Singular differential operators: Titchmarsh-Weyl coefficients and Self-adjoint operators December, 2006, TU Berlin

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1 Singular differential operators: Titchmarsh-Weyl coefficients and Self-adjoint operators December, 2006, TU Berlin Pavel Kurasov Lund-St. Petersburg-Stockholm in collaboration with A.Luger

2 Singular differential operator l(y) = d2 dx 2y + ( q0 x 2 + q ) y x Describes a point particle in the presence of a Coulomb field. Has been studied for several centuries. Our aim: Explore connection to the theory of generalized Nevanlinna functions.

3 Scattered waves in R 3 l(ψ) = ψ + V (x)ψ, acting in L 2 (R 3 ) V (x) = γ + Ṽ (x), radial potential Ṽ (x) = Ṽ ( x ) x Eigenfunction equation: l(ψ) = λψ. () Bound states (discrete spectrum) λ j ψ j are L 2 (R 3 ) solutions to () Scattered waves (abs. continuous spectrum) 0 λ ψ(λ) - generalized solutions to () ψ(λ, x) x ν eik ν, x a(λ, ν, ν ) eik x 4π x, k2 = λ, ν = ν = Spherical harmonics reduction to the set of one dimensional operators l n (ψ) = d2 dx 2ψ + V (r)ψ acting in L 2(R + ) with V (r) = γ r + n(n+) r 2 + Ṽ (r)

4 Regular Schrödinger operator: selected extension l(ψ) = d2 dr2ψ + V (r)ψ, r [0, ) - limit circle case at r = 0 and limit point case at r = 0. L min = l(c0 (,). ) is a symmetric operator with deficiency indices Consider L 0 - one selected extension from the one-dimensional family.

5 Special solutions to the eigenfunction equation l(ψ) = λψ (2) g ± (λ) solutions which satisfy/do not satisfy the boundary condition at the origin (regular/singular at the origin); g(λ),im λ > 0 - solution from L 2 (R + ) (regular at ) (the deficiency element for L 0, L min ) Titchmarsh-Weyl coefficient m(λ) (Nevanlinna-type function) is determined by the following expansion g(λ) = g (λ) m(λ)g + (λ). Scattered waves (absolutely continuous spectrum) 0 λ ψ(λ) = g + (λ), where g + (λ) is the unique (generalized) solution to (2) satisfying the boundary condition at r = 0.

6 Regular Schrödinger operator: One-parameter family of extensions Two possibilities to describe all self-adjoint extensions of L min by boundary conditions at the origin L θ - the extension of L min to the set of W2 2 -functions satisfying the boundary condition u (0) = cot θ u(0), θ [0, π); (3) as H 2 -perturbations of the operator L 0 L α = L 0 + α ϕ, ϕ, α R ϕ = (L 0 λ)g(t)- distribution with support at the origin

7 Krein s formula L θ λ = L 0 λ g(λ), g(λ) m(λ) + cot θ Bound states (discrete spectrum) λ j = λ j (θ) ψ θ j = g(λ j(θ)), where the points λ j (θ) are selected by the condition that g(λ j (θ)) satisfies the boundary condition (3). Scattered waves (absolutely continuous spectrum) 0 λ ψ θ (λ) = g + (λ) a(λ)g(λ + i0) The scattering amplitude a(λ) is calculated from the requirement that ψ θ (λ) satisfies b.c. (3) a(λ) = m(λ) + cot θ

8 Singular Schrödinger operator Singular differential expression ( l(y) := y q0 (r) + r 2 + q ) y(r) on x (0, ) r q 0, q R, q 0 > 3 4. Classical differential expression. Recent interest: A. Dijskma, Yu. Shondin, C. Fulton, H. Langer.

9 We need the following results: The operator L := L min is essentially self-adjoint and semibounded unique self-adjoint operator in L 2 (R + ). There exists one solution g(λ) regular at g(λ, ) L 2 (r 0, ),for any r 0. There exists singular (g ) and regular (g + ) solutions to the equation l(g) = λg possessing the power series expansion g ± (λ, r) = r α± n=0 a ± n(λ)r n where α ± = 2 ± 4 + q 0, g + - regular solution, g - singular. There exists a generalized Nevanlinna function m(λ) N κ, κ = κ(q 0 ) such that g(λ) = g (λ) m(λ)g + (λ).

10 Our results: The function g(λ) belongs to the space H n+2 (L) \ H n+3 (L) from the scale of spaces [ associated with the self-adjoint operator L in L 2 (R + ), q0 ] where n = /4. Hence the element ϕ = (L λ)g(λ) belongs to H n (L) \ H n+ (L) - distribution supported by the origin. All point perturbations at the origin of the self-adjoint operator L can be described as supersingular rank one perturbations by L α = L + α ϕ, ϕ. apply the theory of supersingular perturbations developed by P. K., K.Watanabe, A. Dijksma, and Yu.Shondin. Choose normalization points µ j R ρ(l), j =,2,..., n and form a cascade of less and less singular elements g j = j i= L µ i ϕ H n+2j (L).

