Operator Theory and Krein Spaces TU Wien, December 19-22, 2019 Geometric approximations of point interactions Andrii Khrabustovskyi Graz University of Technology & University of Hradec Kr´ alov´ e Talk is based on G. Cardone, A. K., J. Math. Anal. Appl. 473 (2019), 1320–1342 A. K., O. Post, in preparation 1 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Point interactions: warm-up Physical motivation: quantum particles moving in a potentials localized near a discrete set of points 2 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Point interactions: warm-up Physical motivation: quantum particles moving in a potentials localized near a discrete set of points Example: Kronnig-Penney model describing a nonrelativistic electron moving in a crystal lattice. It is given by the Hamiltonian − d 2 d z 2 + α ∑ δ ( ·− k ) k ∈ Z 2 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Point interactions: warm-up Physical motivation: quantum particles moving in a potentials localized near a discrete set of points Example: Kronnig-Penney model describing a nonrelativistic electron moving in a crystal lattice. It is given by the Hamiltonian − d 2 d z 2 + α ∑ δ ( ·− k ) k ∈ Z In what follows we assume that that discrete set consists of only one point 0 2 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Point interactions: warm-up Physical motivation: quantum particles moving in a potentials localized near a discrete set of points Example: Kronnig-Penney model describing a nonrelativistic electron moving in a crystal lattice. It is given by the Hamiltonian − d 2 d z 2 + α ∑ δ ( ·− k ) k ∈ Z In what follows we assume that that discrete set consists of only one point 0 Mathematical realization: self-adjoint differential operators with the action − d 2 d z 2 on ( − ∞ , 0 ) ∪ ( 0 , ∞ ) subject to certain coupling conditions at 0. 2 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Point interactions: warm-up Physical motivation: quantum particles moving in a potentials localized near a discrete set of points Example: Kronnig-Penney model describing a nonrelativistic electron moving in a crystal lattice. It is given by the Hamiltonian − d 2 d z 2 + α ∑ δ ( ·− k ) k ∈ Z In what follows we assume that that discrete set consists of only one point 0 Mathematical realization: self-adjoint differential operators with the action − d 2 d z 2 on ( − ∞ , 0 ) ∪ ( 0 , ∞ ) subject to certain coupling conditions at 0. Two distinguished examples of such couplings: u ′ (+ 0 ) − u ′ ( − 0 ) = γ u ( ± 0 ) δ -interaction: u (+ 0 ) = u ( − 0 ) , δ ′ -interaction: u ′ (+ 0 ) = u ′ ( − 0 ) , u (+ 0 ) − u ( − 0 ) = γ − 1 u ′ ( ± 0 ) 2 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Approximations by regular Schr¨ odinger operators Example 1 [Albeverio-Høegh-Krohn, 1981] (appr. of δ -interactions) odinger operator − d 2 Schr¨ d z 2 + V ε with the potential V ε ( x ) = γε − 1 V ( x ε − 1 ) , where V : R → R compactly supported smooth function such that � R V ε ( x ) dx = 1 converges in a norm-resolvent topology to the Schr¨ odinger operator with δ -interaction at zero. 3 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Approximations by regular Schr¨ odinger operators Example 1 [Albeverio-Høegh-Krohn, 1981] (appr. of δ -interactions) odinger operator − d 2 Schr¨ d z 2 + V ε with the potential V ε ( x ) = γε − 1 V ( x ε − 1 ) , where V : R → R compactly supported smooth function such that � R V ε ( x ) dx = 1 converges in a norm-resolvent topology to the Schr¨ odinger operator with δ -interaction at zero. Example 2 [Cheon-Shigehara, 1998], [Exner-Neidhardt-Zagrebnov, 1981]: (appr. of δ -interactions) odinger operators with δ ′ -interaction can be approximated by Schr¨ Schr¨ odinger operators with smooth potentials. These smooth potentials have the form of a sum of three suitably scaled δ -like profiles. 