Geometric approximations of point interactions Andrii Khrabustovskyi - - PowerPoint PPT Presentation

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 − d2 dz2 +α ∑

k∈Z

δ(·−k)

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 − d2 dz2 +α ∑

k∈Z

δ(·−k) 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 − d2 dz2 +α ∑

k∈Z

δ(·−k) In what follows we assume that that discrete set consists of only one point 0 Mathematical realization: self-adjoint differential operators with the action − d2

dz2 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 − d2 dz2 +α ∑

k∈Z

δ(·−k) In what follows we assume that that discrete set consists of only one point 0 Mathematical realization: self-adjoint differential operators with the action − d2

dz2 on (−∞,0)∪(0,∞) subject to certain coupling

conditions at 0. Two distinguished examples of such couplings:

δ-interaction: u(+0) = u(−0), u′(+0)−u′(−0) = γu(±0) δ ′-interaction: u′(+0) = u′(−0), u(+0)−u(−0) = γ−1u′(±0)

2 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Introduction

Approximations by regular Schr¨

• dinger operators

Example 1 [Albeverio-Høegh-Krohn, 1981] (appr. of δ-interactions) Schr¨

• dinger operator − d2

dz2 +Vε with the potential

Vε(x) = γε−1V(xε−1), where V : R → R compactly supported smooth function such that

• R Vε(x)dx = 1 converges in a norm-resolvent topology to the

Schr¨

• dinger operator with δ-interaction at zero.

3 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Introduction

Approximations by regular Schr¨

• dinger operators

Example 1 [Albeverio-Høegh-Krohn, 1981] (appr. of δ-interactions) Schr¨

• dinger operator − d2

dz2 +Vε with the potential

Vε(x) = γε−1V(xε−1), where V : R → R compactly supported smooth function such that

• R Vε(x)dx = 1 converges in a norm-resolvent topology to the

Schr¨

• dinger operator with δ-interaction at zero.

Example 2 [Cheon-Shigehara, 1998], [Exner-Neidhardt-Zagrebnov, 1981]: (appr. of δ-interactions) Schr¨

• dinger operators with δ ′-interaction can be approximated by

Schr¨

• dinger 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¨

• dinger 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 ⊂ Rn−1

6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Geometry

Ω+

ε

Ω−

ε

−∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ Rn−1 Ω−

ε = (εS)×(a,0), Ω+ ε = (εS)×(0,b), where ε > 0

6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Geometry

Ω+

ε

Ω−

ε

Dε

✟ ✟ ✙

−∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ Rn−1 Ω−

ε = (εS)×(a,0), Ω+ ε = (εS)×(0,b), where ε > 0

Dε = (dεD)×{0}, where dε ∈ (0,ε]

6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Geometry

Ω+

ε

Ω−

ε

Dε

✟ ✟ ✙

−∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ Rn−1 Ω−

ε = (εS)×(a,0), Ω+ ε = (εS)×(0,b), where ε > 0

Dε = (dεD)×{0}, where dε ∈ (0,ε] Ω = Ω−

ε ∪Dε ∪Ω+ ε

6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Geometry

Ω+

ε

Ω−

ε

Dε

✟ ✟ ✙

−∞ ≤ a < 0 < b ≤ ∞ 0 ∈ D ⊂ S ⊂ Rn−1 Ω−

ε = (εS)×(a,0), Ω+ ε = (εS)×(0,b), where ε > 0

Dε = (dεD)×{0}, where dε ∈ (0,ε] Ω = Ω−

ε ∪Dε ∪Ω+ ε

We assume that the following limit, either finite or infinite, exists: γ = lim

ε→0γε, where γε =

         cap(D) 4|S| · (dε)n−2 εn−1 , n ≥ 3 2π · 1 ε lndε , n = 2

6 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Operators

Operator Aε Let Vε ∈ L∞(Ωε). In L2(Ωε) we introduce the quadratic form aε by aε[u] =

• Ωε
• |∇u|2 +Vε|u|2

dx, dom(aε) = H1(Ωε) By Aε we denote the operator associated with this form. Evidently, Aε = −∆N

Ωε +Vε,

where ∆N

Ωε is the Neumann Laplacian on Ωε.

