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) operators between the defect subspaces V : N i → N − i , where N z is defined by N z := 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. sik, M. Sh. Birman, A.V. ˇ Krasnosel’skii, M.A. Naimark, M. I. Viˇ 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 , Γ 0 , Γ 1 } consisting of a Hilbert space H and two linear mappings Γ 0 and Γ 1 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 ′ ) = (Γ 1 � g ) H − (Γ 0 � f , Γ 0 � f , Γ 1 � g ) H (1.1) � f � � g � holds for all � ∈ S ∗ ; f = , � g = f ′ g ′ � Γ 0 � : S ∗ → H 2 is surjective. (ii) the mapping Γ := Γ 1 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 overcome 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 f ∈ S ∗ : Γ � A = A Θ := { � � 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 : A 0 := ker Γ 0 = A Θ ∞ and A 1 := ker Γ 1 = A Θ 1 (1.4) where Θ ∞ = { 0 } × H and Θ 1 = O . Let N λ := ker ( S ∗ − λ ), λ ∈ ρ ( A 0 ), be the defect subspace of S and let � � f λ � � � � N λ = f λ = : f λ ∈ N λ . (1.5) λ f λ Definition The abstract Weyl function and γ -field of S , corresponding to the ordinary boundary triple Π = {H , Γ 0 , Γ 1 } are defined by M ( λ )Γ 0 � f λ = Γ 1 � γ ( λ )Γ 0 � f λ ∈ � � f λ , f λ = f λ , N λ , λ ∈ ρ ( A 0 ) , (1.6) where � f λ is given by (1.5). The γ -field γ ( λ ) and the Weyl function M ( · ) are holomorphic on ρ ( A 0 ) and satisfy the identities γ ( λ ) = [ I + ( λ − µ )( A 0 − λ ) − 1 ] γ ( µ ) , λ, µ ∈ ρ ( A 0 ) , (1.7) M ( λ ) − M ( µ ) ∗ = ( λ − ¯ µ ) γ ( µ ) ∗ γ ( λ ) , λ, µ ∈ ρ ( A 0 ) , (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 N λ and (1.8) ensures that Im M ( λ ) is positive definite, i.e. M ( · ) belongs to the subclass R u [ H ] of uniformly strict Herglotz-Nevanlinna functions: M ( · ) ∈ R u [ 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 , Γ 0 , Γ 1 } , where H is a Hilbert space and Γ = { Γ 0 , Γ 1 } is a single-valued linear mapping from S ∗ into H 2 , is said to be a B-generalized boundary triple , shortly, a BG-boundary triple for S ∗ , if S ∗ is dense in S ∗ and: � f � � g � (B1) the abstract Green’s identity (1.1) holds for all � f = , � g = ∈ S ∗ ; f ′ g ′ (B2) ran Γ 0 = H ; (B3) A 0 := ker Γ 0 is a selfadjoint relation in H . The Weyl function corresponding to a BG-boundary triple is defined by the same equality (1.6) where � f λ runs through � N λ ∩ S ∗ , a dense part of � N λ . 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 �∈ σ p ( Im M ( i )). We denote by R s [ H ] the class of strict Nevanlinna functions , that is F ( · ) ∈ R s [ H ] ⇐ ⇒ F ( · ) ∈ R [ H ] and 0 �∈ σ p ( Im F ( i )) The class R s [ H ] characterizes BG-boundary triples, in particular, every function from the class R s [ 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 : H 2 �→ H 2 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 , � g , � h } , { � k } ∈ Γ; k } ∈ H 2 × H 2 satisfies (3.1) for every { � g , � f , � (G2) Γ is maximal in the sense that if { � 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: � � { � f , h } : { � f , � h } ∈ Γ , � h = { h , h ′ } Γ 0 = , � � (3.2) { � f , h ′ } : { � f , � h } ∈ Γ , � h = { h , h ′ } Γ 1 = . 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 ( H 2 , J H ) as a Kre˘ ın space with scalar product �� f �� � g � := i ( f ′ , g ) − i ( f , g ′ ) , f ′ g ′ J � 0 � − iI H determined on H 2 = H × H by J H := . iI H 0 Now the condition (G1) can be interpreted as follows: ın space ( H 2 , J H ) to the Kre˘ ın space ( H 2 , J H ): Γ is an isometric multivalued mapping from the Kre˘ ( J H � g ) H 2 = ( J H � h , � { � f , � g , � f , � k ) H 2 , h } , { � k } ∈ Γ . The maximality condition (G2) guarantees that a boundary relation Γ is a unitary relation from ( H 2 , J H ) to ( H 2 , J H ): Γ − 1 = Γ [ ∗ ] . In particular, Γ is closed and linear. Converse is also true: Proposition ın space ( H 2 , J H ) to the Kre˘ ın space ( H 2 , J H ) . Then: Let Γ be a unitary relation from the Kre˘ Γ boundary relation for S ∗ ⇐ ⇒ ker Γ = S . S. Hassi Generalized boundary triplets, Weyl functions, and point interactions Vienna, Dec. 20, 2016 8 / 23
Recommend
More recommend
