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A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

Anton A. Boitsev, I. Yu. Popov ITMO University 20.12.19 Wien

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Introduction

Consider R3, x = (x1, x2, x3) ∈ R3 and two regions Ωin = {x ∈ R3 : x3 > 0}, Ωex = {x ∈ R3 : x3 < 0}. In other words, Ωex = R3 \ Ωin, Ω = Ωin ⊕ Ωex. The boundary for Ω is ∂Ω = {x ∈ R3 : x3 = 0}.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Introduction

Consider Laplace operators in L2(Ωex) and L2(Ωin) with Neumann boundary conditions on ∂Ω that is ∆f ≡ 0 in Ω and ∂f ∂n

• ∂Ω

= 0, n is the external unit normal vector to ∂Ω. Restrict the operators onto the set of smooth functions that vanish in the neighbourhood of x0 ∈ ∂Ω.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Introduction

Theorem The closures ∆in

0 and ∆ex

• f the considered operators are

symmetric operators with deficiency indices (1, 1). Thus, the operator ∆0 = ∆in

0 ⊕ ∆ex

is symmetric and has deficiency indices (2, 2).

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Introduction

But what if we put Dirichlet conditions? ∆f ≡ 0 in Ω and f|∂Ω = 0. The operator obtained via restriction onto the set of smooth functions that vanish in the neighbourhood of x0 ∈ ∂Ω is sufficiently self-adjoint.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Literature

• I. Yu. Popov, The Helmholtz resonator and the theory of
• perator extensions in a space with indefinite metric,

Russian Acad. Sci. Sb. Math. Vol. 75 (1993), No. 2.

• J. F. van Diejen and A. Tip, Scattering from generalized

point interactions using selfadjoint extensions in Pontryagin spaces, Journal of Mathematical Physics 32, 630 (1991).

• J. Behrndt, A. Luger, C. Trunk, On the negative squares of

a class of self-adjoint extensions in Krein spaces, Mathematische Nachrichten 286(2-3):118-148.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Preliminaries

Add the derivative of the Green function Gin(x, x0, k) and Gex(x, x0, k) of the Dirichlet Laplacians ∆in and ∆ex in Ωin and Ωex. Ωin = {x ∈ R3 : x3 > 0}, Ωex = {x ∈ R3 : x3 < 0}, Gin,ex(x, x0, k) = 1 4π · eik|x−x0| |x − x0| . Ain,ex =   f : f ∈ L2(Ωin,ex) :

• Ωin,ex

fdx |x − x0|3 converges    .

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Preliminaries

Consider λ0 = k2

0 – a regular point for the operator −∆in,ex and

hin

−1(x) = ∂

∂nGin(x, x0, k0)

• x0=0

= −ik0r cos θ − cos θ 4πr2 eik0r, hin

1 (x) =

• −∆in

0 − λ0

−1 hin

−1(x).

The index of the function h shows the type of the behaviour of the function in the neighbourhood of x0 = 0, i.e. for hin

−1 it is

1/r and for hin

1 it is r.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Pre-Pontryagin space

• Ain =
• f : f = fin + Cin

1 hin 1 + Cin −1hin −1

• ,

fin ∈ Ain. The scalar product is (f, g)

Ain = (fin, gin)L2 +

• Ωin

finCg,in

1

hin

1 dx +

• Ωin

finCg,in

−1 hin −1dx+

+

• Ωin

Cf,in

1

hin

1 gindx+

• Ωin

Cf,in

−1 hin −1gindx+Cf,in 1

Cg,in

−1

• Ωin

hin

1 hin −1dx+

+Cf,in

−1 Cg,in 1

• Ωin

hin

−1hin 1 dx + Cf,in 1

Cg,in

1

• Ωin

hin

1 hin 1 dx.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Laplace operator construction

The domain of the operator ∆in

0 is as follows:

dom ∆in

0 = {f : f ∈

Ain, fin ∈ W 2,loc

2

(Ωin), fin = fin

+1 + Chin 1 },

where fin

+1 is such a function that (−∆in 0 − λ0)fin +1 ∈ Ain.

Theorem The operator −∆in

0 is self-adjoint in Π1.

How do we define the action of ∆in

0 on hin −1?