3 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Geometrical approximations (using Laplace-type operators on tubular waveguides) Example [Kuchment-Zeng, 2003], [Exner-Post, 2005] 4 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Geometrical approximations (using Laplace-type operators on tubular waveguides) Example [Kuchment-Zeng, 2003], [Exner-Post, 2005] Possible limits: Dirichlet decoupling (large connectors) Trivial coupling (small connectors) Kind of a δ -coupling (coupling constant depends on a spectral parameter) (borderline regime) 4 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction Geometrical approximations (using Laplace-type operators on tubular waveguides) Example [Kuchment-Zeng, 2003], [Exner-Post, 2005] Possible limits: Dirichlet decoupling (large connectors) Trivial coupling (small connectors) Kind of a δ -coupling (coupling constant depends on a spectral parameter) (borderline regime) In [Exner-Post, 2013] an approximation of all possible couplings (on a graph !!!) by suitable magnetic Schr¨ odinger operators on tubular domains was realised (see also the important preliminary work [Cheon-Exner-Turek, 2010]) 4 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Introduction In the talk we address the problem of approximation of δ and δ ′ interactions using only geometrical tools. 5 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Geometry 6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Geometry a b − ∞ ≤ a < 0 < b ≤ ∞ 6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Geometry a b − ∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ R n − 1 6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Geometry Ω − Ω + ε ε − ∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ R n − 1 Ω − ε = ( ε S ) × ( a , 0 ) , Ω + ε = ( ε S ) × ( 0 , b ) , where ε > 0 6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Geometry D ε Ω − Ω + ✟ ✟ ✙ ε ε − ∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ R n − 1 Ω − ε = ( ε S ) × ( a , 0 ) , Ω + ε = ( ε S ) × ( 0 , b ) , where ε > 0 D ε = ( d ε D ) ×{ 0 } , where d ε ∈ ( 0 , ε ] 6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Geometry D ε Ω − Ω + ✟ ✟ ✙ ε ε − ∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ R n − 1 Ω − ε = ( ε S ) × ( a , 0 ) , Ω + ε = ( ε S ) × ( 0 , b ) , where ε > 0 D ε = ( d ε D ) ×{ 0 } , where d ε ∈ ( 0 , ε ] Ω = Ω − ε ∪ D ε ∪ Ω + ε 6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Geometry D ε Ω − Ω + ✟ ✟ ✙ ε ε − ∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ R n − 1 Ω − ε = ( ε S ) × ( a , 0 ) , Ω + ε = ( ε S ) × ( 0 , b ) , where ε > 0 D ε = ( d ε D ) ×{ 0 } , where d ε ∈ ( 0 , ε ] Ω = Ω − ε ∪ D ε ∪ Ω + ε We assume that the following limit, either finite or infinite, exists: · ( d ε ) n − 2 cap ( D ) ε n − 1 , n ≥ 3 4 | S | γ = lim ε → 0 γ ε , where γ ε = 1 2 π · , n = 2 ε ln d ε 6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Operators Operator A ε Let V ε ∈ L ∞ (Ω ε ) . In L 2 (Ω ε ) we introduce the quadratic form a ε by � � | ∇ u | 2 + V ε | u | 2 � dom ( a ε ) = H 1 (Ω ε ) a ε [ u ] = dx , Ω ε By A ε we denote the operator associated with this form. Evidently, A ε = − ∆ N Ω ε + V ε , where ∆ N Ω ε is the Neumann Laplacian on Ω ε . 7 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Operators Operator A γ Let V ∈ L ∞ ( a , b ) . In L 2 ( a , b ) we introduce the sesquilinear form a γ by � b � | u ′ | 2 + V | u | 2 � dx + γ | u (+ 0 ) − u ( − 0 ) | 2 , a γ [ u ] = γ < ∞ : a dom ( a γ ) = H 1 (( a , b ) \{ 0 } ) 8 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
Approximation of δ ′ -interaction Operators Operator A γ Let V ∈ L ∞ ( a , b ) . In L 2 ( a , b ) we introduce the sesquilinear form a γ by � b � | u ′ | 2 + V | u | 2 � dx + γ | u (+ 0 ) − u ( − 0 ) | 2 , a γ [ u ] = γ < ∞ : a dom ( a γ ) = H 1 (( a , b ) \{ 0 } ) � b � | u ′ | 2 + V | u | 2 � a ∞ [ u ] = dom ( a γ ) = H 1 ( a , b ) γ = ∞ : dx , a 8 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions
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