7 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Operators

Operator A γ Let V ∈ L∞(a,b). In L2(a,b) we introduce the sesquilinear form aγ by γ < ∞ : aγ[u] =

b

a

• |u′|2 +V|u|2

dx + γ |u(+0)−u(−0)|2, dom(aγ) = H1((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 L2(a,b) we introduce the sesquilinear form aγ by γ < ∞ : aγ[u] =

b

a

• |u′|2 +V|u|2

dx + γ |u(+0)−u(−0)|2, dom(aγ) = H1((a,b)\{0}) γ = ∞ : a∞[u] =

b

a

• |u′|2 +V|u|2

dx, dom(aγ) = H1(a,b)

8 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Operators

Operator A γ Let V ∈ L∞(a,b). In L2(a,b) we introduce the sesquilinear form aγ by γ < ∞ : aγ[u] =

b

a

• |u′|2 +V|u|2

dx + γ |u(+0)−u(−0)|2, dom(aγ) = H1((a,b)\{0}) γ = ∞ : a∞[u] =

b

a

• |u′|2 +V|u|2

dx, dom(aγ) = H1(a,b) By A γ we denote the operator associated with this form.

8 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Operators

Operator A γ Let V ∈ L∞(a,b). In L2(a,b) we introduce the sesquilinear form aγ by γ < ∞ : aγ[u] =

b

a

• |u′|2 +V|u|2

dx + γ |u(+0)−u(−0)|2, dom(aγ) = H1((a,b)\{0}) γ = ∞ : a∞[u] =

b

a

• |u′|2 +V|u|2

dx, dom(aγ) = H1(a,b) By A γ we denote the operator associated with this form. One has: (A γu) ↾(a,0)= −(u ↾(a,0))′′, (A γu) ↾(0,b)= −(u ↾(0,b))′′,

8 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Operators

Operator A γ Let V ∈ L∞(a,b). In L2(a,b) we introduce the sesquilinear form aγ by γ < ∞ : aγ[u] =

b

a

• |u′|2 +V|u|2

dx + γ |u(+0)−u(−0)|2, dom(aγ) = H1((a,b)\{0}) γ = ∞ : a∞[u] =

b

a

• |u′|2 +V|u|2

dx, dom(aγ) = H1(a,b) By A γ we denote the operator associated with this form. One has: (A γu) ↾(a,0)= −(u ↾(a,0))′′, (A γu) ↾(0,b)= −(u ↾(0,b))′′, γ < ∞: dom(A γ) =

• u ∈ H2((a,b)\{0}) : u′(a) = u′(b) = 0,

u′(−0) = u′(+0) = γ (u(+0)−u(−0))

• 8 / 20

Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Operators

Operator A γ Let V ∈ L∞(a,b). In L2(a,b) we introduce the sesquilinear form aγ by γ < ∞ : aγ[u] =

b

a

• |u′|2 +V|u|2

dx + γ |u(+0)−u(−0)|2, dom(aγ) = H1((a,b)\{0}) γ = ∞ : a∞[u] =

b

a

• |u′|2 +V|u|2

dx, dom(aγ) = H1(a,b) By A γ we denote the operator associated with this form. One has: (A γu) ↾(a,0)= −(u ↾(a,0))′′, (A γu) ↾(0,b)= −(u ↾(0,b))′′, γ < ∞: dom(A γ) =

• u ∈ H2((a,b)\{0}) : u′(a) = u′(b) = 0,

u′(−0) = u′(+0) = γ (u(+0)−u(−0))

• γ = ∞:

dom(A ∞) =

• u ∈ H2(a,b) : u′(a) = u′(b) = 0
• 8 / 20

Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Identification operators

In the following we assume for simplicity that Vε ≥ 0, V ≥ 0.

9 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Identification operators

In the following we assume for simplicity that Vε ≥ 0, V ≥ 0. Operators Aε and A γ act in different Hilbert spaces. How can one compare their resolvents Rε := (Aε +I)−1 and Rγ := (A γ +I)−1 ?

