Limit-point / limit-circle classification of second-order differential operators and PT -QM

Transcription

1 Limit-point / limit-circle classification of second-order differential operators and PT -QM TU Ilmenau 19th December 2016 TU Ilmenau Seite 1 / 24

2 TU Ilmenau Seite 2 / 24

3 PT -symmetry Definition (Bender Boettcher 1998) Closed, densely defined operator H on L 2 (R) is PT -symmetric, if for all f D(H) PT f D(H) and PT Hf = HPT f. Parity P: Pf (x) = f ( x) Time reversal T : T f (x) = f (x) TU Ilmenau Seite 3 / 24

4 PT -symmetric Hamiltonian Hamiltonian H := p 2 + z 2 (iz) ɛ, z Γ, ɛ R Eigenvalue problem Hφ = λφ Schrödinger eigenvalue differential expression φ (z) + z 2 (iz) ɛ φ(z) = λφ(z), z Γ Here: ɛ = N N: Schrödinger eigenvalue differential expression φ (z) (iz) N+2 φ(z) = λφ(z), z Γ TU Ilmenau Seite 4 / 24

5 Back to the real line Today Γ := {xe iφsgnx : x R}: Im 4 Γ Re TU Ilmenau Seite 5 / 24

6 Back to the real line Parameterize Γ: z(x) := xe iφsgnx, w(x) := y(z(x)), dom A ± := { w, A ± w L 2 (R ± ) : w, w AC(R ± ), w(0±) = 0 } and A + w(x) := e 2iφ w (x) e (N+2)iφ (ix) N+2 w(x) = λw(x), x 0, A w(x) := e 2iφ w (x) e (N+2)iφ (ix) N+2 w(x) = λw(x), x 0. Sturm-Liouville Problem A ± w(x) := p ± (x)w (x) + q ± (x)w(x) = λw(x), x R, with p ± (x) = e 2iφ and q ± (x) = e ±(N+2)iφ (ix) N+2. TU Ilmenau Seite 6 / 24

7 Plan Consider operator with boundary conditions on the semi axis Study operator with matching conditions on R Spectrum TU Ilmenau Seite 7 / 24

8 Recall: Complex Sturm-Liouville Problems with τ[y](x) := ( p(x)y (x) ) + q(x)y(x) = λy(x) on [0, ) p : [0, ) C with 1/p L 1 loc [0, ), p(x) 0 potential function q : [0, ) C with q L 1 loc [0, ) initial values: y(0) = x 0,1, p(0)y (0) = x 0,2 TU Ilmenau Seite 8 / 24

9 Some definitions: [Brown et. al 99] Q = clconv {rp(x) + q(x) : x [0, ), 0 < r < }, Q C For λ C\Q define K such that K λ = min{ q λ }. q Q Λ K,η L Q K λ Let L be the tangent on Q in K. Translate z z K and rotate L with η ( π, π] on imaginary axis. Define Λ K,η = {λ C : Re(λ K)e iη < 0}. Here: N = 3, φ = π/10, p = e iπ/5 and q(x) = x 5. TU Ilmenau Seite 9 / 24

10 τ[y] = λy, λ C (1) 0 Re [ e iη ( p y 2 + (q K) y 2)] dx < y L 2 (0, ) ( ) ( ) Theorem (Brown et. al 99) Exactly one of the following holds I There exists a unique solution of (1) satisfying ( ) and this is the only solution satisfying ( ) (LPC). II There exists a unique solution of (1) satisfying ( ) but all solutions satisfy ( ) (LPC). III All solutions of (1) satisfy ( ) and ( ) (LCC). TU Ilmenau Seite 10 / 24

11 Operator realisation Let ψ be the solution of τ[y] = λy satisfying ( ) and ( ). Definition In I set D(τ)=D 1 := { u : u, pu AC(0, ), u, τ[u] L 2 (0, ), u(0)=0 } In II and III set { } D(τ)= u D 1 : lim b p(b)ψ (b)u(b) p(b)u (b)ψ(b)=0 TU Ilmenau Seite 11 / 24

12 Spectrum Theorem (Brown et. al 99) In I σ(τ) Q (with only isolated eigenvalues of finite algebraic multiplicity in Q). In II and III σ(τ) consists in C\Λ K,η only of isolated eigenvalues with finite algebraic multiplicity. Q = clconv {rp(x) + q(x) : x [0, ), 0 < r < } Λ K,η = {λ C : Re(λ K)e iη < 0} TU Ilmenau Seite 12 / 24

