unitary boundary triplets. The notion of SG-boundary triples provides a wider subclass of unitary boundary triples; it has been initially studied in Derkach-H-Malamud-de Snoo [4], [5].

Definition ([4])

A unitary boundary triple Π = {H, Γ0, Γ1} is said to be an S-generalized (or SG-)boundary triple for S∗, if condition (B3) holds, i.e., A0 := ker Γ0 is a selfadjoint extension of S. One has the following characterizations of SG-boundary triples; see [4] and [5, Theorem 7.39].

Theorem ([5])

Let Π = {H, Γ0, Γ1} be a unitary boundary triple for S∗ and let M(·) and γ(·) be the corresponding Weyl function and γ-field. Then TFAE: (i) A0 = ker Γ0 is selfadjoint, i.e. Π is a SG-boundary triple; (ii) ran Γ0 = dom M(λ) for all λ ∈ C \ R; (iii) dom γ(λ) is dense in H and γ(λ) is bounded for all λ ∈ C \ R; (iv) the Weyl function M(·) ∈ Rs(H) and it admits the representation M(λ) = E + M0(λ), M0(·) ∈ R[H], λ ∈ C±, (4.1) where E = E ∗ is a selfadjoint (in general unbounded) operator in H.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 12 / 23

• 5. ESG-boundary triples and their Weyl functions

ESG-boundary triples and their Weyl functions

Definition

A unitary boundary triple {H, Γ0, Γ1} for S∗ is said to be an essentially selfadjoint generalized boundary triple (in short, ESG-boundary triple) for S∗, if: (B3′) A0 := ker Γ0 is essentially selfadjoint linear relation in H. Associate with M(·) a family of non-negative quadratic forms tM(λ) in H: tM(λ)[u, v] := 1 λ − ¯ λ [(M(λ)u, v) − (u, M(λ)v)], u, v ∈ dom (M(λ)), λ ∈ C \ R. (5.1) The forms tM(λ) are not necessarily closable. However, if tM(λ0) is closable at one point λ0 ∈ C±, then tM(λ) is closable for each λ ∈ C±. In this case the domain of the closure tM(λ) does not depend on λ ∈ C±, i.e., it is form domain invariant.

Theorem

Let Π = {H, Γ0, Γ1} be a unitary boundary triple for S∗ and let M(·) and γ(·) be the corresponding Weyl function and γ-field. Then TFAE: (i) Π is an ESG-boundary triple for S∗; (ii) γ(±i) is closable; (iii) γ(λ) is closable for every λ ∈ C± and dom γ(λ) = dom γ(±i), λ ∈ C±; (iv) the form tM(±i) is closable; (v) the form tM(λ) is closable for each λ ∈ C± and dom tM(λ) = dom tM(±i), λ ∈ C±; (vi) the Weyl function M(·) belongs to Rs(H) and is form-domain invariant in C±.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 13 / 23

• 6. Laplacian on a bounded domain

Laplacian on a bounded domain

Ω a bounded domain in Rn with smooth or, more generally, Lipschitz boundary ∂Ω. Let S∗ = −∆ be the maximal Laplace operator in L2(Ω) and let a γD and γN be the Dirichlet and Neumann traces. It is known that the mappings γD : H3/2(Ω) → H1(∂Ω), γN : H3/2(Ω) → H0(∂Ω) are well defined and surjective; see e.g. Agranovic (2015), Lions-Magenes (1972), Moren (1965), Grubb (1968) (smooth case) and Jerison - Kenig (1981, 1995), Gesztesy - Mitrea (2011) (Lipschitz case). Let S∗ be the restriction of Smax to the domain dom S∗ = H3/2

∆ (Ω) := H3/2(Ω) ∩ dom Smax =

• f ∈ H3/2(Ω) : ∆f ∈ L2(Ω)
• ,

(6.1) Following Viˇ sik [15] and Grubb [10] introduce the regularized trace operators via

• Γ0,Ωy = (γN − Λ(0)γD)y,
• Γ1,Ωy = γDy,

y ∈ dom S∗, (6.2) where Λ := Λ(0) is the Dirichlet-to-Neumann map, an (unbounded) selfadjoint operator in L2(∂Ω) on the domain dom Λ = H1(∂Ω). It is known (see [10]) that the mappings Γ0,Ω and Γ1,Ω are well defined and

• Γ0,Ω : dom Smax → H1/2(∂Ω),
• Γ1,Ω : dom Smax → H−1/2(∂Ω).