9 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Identification operators

In the following we assume for simplicity that Vε ≥ 0, V ≥ 0. Operators Aε and A γ act in different Hilbert spaces. How can one compare their resolvents Rε := (Aε +I)−1 and Rγ := (A γ +I)−1 ? Idea Introduce suitable maps Jε : L2(a,b) → L2(Ωε),

• Jε : L2(Ωε) → L2(a,b)

such that JεfL2(Ωε) = fL2(a,b) +o(1), JεuL2(a,b) = uL2(Ωε) +o(1)

9 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Identification operators

In the following we assume for simplicity that Vε ≥ 0, V ≥ 0. Operators Aε and A γ act in different Hilbert spaces. How can one compare their resolvents Rε := (Aε +I)−1 and Rγ := (A γ +I)−1 ? Idea Introduce suitable maps Jε : L2(a,b) → L2(Ωε),

• Jε : L2(Ωε) → L2(a,b)

such that JεfL2(Ωε) = fL2(a,b) +o(1), JεuL2(a,b) = uL2(Ωε) +o(1) and then investigate the asymptotic behaviour of RεJε −JεRγ and

• JεRε −Rγ

Jε

9 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Identification operators

In the following we assume for simplicity that Vε ≥ 0, V ≥ 0. Operators Aε and A γ act in different Hilbert spaces. How can one compare their resolvents Rε := (Aε +I)−1 and Rγ := (A γ +I)−1 ? Idea Introduce suitable maps Jε : L2(a,b) → L2(Ωε),

• Jε : L2(Ωε) → L2(a,b)

such that JεfL2(Ωε) = fL2(a,b) +o(1), JεuL2(a,b) = uL2(Ωε) +o(1) and then investigate the asymptotic behaviour of RεJε −JεRγ and

• JεRε −Rγ

Jε Our choice for the above maps (Jεf)(x,z) = 1

• εn−1|S|

f(z), (x,z) ∈ Rn−1 ×R

9 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Identification operators

In the following we assume for simplicity that Vε ≥ 0, V ≥ 0. Operators Aε and A γ act in different Hilbert spaces. How can one compare their resolvents Rε := (Aε +I)−1 and Rγ := (A γ +I)−1 ? Idea Introduce suitable maps Jε : L2(a,b) → L2(Ωε),

• Jε : L2(Ωε) → L2(a,b)

such that JεfL2(Ωε) = fL2(a,b) +o(1), JεuL2(a,b) = uL2(Ωε) +o(1) and then investigate the asymptotic behaviour of RεJε −JεRγ and

• JεRε −Rγ

Jε Our choice for the above maps (Jεf)(x,z) = 1

• εn−1|S|

f(z), (x,z) ∈ Rn−1 ×R ( Jεu)(z) = 1

• εn−1|S|
• εS u(x,z)dx, z ∈ R

9 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Main result

We assume that Vε converges to V in the following sense as ε → 0: Vε −

• εn−1|S|JεVL∞(Ωε) → 0, ε → 0

10 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Main result

We assume that Vε converges to V in the following sense as ε → 0: Vε −

• εn−1|S|JεVL∞(Ωε) → 0, ε → 0

Theorem [Cardone - K., 2019] One has RεJε −JεRγL2(a,b)→L2(Ωε) ≤ Cδ(ε),

• JεRε −Rγ

Jε

• L2(Ωε)→L2(a,b) ≤ Cδ(ε),

where δ(ε) = Vε −

• εn−1|S|JεVL∞(Ωε) +|γε −γ|+

  

ε1/2|lnε|, γ < ∞, n = 2, ε1/2, γ < ∞, n ≥ 3, ε1/2 +γ−1/2

ε

, γ = ∞,

the constant C depends on n, S, D, γ, supε γε and supε VεL∞(Ωε).

10 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Assume for a moment that Aε and A γ act in the same Hilbert space (we denote it H ) and, moreover, dom(aε) = dom(aγ).

