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 R 3 , x = ( x 1 , x 2 , x 3 ) ∈ R 3 and two regions Ω in = { x ∈ R 3 : x 3 > 0 } , Ω ex = { x ∈ R 3 : x 3 < 0 } . In other words, Ω ex = R 3 \ Ω in , Ω = Ω in ⊕ Ω ex . The boundary for Ω is ∂ Ω = { x ∈ R 3 : x 3 = 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 L 2 ( Ω ex ) and L 2 ( Ω in ) with Neumann boundary conditions on ∂ Ω that is � � ∆ f ≡ 0 in Ω and ∂ f � = 0 , � ∂ n ∂ Ω n is the external unit normal vector to ∂ Ω . Restrict the operators onto the set of smooth functions that vanish in the neighbourhood of x 0 ∈ ∂ 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 of the considered operators are 0 symmetric operators with deficiency indices (1 , 1) . Thus, the operator ∆ 0 = ∆ in 0 ⊕ ∆ ex 0 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 x 0 ∈ ∂ Ω is su ffi ciently 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 operator 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 G in ( x, x 0 , k ) and G ex ( x, x 0 , k ) of the Dirichlet Laplacians ∆ in and ∆ ex in Ω in and Ω ex . Ω in = { x ∈ R 3 : x 3 > 0 } , Ω ex = { x ∈ R 3 : x 3 < 0 } , 4 π · e ik | x − x 0 | G in,ex ( x, x 0 , k ) = 1 | x − x 0 | . � fdx A in,ex = f : f ∈ L 2 ( Ω in,ex ) : | x − x 0 | 3 converges . Ω in,ex A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition A. A. Boitsev, I. Yu. Popov
Preliminaries 0 – a regular point for the operator − ∆ in,ex and Consider λ 0 = k 2 � � − 1 ( x ) = ∂ = − ik 0 r cos θ − cos θ h in ∂ nG in ( x, x 0 , k 0 ) � e ik 0 r , � 4 π r 2 x 0 =0 � � − 1 h in h in − ∆ in 1 ( x ) = 0 − λ 0 − 1 ( x ) . The index of the function h shows the type of the behaviour of the function in the neighbourhood of x 0 = 0, i.e. for h in − 1 it is 1 /r and for h in 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 � � � f : f = f in + C in f in ∈ A in . A in = 1 h in 1 + C in − 1 h in , − 1 The scalar product is � � f in C g,in f in C g,in A in = ( f in , g in ) L 2 + h in − 1 h in ( f, g ) � 1 dx + − 1 dx + 1 Ω in Ω in � � � C f,in C f,in − 1 g in dx + C f,in C g,in h in − 1 h in h in 1 h in 1 g in dx + + − 1 dx + 1 1 − 1 Ω in Ω in Ω in � � + C f,in − 1 C g,in 1 dx + C f,in C g,in h in h in − 1 h in 1 h in 1 dx. 1 1 1 Ω in Ω in 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: 0 = { f : f ∈ � A in , f in ∈ W 2 ,loc ( Ω in ) , f in = f in dom ∆ in +1 + Ch in 1 } , 2 where f in +1 is such a function that ( − ∆ in 0 − λ 0 ) f in +1 ∈ A in . Theorem The operator − ∆ in 0 is self-adjoint in Π 1 . How do we define the action of ∆ in 0 on h in − 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 h in − 1 On the domain dom ∆ in 0 we consider the functional χ as � � ( − ∆ in 0 − λ 0 ) f, h in ( f, χ ) = . − 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 (( − ∆ ∗ − λ 0 ) f, ϕ ) − ( f, ( − ∆ ∗ − λ 0 ) ϕ ) = � � = 1 C f,in C ϕ ,in − C f,in − 1 C ϕ ,in + C f,ex C ϕ ,ex − C f,ex − 1 C ϕ ,ex . 1 − 1 1 1 − 1 1 2 A self-adjoint extension can be constructed via � C in 1 = C ex 1 C in − 1 = − C ex − 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 � � � � 0 0 G 0 ( x, y, k ) = − + G ex ( x, y, k ) G ex ( x, y ∗ , k ) � ∂ G in ( x,x 0 ,k ) � � � 0 a 0 ,in ∂ n x 0 + a 0 ,ex ∂ G ex ( x,x 0 ,k ) in 0 ∂ n x 0 ex in,ex is the external normal for ∂ Ω in,ex at point x 0 , where n x 0 y = ( y 1 , y 2 , y 3 ) , y ∗ = ( y 1 , y 2 , − y 3 ). To construct the Green function, we should determine C in,ex − 1 ,x 0 , C in,ex 1 ,x i . A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition A. A. Boitsev, I. Yu. Popov
Green function � � � � 0 0 G 0 ( x, y, k ) = − + G ex ( x, y ∗ , k ) G ex ( x, y, k ) � ∂ G in ( x,x 0 ,k 0 ) � � � 0 a 0 ,in ∂ n x 0 + a 0 ,ex + ∂ G ex ( x,x 0 ,k 0 ) in 0 ∂ n x 0 ex � � � � ∂ G in ( x, x 0 , k ) − G in ( x, x 0 , k 0 ) ∂ n x 0 a 0 ,in + in 0 � � 0 � � a 0 ,ex . ∂ G ex ( x, x 0 , k ) − G ex ( x, x 0 , k 0 ) ∂ n x 0 ex A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition A. A. Boitsev, I. Yu. Popov
Determining the coe ffi cients C in − 1 ,x 0 = a 0 ,in , C ex − 1 ,x 0 = a 0 ,ex . � �� � � � ∂ ∂ � C in 1 ,x 0 = a 0 ,in G in ( x, x 0 , k ) − G in ( x, x 0 , k 0 ) , � ∂ n x ∂ n x 0 � in in x = x 0 1 ,x 0 = ∂ G ex ( x, x 0 , k ) C ex + ∂ n x 0 ex ��� � � � ∂ ∂ � a 0 ,ex G ex ( x, x 0 , k ) − G ex ( x, x 0 , k 0 ) . � ∂ n x ∂ n x 0 ex ex x = x 0 A model of several point-like windows in the resonator boundary with the Dirichlet boundary condition A. A. Boitsev, I. Yu. Popov
Coe ffi cients c ( x 0 ) a 0 ,ex = − b ex ( x 0 ) + b in ( x 0 ) , c ( x 0 ) a 0 ,in = b ex ( x 0 ) + b in ( x 0 ) , where b in,ex ( x 0 ) = � �� � � � ∂ ∂ � G in,ex ( x, x 0 , k ) − G in,ex ( x, x 0 , k 0 ) = , � ∂ n x ∂ n x 0 � in,ex in,ex x = x 0 c ( x 0 ) = ∂ G ex ( x, x 0 , k ) . ∂ n x 0 ex Equating the denominator to zero we get that k 0 = 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 x 0 and x 1 In the case of 2 points x 0 and x 1 the construction is very similar and after calculations we get the following equation on resonances ik − 1 ± i ( k 3 − k 3 0 ) = 3 e ik | x 0 − x 1 | · | x 0 − x 1 | − | x 0 − x 1 | 3 , → 0 . It is possible to choose the model parameter k 0 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 k 0 / 2 instead of k 0 : 3 e ik 0 L/ 2 · ik 0 / 2 − 1 + 9 k 3 0 / 8 = 0 . L 3 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
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