11 Cascade model Supersingular perturbation L α = L + α ϕ, ϕ, ϕ H n (L) \ H 2 (L). Two at the first glance contradicting requirements: The restriction L {ψ dom(l): ϕ,ψ =0} is not essentially self-adjoint only if it is defined in the Hilbert space H n 2 where L has domain H n (L); The formal resolvent of the operator L α contains elements of the form (L µ) ϕ H n+2 (L) \ H n+3 (L). In order to meet both requirements it was suggested to define L α on the functions possessing the representation U = n 2 j= u j g j + U, U H n 2 (L).

12 Extended Hilbert space H = C n 2 H n 2 with the scalar product U,V H = u,γ v C n 2 + U, b n 2 (L)V, where Γ is a certain Gram matrix (to be specified later); b n 2 (x) = (x µ )(x µ 2 )...(x µ n 2 ). Natural embedding ρu = n 2 j= u j g j + U H n+2 (L), - every element U from H can be viewed as an element from H n+2 (L).

13 The maximal operator L max in H - acts on functions ρu as the original differential expression. Uniquely determined by one of the two conditions ρl max modϕ = Lρ or L max = ρ L max ρ, where L max is the triplet adjoint to L min acting in H n 2 (L). Domain Dom(L max ) = {U = ( u, U) : U = u n g n + U r, U r H n (L)} Action L max ( u U ) = where e n 2 = (0,...,0,) and M = ( M u + u n e n 2 LU r + µ n u n g n µ µ µ µ n 2 ),.

14 The Gram matrix Γ is chosen so that M is Hermitian in C n 2 ΓM M Γ = 0. The self-adjoint restrictions of L max can be defined using the generalized boundary condition u n + cot θ ( ϕ, U r e n 2,Γ u C n 2) = 0, θ [0, π). (4) Theorem The operator L θ is self-adjoint in H. Selected extension L 0 = M L.

15 Krein s formula for L θ where L θ λ F = L 0 λ F Q(λ) + cot θ Φ(λ) = ( M λ e n 2, L λ g n 2) - the deficiency element, ρφ(λ) = b n 2 (λ) L λ ϕ; Q(λ) = Q L (λ) + Q M (λ) = g n 2, b n 2 (L) λ µ n (L λ)(l µ n ) g n 2 + Φ(λ),F e n 2,Γ M λ e n 2 C n 2 = (λ µ n ) Φ(µ ),Φ(λ) H + e n 2,Γ M µ e n 2 C n 2 H Φ(λ),

16 Generalized Krein s formula l ρ L θ λ H n 2 = L λ b n 2 (λ)(q(λ) + cot θ) L λ ϕ, L λ ϕ. Theorem The (generalized) Titchmarsch-Weyl coefficient m(λ) and the Q-function D(λ) = b n 2 (λ)(q(λ) + cot θ) differ by a polynomial of degree n 2.

17 Eigenfunction expansion in H (standard) Bound states (discrete spectrum) λ j : Φ(λ j )satisfies the boundary condition (4) Corresponding eigenfunctions are Ψ j = Φ(λ j ) Scattered waves (absolutely continuous spectrum) Ψ(λ) = g + (λ) a(λ)φ(λ + i0) Boundary condition a(λ) = (Q(λ + i0) + cot θ). Ψ(λ) = g + (λ) Φ(λ + i0) Q(λ + i0) + cot θ F = N bs j= Q (λ j ) Ψj,F H Ψ j+ π 0 Im m(λ + i0) b n 2 (λ) Ψ(λ),F H Ψ(λ)dλ L θ F = N bs j= λ j Q (λ j ) Ψj,F H Ψ j+ π 0 λim m(λ + i0) b n 2 (λ) Ψ(λ),F H Ψ(λ)dλ

18 Modified eigenfunction expansion Use the standard spectral representation with F = (0, F), F [0, ) and remember that ρf = F. C 0 F(x) = N bs D (λ j ) j= + π 0 0 g(x, λ j)g(y, λ j )F(y)dy ( Im m(λ + i0) g + (x, λ) g(x, λ + i0) D(λ + i0) ( ) g + (y, λ) g(y, λ i0) F(y)dydλ. 0 D(λ i0) Here the functions g(λ j ) and g + (λ) g(λ + i0) play the D(λ+i0) role of the discrete and continuous spectrum eigenfunctions respectively. )

19 Spectral representation l(f)(x) = N bs λ j D (λ j ) j= + π 0 0 g(x, λ j)g(y, λ j )F(y)dy ( λim m(λ + i0) g + (x, λ) g(x, λ + i0) D(λ + i0) ( ) g + (y, λ) g(y, λ i0) F(y)dydλ. 0 D(λ i0) )