11 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Assume for a moment that Aε and A γ act in the same Hilbert space (we denote it H ) and, moreover, dom(aε) = dom(aγ). Then one has the following well-known result: the estimate Rε −Rγ ≤ δ(ε) holds provided |aε[u,v]−aγ[u,v]| ≤ δ(ε)·Aεu +uH ·A γv +vH .

11 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Assume for a moment that Aε and A γ act in the same Hilbert space (we denote it H ) and, moreover, dom(aε) = dom(aγ). Then one has the following well-known result: the estimate Rε −Rγ ≤ δ(ε) holds provided |aε[u,v]−aγ[u,v]| ≤ δ(ε)·Aεu +uH ·A γv +vH . How to modify this result for operators acting in different spaces?

11 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Theorem [Post, 2006] One has RJε −JεRγ ≤ 6δ(ε), JεRε −Rγ Jε ≤ 6δ(ε) provided

12 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Theorem [Post, 2006] One has RJε −JεRγ ≤ 6δ(ε), JεRε −Rγ Jε ≤ 6δ(ε) provided 1. |(Jεu,v)L2(Ωε) −(u, Jεv)L2(a,b)| ≤ δ(ε)uL2(a,b)vL2(Ωε)

12 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Theorem [Post, 2006] One has RJε −JεRγ ≤ 6δ(ε), JεRε −Rγ Jε ≤ 6δ(ε) provided 1. |(Jεu,v)L2(Ωε) −(u, Jεv)L2(a,b)| ≤ δ(ε)uL2(a,b)vL2(Ωε) and the estimate 2. |aε[J1

ε u,v]−aγ[u,

J1

ε v]| ≤ δ(ε)·A γu +uL2(a,b) ·Aεv +vL2(Ωε).

hold with bounded operators J1

ε : dom(aγ) → dom(aε),

• J1

ε : dom(aε) → dom(aγ)

12 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Theorem [Post, 2006] One has RJε −JεRγ ≤ 6δ(ε), JεRε −Rγ Jε ≤ 6δ(ε) provided 1. |(Jεu,v)L2(Ωε) −(u, Jεv)L2(a,b)| ≤ δ(ε)uL2(a,b)vL2(Ωε) and the estimate 2. |aε[J1

ε u,v]−aγ[u,

J1

ε v]| ≤ δ(ε)·A γu +uL2(a,b) ·Aεv +vL2(Ωε).

hold with bounded operators J1

ε : dom(aγ) → dom(aε),

• J1

ε : dom(aε) → dom(aγ)

satisfying 3a. ( J1

ε −

Jε)uL2(a,b) ≤ δ(ε)

• aε[u]+u2

L2(Ωε),

3b. (J1

ε −Jε)uL2(a,b) ≤ δ(ε)

• aγ[u]+u2

L2(a,b)

12 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Choice of J1

ε and

J1

ε for γ < ∞

• J1

ε : H1(Ωε) → H1((a,b)\{0}),

J1

ε u =

Jεu

13 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Choice of J1

ε and

J1

ε for γ < ∞

• J1

ε : H1(Ωε) → H1((a,b)\{0}),

J1

ε u =

Jεu J1

ε : H1((a,b)\{0}) → H1(Ωε), J1 ε u = Jεu +Gεu

13 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Choice of J1

ε and

J1

ε for γ < ∞

• J1

ε : H1(Ωε) → H1((a,b)\{0}),

J1

ε u =

Jεu J1

ε : H1((a,b)\{0}) → H1(Ωε), J1 ε u = Jεu +Gεu

The correcting term Gεu is localized in a small neighbourhood of the window Dε; for its construction we use a function φε such that ∆φε = 0, φε = 0 on Dε, φε → 0 as |x| → ∞

13 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Choice of J1

ε and

J1

ε for γ < ∞

• J1

ε : H1(Ωε) → H1((a,b)\{0}),

J1

ε u =

Jεu J1

ε : H1((a,b)\{0}) → H1(Ωε), J1 ε u = Jεu +Gεu

The correcting term Gεu is localized in a small neighbourhood of the window Dε; for its construction we use a function φε such that ∆φε = 0, φε = 0 on Dε, φε → 0 as |x| → ∞ (that is, cap(Dε) = ∇φε2

L2(Rn)).