(6.3) Let S∗ be a restriction of Smax to the domain dom S∗ = {f ∈ dom Smax : γDf ∈ L2(∂Ω)}. (6.4)

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 14 / 23

• 6. Laplacian on a bounded domain

Theorem

Let Ω be smooth and let γN, γD, Λ(z), S∗ and S∗ be as above. Then:

1 {L2(∂Ω), γD↾ dom S∗, −γN↾ dom S∗} is an S-generalized boundary triple for S∗, and the

corresponding Weyl function M(·) coincides with −Λ(·), (Λ(·) Dirichlet-to-Neumann map);

2 the transposed triple Π = {H0(∂Ω), γN↾ dom S∗, γD↾ dom S∗} is a BG-boundary triple for S∗; 3 {L2(∂Ω),

Γ0,Ω↾ dom S∗, Γ1,Ω↾ dom S∗} is an ESG-boundary triple for S∗, the corresponding Weyl function is the L2(∂Ω)-closure

• M(z) = clos (Λ(z) − Λ(0))−1;

4 the extension

A0 := S∗↾ ker Γ0,Ω is essentially selfadjoint and its closure coincides with the Kre˘ ın - von Neumann extension of the operator Smin;

5 the γ-field is closable and its closure

• γ(z) = (

Γ0,Ω↾ Nz(Smax))−1, is an unbounded domain invariant operator function with dom γ(z) = H1/2(∂Ω);

6 the Weyl function M(z) = −Λ(z) is domain invariant with dom M(z) = H1(∂Ω) and belongs

to the class of inverse Stieltjes functions while the inverse Λ(z)−1 belongs to the class of Stieltjes functions of compact operators;

7 the Weyl function

M(·) is form domain invariant with form domain H1/2(∂Ω) and it belongs to the class of Stieltjes functions of unbounded operators while the inverse − M(·)−1 = clos (Λ(0) − Λ(·)) is in the class of inverse Stieltjes functions of bounded

• perators.
• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 15 / 23

• 7. Direct sums of boundary triples and local point interactions

Direct sums of boundary triples

Sn a densely defined symmetric operator with equal defect numbers n+(Sn) = n+(Sn). Consider A =

∞

• n=1

Sn in the Hilbert space H := ∞

n=1 Hn = { ⊕∞ n=1fn : fn ∈ Hn, ∞ n=1 fn2 < ∞}. Then A is

symmetric with equal defect numbers and A∗ =

∞

• n=1

S∗

n ,

dom A∗ =

• ⊕∞

n=1fn ∈ H : fn ∈ dom S∗ n , ∞

• n=1

S∗

n fn2 < ∞

• .

Let Πn = {Hn, Γ(n)

0 , Γ(n) 1 } be an ordinary boundary triple for S∗ n , n ∈ N. Let H = ∞ n=1 Hn and

let the mappings Γ0 and Γ1 be defined by Γj :=

∞

• n=1

Γ(n)

j

, dom Γj =

• f = ⊕∞

n=1fn ∈ dom A∗ :

• n∈N

Γ(n)

j

fn2

Hn < ∞

• ,

j ∈ {0, 1}. (7.1) Clearly, dom Γ := dom Γ1 ∩ dom Γ0 is dense in dom A∗ w.r.t. the graph norm of A∗. Define Sn,j := S∗

n ↾ ker Γ(n) j

and Aj :=

∞

• n=1

Sn,j, j ∈ {0, 1}. A0 and A1 are self-adjoint extensions of A. Finally, let A∗ := A∗ ↾ dom Γ and A∗j := A∗ ↾ ker Γj, j ∈ {0, 1}. (7.2) Clearly, A∗j = Aj, hence A∗j is essentially selfadjoint, j ∈ {0, 1}.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 16 / 23

• 7. Direct sums of boundary triples and local point interactions

Abstract results

The following result is contained in Kostenko - Malamud [13, Theorem 3.2].