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

The definition on hin

−1

On the domain dom ∆in

0 we consider the functional χ as

(f, χ) =

• (−∆in

0 − λ0)f, hin −1

• .

We restrict the operator ∆in

0 on the set of functions

f ∈ dom ∆in

0 so that

(f, χ) = 0 and obtain the operator ∆in. Theorem The operator −∆in is symmetric.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Final operator

Let now ∆0 = ∆in

0 ⊕ ∆ex 0 .

By the described procedure, we can obtain symmetric operator ∆ = ∆in ⊕ ∆ex with the deficiency indices (2, 2).

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

The boundary form

((−∆∗ − λ0)f, ϕ) − (f, (−∆∗ − λ0)ϕ) = = 1 2

• Cf,in

1

Cϕ,in

−1

− Cf,in

−1 Cϕ,in 1

+ Cf,ex

1

Cϕ,ex

−1

− Cf,ex

−1 Cϕ,ex 1

• .

A self-adjoint extension can be constructed via

• Cin

1 = Cex 1

Cin

−1 = −Cex −1.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Green function

G0(x, y, k) =

• Gex(x, y, k)
• −
• Gex(x, y∗, k)
• +

a0,in ∂Gin(x,x0,k)

∂nx0

in

• + a0,ex
• ∂Gex(x,x0,k)

∂nx0

ex

• where nx0

in,ex is the external normal for ∂Ωin,ex at point x0,

y = (y1, y2, y3), y∗ = (y1, y2, −y3). To construct the Green function, we should determine Cin,ex

−1,x0,

Cin,ex

1,xi .

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Green function

G0(x, y, k) =

• Gex(x, y, k)
• −
• Gex(x, y∗, k)
• +

a0,in ∂Gin(x,x0,k0)

∂nx0

in

• + a0,ex
• ∂Gex(x,x0,k0)

∂nx0

ex

• +

a0,in

• ∂

∂nx0

in

• Gin(x, x0, k) − Gin(x, x0, k0)
• +

a0,ex

• ∂

∂nx0

ex

• Gex(x, x0, k) − Gex(x, x0, k0)
• .

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Determining the coefficients

Cin

−1,x0 = a0,in, Cex −1,x0 = a0,ex.

Cin

1,x0 = a0,in

∂ ∂nx

in

• ∂

∂nx0

in

• Gin(x, x0, k) − Gin(x, x0, k0)
• x=x0

, Cex

1,x0 = ∂Gex(x, x0, k)

∂nx0

ex

+ a0,ex ∂ ∂nx

ex

• ∂

∂nx0

ex

• Gex(x, x0, k) − Gex(x, x0, k0)
• x=x0

.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Coefficients

a0,ex = − c(x0) bex(x0) + bin(x0), a0,in = c(x0) bex(x0) + bin(x0), where bin,ex(x0) = = ∂ ∂nx

in,ex

• ∂

∂nx0

in,ex

• Gin,ex(x, x0, k) − Gin,ex(x, x0, k0)
• x=x0

, c(x0) = ∂Gex(x, x0, k) ∂nx0

ex

. Equating the denominator to zero we get that k0 = k – the eigenvalue.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

The case of two points x0 and x1

In the case of 2 points x0 and x1 the construction is very similar and after calculations we get the following equation on resonances ±i(k3 − k3

0) = 3eik|x0−x1| ·

ik − 1 |x0 − x1|3 , |x0 − x1| − → 0. It is possible to choose the model parameter k0 in such a way that the resonance for two close point-like windows would be close to the eigenvalue for one window with the model parameter k0/2 instead of k0: 3eik0L/2 · ik0/2 − 1 L3 + 9k3

0/8 = 0.

The considered normalization gives one window which is twice wider than each of the two close windows.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

The case of two points x0 and x1

When comparing the Green function for the point-like opening and for small window of radius a, one can find the value of the model parameter ensuring the coincidence of the main term of the asymptotics in a of ”realistic” solution and the model one: k0 = 2i/a. Correspondingly, the described regularization gives

• ne window which is twice wider than each of the two close

windows.

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov

Thank you for your attention!

A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition

• A. A. Boitsev, I. Yu. Popov