13 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Choice of J1

ε and

J1

ε for γ < ∞

• J1

ε : H1(Ωε) → H1((a,b)\{0}),

J1

ε u =

Jεu J1

ε : H1((a,b)\{0}) → H1(Ωε), J1 ε u = Jεu +Gεu

The correcting term Gεu is localized in a small neighbourhood of the window Dε; for its construction we use a function φε such that ∆φε = 0, φε = 0 on Dε, φε → 0 as |x| → ∞ (that is, cap(Dε) = ∇φε2

L2(Rn)).

Choice of J1

ε and

J1

ε for γ = ∞

J1

ε : H1(a,b) → H1(Ωε), J1 ε u = Jεu

13 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Choice of J1

ε and

J1

ε for γ < ∞

• J1

ε : H1(Ωε) → H1((a,b)\{0}),

J1

ε u =

Jεu J1

ε : H1((a,b)\{0}) → H1(Ωε), J1 ε u = Jεu +Gεu

The correcting term Gεu is localized in a small neighbourhood of the window Dε; for its construction we use a function φε such that ∆φε = 0, φε = 0 on Dε, φε → 0 as |x| → ∞ (that is, cap(Dε) = ∇φε2

L2(Rn)).

Choice of J1

ε and

J1

ε for γ = ∞

J1

ε : H1(a,b) → H1(Ωε), J1 ε u = Jεu

• J1

ε : H1(Ωε) → H1(a,b),

J1

ε u =

Jεu + Hεu

13 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Idea of the proof

Choice of J1

ε and

J1

ε for γ < ∞

• J1

ε : H1(Ωε) → H1((a,b)\{0}),

J1

ε u =

Jεu J1

ε : H1((a,b)\{0}) → H1(Ωε), J1 ε u = Jεu +Gεu

The correcting term Gεu is localized in a small neighbourhood of the window Dε; for its construction we use a function φε such that ∆φε = 0, φε = 0 on Dε, φε → 0 as |x| → ∞ (that is, cap(Dε) = ∇φε2

L2(Rn)).

Choice of J1

ε and

J1

ε for γ = ∞

J1

ε : H1(a,b) → H1(Ωε), J1 ε u = Jεu

• J1

ε : H1(Ωε) → H1(a,b),

J1

ε u =

Jεu + Hεu The correcting term Hεu is localized in a small neighbourhood of zero.

13 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Definition: Hausdorff distance Let X, Y ⊂ R be compact sets. We define distH(X,Y) = max

• sup

x∈X

dist(x,Y); sup

y∈Y

dist(y,X)

• .

14 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Definition: Hausdorff distance Let X, Y ⊂ R be compact sets. We define distH(X,Y) = max

• sup

x∈X

dist(x,Y); sup

y∈Y

dist(y,X)

• .

Theorem [Cardone - K., 2019] distH(σ(Rε),σ(Rγ)) ≤ C1δ(ε)

14 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Definition: Hausdorff distance Let X, Y ⊂ R be compact sets. We define distH(X,Y) = max

• sup

x∈X

dist(x,Y); sup

y∈Y

dist(y,X)

• .

Theorem [Cardone - K., 2019] distH(σ(Rε),σ(Rγ)) ≤ C1δ(ε) To prove the above theorem we use the following abstract result.

14 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Definition: Hausdorff distance Let X, Y ⊂ R be compact sets. We define distH(X,Y) = max

• sup

x∈X

dist(x,Y); sup

y∈Y

dist(y,X)

• .

Theorem [Cardone - K., 2019] distH(σ(Rε),σ(Rγ)) ≤ C1δ(ε) To prove the above theorem we use the following abstract result. Let Hε and H be two Hilbert spaces

14 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Definition: Hausdorff distance Let X, Y ⊂ R be compact sets. We define distH(X,Y) = max

• sup

x∈X

dist(x,Y); sup

y∈Y

dist(y,X)

• .