Theorem ([13])

Let Πn = {Hn, Γ(n)

0 , Γ(n) 1 } be an ordinary boundary triple for S∗ n , let Sn,j = S∗ n ↾ ker Γ(n) j

, j ∈ {0, 1}, and let Mn(·), n ∈ N, be the Weyl function. Let A∗ = ∞

n=1 S∗ n and Γj = ∞ n=1 Γ(n) j

, j ∈ {0, 1}. Then:

1 Π = {H, Γ0, Γ1} is a unitary boundary triple for A∗; 2 the corresponding Weyl function is the orthogonal sum M(z) = ∞

n=1 Mn(z);

3 the mapping Γj : H+ → H, j ∈ {0, 1}, is closable; 4 The operator A∗j given by (7.2) is essentially selfadjoint and A∗j = ∞

n=1 Sn,j = Aj,

j ∈ {0, 1}. The following result characterizes selfadjointness of Aj = ker Γj, j ∈ {0, 1}.

Proposition

Let the assumptions be as in Theorem 14 and let Aj = ker Γj, j ∈ {0, 1}. Then Aj =

∞

• n=1

Sn,j ⇔ Γj′↾ Aj is bounded (j′ = 1 − j ∈ {0, 1}). (7.3)

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 17 / 23

• 7. Direct sums of boundary triples and local point interactions

Abstract results via Weyl function

The following criteria for a direct sum of ordinary triples to be an ordinary or B-generalized boundary triple in terms of the corresponding Weyl function can be found from Kostenko - Malamud (2010), Malamud - Neidhardt (2012) and Carlone - Malamud - Posilicano (2013).

Theorem

Let Πn = {Hn, Γ(n)

0 , Γ(n) 1 } be a boundary triple for S∗ n and let Mn(·) be the corresponding Weyl

function, n ∈ N.

1 The direct sum Π = ⊕∞

n=1Πn forms an ordinary boundary triple for the operator

A∗ = ⊕∞

n=1S∗ n if and only if

C1 = sup

n

Mn(i)Hn < ∞ and C2 = sup

n

(Im Mn(i))−1Hn < ∞.

2 The direct sum Π = ⊕∞

n=1Πn is a B-generalized boundary triple for the operator

A∗ = ⊕∞

n=1S∗ n if and only if C1 < ∞.

3 If, in addition, the operators {Sn0}n∈N have a common gap (a − ε, a + ε), then the direct

sum Π = ∞

n=1 Πn is a B-generalized boundary triple for A∗ = ∞ n=1 S∗ n if and only if

C3 := sup

n∈N

Mn(a)Hn < ∞ and C4 := sup

n∈N

M′

n(a)Hn < ∞,

(7.4) where M′

n(a) := (dMn(z)/dz)|z=a.

4 Π = ∞

n=1 Πn is an ordinary boundary triple for A∗ if and only if in addition to (7.4) one has

C5 := sup

n∈N

• M′

n(a)

−1Hn < ∞. (7.5)

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 18 / 23

• 7. Direct sums of boundary triples and local point interactions

The next statement contains analogous characterization for S-generalized boundary triples.

Proposition

The direct sum Π = ∞

n=1 Πn forms an S-generalized boundary triple for A∗ = ∞ n=1 S∗ n if and

• nly if

sup

n

Im Mn(i)Hn < ∞. (7.6) Similarly, if the operators (Sn0) have a common gap (a − ε, a + ε), then Π forms an S-generalized boundary triple for A∗ if and only if C4 := sup

n∈N

M′

n(a)Hn < ∞.