13 Back to PT -symmetric Problem (Bender Boettcher): φ (z) (iz) N+2 φ(z) = λφ(z), z Γ TU Ilmenau Seite 13 / 24

14 A + w(x) := e 2iφ w (x) e (N+2)iφ (ix) N+2 w(x) = λw(x), x 0 A w(x) := e 2iφ w (x) e (N+2)iφ (ix) N+2 w(x) = λw(x), x 0 with dom A ± = { w, A ± w L 2 (R ± ) : w, w AC(R ± ), w(0±) = 0 } TU Ilmenau Seite 14 / 24

15 Limit-point/limit-circle classification of A + Linear independent solutions y ± for A + w(x) = λw(x). Then we have ( x ) y ± [e (N+4)iφ s(x)] 1/4 exp ± Re s(t) 1/2 dt, 0 where s(t) := e (N+4)iφ (ix) (N+2) e 2iφ λ. If φ 4k N 2 2N+8 π, then LPC. (Re s(t)1/2 0, Case I) If φ = 4k N 2 2N+8 π, then LCC. (Re s(t)1/2 = 0, Case III) Γ on Stokes-wedges = LPC and Γ on Stokes-lines = LCC TU Ilmenau Seite 15 / 24

16 Stokes-Wedges for N = 3: 3 Im Re TU Ilmenau Seite 16 / 24

17 e 2iφ w (x) ie 5iφ x 5 w(x)=λw(x), x 0 with φ=π/10, e.g. Case I: Im Re 1 2 Q 3 Q = clconv { re iπ/5 + x 5 : x [0, ), 0 < r < } TU Ilmenau Seite 17 / 24

18 Full line operator A: Aw := { e 2iφ w (x) (ix) N+2 e (N+2)iφ w(x), x > 0 e 2iφ w (x) (ix) N+2 e (N+2)iφ w(x), x < 0 with domain dom A := w, Aw L2 (R) : Lemma y is continous if and only if α = e 2iφ. A is PT -symmetric if and only if α = 1. w R ±, w R ± AC(R ± ), w(0+) = w(0 ), w (0+) = αw (0 ) A is [, ]-selfadjoint if and only if α = e 4iφ. ([, ] := (P, ) Krein space inner produkt in L 2 (R)) TU Ilmenau Seite 18 / 24

19 Full line operator A: Essential spectrum Aw := { e 2iφ w (x) (ix) N+2 e (N+2)iφ w(x), x > 0 e 2iφ w (x) (ix) N+2 e (N+2)iφ w(x), x < 0 Theorem ρ(a). σ ess (A) =. σ(a) consists only of isolated eigenvalues acc. to with dim ker (A λ) = 1 (this is due to limit point). TU Ilmenau Seite 19 / 24

20 Eigenvalues of the full line operator A Let w ± be a solution A λ = 0 restricted to x > 0 (resp. x < 0). Lemma Then w ± are unique up to a constant due to limit point case (I) and λ σ p (A) w +(0) w + (0) = αw (0) w (0), Proof. w(x) := { w + (x), x > 0 w +(0) w (0) w (x), x < 0 TU Ilmenau Seite 20 / 24

21 Main result: Spectrum of A Theorem Let α = 1. Then we know: A is PT -symmetric and σ ess (A) =. σ(a) = eigenvalues in a circle and two small sectors acc. to. E.g., if N = 3 and φ = π 10. Then σ(a) is in a neighbourhood of σ(a) {x : x > M} K M (0) K M (0) If N = 4 and φ = π 7. Then σ(a) is in a neighbourhood of {xe ± 6πi 7 : x > M} K M (0) σ(a) K M (0) TU Ilmenau Seite 21 / 24

22 How to show this? Idea of proof: We use WKB-asymptotics: For a solution w of we have w(x) = p(x) 1/4 exp w (x) = (p(x) + q(x)) w(x) ( x ) ± p(y) 1/2 dy (1 + R(x)) 0 with ( x ) R(x) exp E(y) dy 1. E contains p, q. In our case p =const and q contains λ. Observe: We can let λ and get an estimate for w(0). Apply to λ σ p (A) w +(0) w + (0) = αw (0) w (0), TU Ilmenau Seite 22 / 24

23 Literature T.Azizov, C.Trunk, On the limit point and limit circle classification for PT symmetric operators T.Azizov, C.Trunk, On domains of PT symmetric operators related to y (x) + ( 1) n x 2n y(x), J. Phys. A, Math. Theor. 43 (2010), C.Bender, S.Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80 (1998), B.Brown et.al., On the spectrum of second-order differential operators with complex coefficients, Proc. R. Soc. Lond. 455 (1999), A.R.Sims, Secondary Conditions for Linear Differential Operators of the Second Order, J. of Math. and Mech., Vol. 6 No. 2 (1957), F.W.J.Olver, Asymptotics and special functions, Academic Press New York, 1974 TU Ilmenau Seite 23 / 24

24 Thank You! TU Ilmenau Seite 24 / 24