Theorem [Cardone - K., 2019] distH(σ(Rε),σ(Rγ)) ≤ C1δ(ε) To prove the above theorem we use the following abstract result. Let Hε and H be two Hilbert spaces aε and a be two densely defined non-negative closed forms in Hε and H , respectively

14 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Definition: Hausdorff distance Let X, Y ⊂ R be compact sets. We define distH(X,Y) = max

• sup

x∈X

dist(x,Y); sup

y∈Y

dist(y,X)

• .

Theorem [Cardone - K., 2019] distH(σ(Rε),σ(Rγ)) ≤ C1δ(ε) To prove the above theorem we use the following abstract result. Let Hε and H be two Hilbert spaces aε and a be two densely defined non-negative closed forms in Hε and H , respectively Aε and A be the operators associated with aε and a

14 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Definition: Hausdorff distance Let X, Y ⊂ R be compact sets. We define distH(X,Y) = max

• sup

x∈X

dist(x,Y); sup

y∈Y

dist(y,X)

• .

Theorem [Cardone - K., 2019] distH(σ(Rε),σ(Rγ)) ≤ C1δ(ε) To prove the above theorem we use the following abstract result. Let Hε and H be two Hilbert spaces aε and a be two densely defined non-negative closed forms in Hε and H , respectively Aε and A be the operators associated with aε and a Rε := (Aε +I)−1, R := (A +I)−1

14 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Theorem [Cardone - K., 2019] Let RεJε −JεRH →Hε ≤ αε,

• JεRε −R

Jε

• Hε→H ≤

αε for bounded linear operators Jε : H → Hε, Jε : Hε → H

15 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Theorem [Cardone - K., 2019] Let RεJε −JεRH →Hε ≤ αε,

• JεRε −R

Jε

• Hε→H ≤

αε for bounded linear operators Jε : H → Hε, Jε : Hε → H satisfying f2

H ≤ βεJεf2 Hε +γεa[f,f],

∀f ∈ dom(a), u2

Hε ≤

βε Jεu2

H +

γεaε[u,u], ∀u ∈ dom(aε). Here αε, αε, βε, βε, γε, γε are nonnegative constants.

15 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ′-interaction Spectral convergence

Theorem [Cardone - K., 2019] Let RεJε −JεRH →Hε ≤ αε,

• JεRε −R

Jε

• Hε→H ≤

αε for bounded linear operators Jε : H → Hε, Jε : Hε → H satisfying f2

H ≤ βεJεf2 Hε +γεa[f,f],

∀f ∈ dom(a), u2

Hε ≤

βε Jεu2

H +

γεaε[u,u], ∀u ∈ dom(aε). Here αε, αε, βε, βε, γε, γε are nonnegative constants. Then for any 0 < κ < 1 one has distH (σ(Rε), σ(R)) ≤ max

• γε

κ +γε ; αε βε √ 1−κ ;

• γε

κ + γε ;

• αε

βε √ 1−κ

• .

15 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry 16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry 16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

ε, dε, hε, bε > 0

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0)

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ Bε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

Bε ∼ =

• −dε

2 , dε 2

• ×
• −dε

2 , dε 2

• 16 / 20

Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ Bε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

Bε ∼ =

• −dε

2 , dε 2

• ×
• −dε

2 , dε 2

• Ω = Πε ∪Pε ∪Bε

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ Bε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

Bε ∼ =

• −dε

2 , dε 2

• ×
• −dε

2 , dε 2

• Ω = Πε ∪Pε ∪Bε

Assumptions

1

dε, hε, bε → 0 as ε → 0

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ Bε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

Bε ∼ =

• −dε

2 , dε 2

• ×
• −dε

2 , dε 2

• Ω = Πε ∪Pε ∪Bε

Assumptions

1

dε, hε, bε → 0 as ε → 0

2

∃ lim

ε→0

dε εhε =: γ < ∞

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ Bε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

Bε ∼ =

• −dε

2 , dε 2

• ×
• −dε

2 , dε 2

• Ω = Πε ∪Pε ∪Bε

Assumptions

1

dε, hε, bε → 0 as ε → 0

2

∃ lim

ε→0

dε εhε =: γ < ∞

3

lim

ε→0

ε (bε)2 = 0

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ Bε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