Proposition

The following conditions are equivalent: (i) Γ0 : A∗ → H is bounded; (ii) C2 = supn (Im Mn(i))−1Hn < ∞. In this case the transposed boundary triple Π⊤ = {H, Γ1, −Γ0} is B-generalized. Similarly, the following conditions are equivalent: (i)′ Γ1 : A∗ → H is bounded; (ii)′ the following condition is satisfied C ⊤

2 := supn (Im (M−1 n

(i)))−1Hn < ∞. In this case Π is a B-generalized boundary triple.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 19 / 23

• 7. Direct sums of boundary triples and local point interactions

Schr¨

• dinger operators with local point interactions

Differential operators with point interactions arise in various physical applications as exactly solvable models that describe complicated physical phenomena. Numerous results may be found in monographs Albeverio - Gesztesy - Hoegh-Krohn - Holden, H. Solvable Models in Quantum Mechanics, Sec. Ed. 2005; with Appendix K by P. Exner, Seize ans apr` es, and Albeverio - Kurasov, Singular Perturbations of Differential Operators and Schr¨

• dinger Type Operators, 2000.

The following has its origin in Kronig–Penney model. Let ℓX be a formal differential expression ℓX,α := − d2 dx2 +

• xn∈X

αnδ(x − xn), (7.7) where α = (αn)∞

n=0 and

0 = x0 < x1 < x2 < · · · < xn < · · · < +∞, and lim

n→∞ xn ≤ ∞.

(7.8) With the sequence X we associate the following numbers dn := xn − xn−1 > 0, 0 ≤ d∗(X) := inf

n∈N dn,

d∗(X) := sup

n∈N

dn ≤ ∞. (7.9) Define the operator H0

X,α in L2(R+) by H0 X,αf (x) := − d2f dx2 on the set of functions

f ∈ W 2,2

comp(R+ \ X) such that

f ′(0) = 0, f (xn+) = f (xn−), f ′(xn+) − f ′(xn−) = αnf (xn) (n ∈ N). (7.10) Let HX,α be the closure of H0

X,α. In general, the operator HX,α is symmetric but not

automatically selfadjoint, even in the case q ≡ 0.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 20 / 23

• 7. Direct sums of boundary triples and local point interactions

To include the operator HX,α in the framework of extension theory, consider the following symmetric operator in L2(R+) H := Hmin = − d2 dx2 , dom (Hmin) = W 2,2 (R+ \ X) =

∞

• n=1

W 2,2 [xn−1, xn]. Clearly, H = Hmin is closed and H = Hmin = ⊕∞

n=1Hn,

where Hn = − d2 dx2 , dom (Hn) = W 2,2 [xn−1, xn]. Then Πn = {C2, Γ(n)

0 , Γ(n) 1 } given by

Γ(n)

0 f :=

f ′(xn−1+) f ′(xn−)

• ,

Γ(n)

1 f :=

−f (xn−1+) f (xn−)

• (f ∈ W 2

2 [xn−1, xn])

(7.11) forms a boundary triple for H∗

n satisfying ker Γ(n)

= dom HF

n . Moreover, the corresponding Weyl

function Mn is given by Mn(z) = −1 √z

• cot(√zdn)

−

1 sin(√zdn)

−

1 sin(√zdn)

cot(√zdn)

• .

(7.12) As was shown by Kochubei (1989) [12], the triple Π = ⊕n∈NΠn forms an ordinary boundary triple for the operator H∗

min := (Hmin)∗ = Hmax whenever

0 < d∗ = inf

n∈N dn

≤ d∗ = sup

n∈N

dn < +∞. (7.13) Moreover, [13] showed that the converse is also true: if Π = ⊕n∈NΠn is an ordinary boundary triple for H∗

min := Hmax, then d∗ > 0 holds.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 21 / 23

• 7. Direct sums of boundary triples and local point interactions

The next result completes the earlier results by Kostenko - Malamud, Malamud - Neidhardt, and Carlone - Malamud - Posilicano on the non-regularized boundary triple Π in the general case 0 ≤ d∗ ≤ d∗ ≤ ∞.