Bε ∼ =

• −dε

2 , dε 2

• ×
• −dε

2 , dε 2

• Ω = Πε ∪Pε ∪Bε

Assumptions

1

dε, hε, bε → 0 as ε → 0

2

∃ lim

ε→0

dε εhε =: γ < ∞

3

lim

ε→0

ε (bε)2 = 0

4

lim

ε→0ε lndε = 0

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Geometry

Πε Pε ✲ Bε ε, dε, hε, bε > 0 −∞ ≤ a < 0 < b ≤ ∞ Πε = (a,b)×(−ε,0) Pε =

• −dε

2 , dε 2

• ×[0,hε]

Bε ∼ =

• −dε

2 , dε 2

• ×
• −dε

2 , dε 2

• Ω = Πε ∪Pε ∪Bε

Assumptions

1

dε, hε, bε → 0 as ε → 0

2

∃ lim

ε→0

dε εhε =: γ < ∞

3

lim

ε→0

ε (bε)2 = 0

4

lim

ε→0ε lndε = 0

Ex.: dε = εα+1, hε = εα, βε = εβ with α > 0, β ∈ (0,1/2)

16 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Operators

Vε ∈ L∞(Ωε) with supp(Vε) ⊂ Πε.

17 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Operators

Vε ∈ L∞(Ωε) with supp(Vε) ⊂ Πε. Vε converges to V ∈ L∞(a,b) in the following sense as ε → 0: Vε −√εJεVL∞(Ωε) → 0, ε → 0.

17 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Operators

Vε ∈ L∞(Ωε) with supp(Vε) ⊂ Πε. Vε converges to V ∈ L∞(a,b) in the following sense as ε → 0: Vε −√εJεVL∞(Ωε) → 0, ε → 0. Aε = −∆N

Ωε +Vε, where ∆N Ωε is the Neumann Laplacian on Ωε.

17 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Operators

Vε ∈ L∞(Ωε) with supp(Vε) ⊂ Πε. Vε converges to V ∈ L∞(a,b) in the following sense as ε → 0: Vε −√εJεVL∞(Ωε) → 0, ε → 0. Aε = −∆N

Ωε +Vε, where ∆N Ωε is the Neumann Laplacian on Ωε.

A γ acts in L2(a,b) and is defined as follows,

(A γu) ↾(a,0)= −(u ↾(a,0))′′, (A γu) ↾(0,b)= −(u ↾(0,b))′′, dom(A γ) =

• u ∈ H2((a,b)\{0}) : u′(a) = u′(b) = 0,

u(−0) = u(+0), u′(+0)−u′(−0) = γu(±0)

• 17 / 20

Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Norm resolvent convergence

Identification operators (Jεf)(x,z) = 1 √ε f(z), (x,z) ∈ R×R

18 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Norm resolvent convergence

Identification operators (Jεf)(x,z) = 1 √ε f(z), (x,z) ∈ R×R ( Jεu)(z) = 1 √ε

−ε u(x,z)dx, z ∈ R

18 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Norm resolvent convergence

Identification operators (Jεf)(x,z) = 1 √ε f(z), (x,z) ∈ R×R ( Jεu)(z) = 1 √ε

−ε u(x,z)dx, z ∈ R

Theorem [Post - K., 2020] One has

• (Aε +I)−1Jε −Jε(A γ +I)−1
• L2(a,b)→L2(Ωε) ≤ δ(ε),
• Jε(Aε +I)−1 −(A γ +I)−1

Jε

• L2(Ωε)→L2(a,b) ≤ δ(ε),

where δ(ε) = C ·max

• |γε −γ|,
• ε lndε,
• ε

(bε)2 , hε, bε,Vε −√εJεVL∞(Ωε)

• .

18 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Spectral convergence

We denote Sγ = σ(Rγ)∪{1} Theorem [K. - Post, 2020] distH(σ(Rε),Sγ) ≤ C1δ(ε).

19 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions

Approximation of δ-interaction Spectral convergence

Thank you for your attention!

20 / 20 Andrii Khrabustovskyi Geometric approximations of point interactions