Theorem

Let Π := ⊕∞

n=1Πn = {H, Γ0, Γ1} be the direct sum of boundary triples Πn defined by (7.11) and

let M(·) be the corresponding Weyl function. Then: (i) Π is an ES-generalized boundary triple for the operator H∗ = ⊕∞

n=1H∗ n, such that the

extension A0 is essentially selfadjoint; (ii) If d∗ > 0, then Π is an ordinary boundary triple for the operator H∗; (iii) If d∗ = 0 (and d∗ ≤ ∞), then Π is an ES-generalized boundary triple for the operator H∗

min

such that A0 = A∗

0.

(iv) If d∗ = 0 (and d∗ ≤ ∞), then the ES-generalized boundary triple Π satisfies A1 = A∗

1 and

the transposed triple Π⊤ is an S-generalized, but not B-generalized, boundary triple for the

• perator H∗

min.

(v) If d∗ = 0 (and d∗ ≤ ∞), then the Weyl functions M(·) and −M(·)−1 are both unbounded and domain invariant. (vi) If d∗ = 0 (and d∗ ≤ ∞), then the imaginary part Im M(z) of the Weyl function M(·) is unbounded while the imaginary part Im (M(z)−1) is bounded for every z ∈ C \ R.

• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 22 / 23

Bibliography Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H. Solvable Models in Quantum Mechanics, 2 Ed., 2005. V.M. Bruk, ”On a class of problems with the spectral parameter in the boundary conditions”, Mat. Sb., 100 (1976), 210–216. J.W. Calkin, Abstract symmetric boundary conditions, TAMS, 1939, 45, no. 3, 369-442. V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, ”Boundary relations and Weyl families”, TAMS, 358 (2006), 5351–5400. V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. de Snoo, Boundary triples and Weyl functions. Recent developments, London Mathematical Society Lecture Notes, 404, (2012), 161–220. V.A. Derkach and M.M. Malamud, ”Generalized resolvents and the boundary value problems for hermitian operators with gaps”, J. Funct. Anal., 95 (1991), 1–95. V.A. Derkach and M.M. Malamud, ”The extension theory of hermitian operators and the moment problem”, J. Math. Sciences, 73 (1995), 141–242.

• P. Exner, Seize ans apr`

es, Appendix K to ”Solvable Models in Quantum Mechanics” by Albeverio, Gesztesy, Hoegh-Krohn, Holden, 2. Ed., 2004.

• M. L. Gorbachuk, Self-adjoint boundary value problems for differential equation of the secoind order with unbounded operator coefficient,

Functional Anal. Appl., 5 (1971), no.1, 10-21.

• G. Grubb, A characterization of the non local boundary value problems associated with an elliptic operator, Ann. Scuola Normale Superiore de Pisa,

22 (1968), no. 3, 425-513. A.N. Kochubei, On extentions of symmetric operators and symmetric binary relations, Matem. Z., 17 (1975), 41–48. A.N. Kochubei, One-dimensional point interactions, Ukrain. Math. J. 41 (1989), 1391–1395. A.S. Kostenko, M.M. Malamud, 1–D Schr¨

• dinger operators with local point interactions on a discrete set. J. Differential Equations 249 (2010), no.

2, 253–304.

• K. Moren, Hilbert space methods, Moscow: Mir, 1965, 570 p.
• M. I. Viˇ

sik, On general boundary problems for elliptic differential equations. (Russian) Trudy Moskov. Mat. Obˇ

• sc. 1, (1952). 187–246.
• S. Hassi

Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 23 / 23