Surprising spectra of PT -symmetric point interactions

Transcription

1 Surprising spectra of PT -symmetric point interactions Petr Siegl Nuclear Physics Institute, Řež Faculty of Nuclear Sciences and Physical Engineering, Prague Laboratoire Astroparticules et Cosmologie, Université Paris 7, Paris

2 PT -symmetry Origins of PT -symmetry Hamiltonian d2 dx + ix 3 has real, positive, discrete spectrum 2 Bender, Boettcher 1998 original hypothesis - the reality of spectrum due to PT -symmetry [PT, H] = 0 parity P, (Pψ)(x) = ψ( x) complex conjugation T, (T ψ)(x) = ψ(x) PT -symmetry is not sufficient for reality of the spectrum some PT -symmetric operators are similar to the self-adjoint ones h = ρ 1 Hρ = h

3 Antilinear symmetry Definition Let A C (H). We say that A possesses an antilinear symmetry if there exists an antilinear bijective operator C and the relation holds for all ψ Dom(A). ACψ = CAψ λ σ p,c,r (A) λ σ p,c,r (A) example: C = PT, H = d2 dx + V (x), V ( x) = V (x) 2

4 Pseudo-Hermiticity Definition Let A L (H) be densely defined. A is called pseudo-hermitian, if there exists an operator η with properties (i) η, η 1 B(H), (ii) η = η (iii) A = η 1 A η. σ p,c,r (A) = σ p,c,r (A ) example: η = P, H = d2 dx + V (x), V ( x) = V (x) 2 A is a self-adjoint operator in a Krein space [, ] J = J, fundamental symmetry J = η η 1

5 Relations between the operator classes finite dimension: antilinear symmetry pseudo-hermiticity essential fact: C-symmetric operators C antilinear isometric involution, C 2 = I, Cx, Cy = y, x, A = CA C assumption of spectral decomposition (spectral operators of scalar and finite type): AS P-H 2002 Scolarici, Solombrino, 2009 Siegl bounded operators AS is not equivalent to P-H! 2009 Siegl

6 Antilinear symmetry without pseudo-hermiticity Example {e n } n=1 standard orthonormal basis of H = l 2 (N), e n (m) = δ mn T e n := e n 1, n N, e 0 := 0 T e n := e n+1, n N T = antilinear symmetry T for λ < 1, λ σ p (T ) but λ σ r (T )

7 Pseudo-Hermiticity without antilinear symmetry Example {e i } orthonormal basis of H = l2 (Z), e n (m) = δ mn λ 0 e i + e i+1, i 1, 0, i = 0, T e i := λ 0 e 1, i = 1, λ 0 e i + e i+1, i < 1, λ 0 C, Im λ 0 > 1 2

8 Pseudo-Hermiticity without antilinear symmetry Example T = λ λ λ λ pseudo-hermiticity η = P, Pe i = e i λ 0 σ r (T ) but λ 0 σ p (T ) λ λ 0 < 1 σ p (T )

9 Counterexamples Counterexamples - properties both examples - not spectral - uncountable point spectrum AS+P-H C-symmetric operator σ r = Related questions? what are equivalent subclasses (AS, P-H)? is AS+P-H related to existence of spectral decomposition?? at least for special classes of operators?? point interactions?

10 PT -symmetric point interactions Definitions of operators line L 2 (R) or finite interval (circle) L 2 (a, b) H = d2 dx 2 Dom(H) = AC 1 + boundary conditions at x = 0 or at x = a, b 0 l -l 0

11 PT -symmetric point interactions PT -symmetric boundary conditions 2002 Albeverio, Fei, Kurasov ( ) ( ) ψ(0+) ψ(0 ) ψ = B (0+) ψ (0 ) ( ) 1 + bce iφ b b 0, c 1/b B =, c 1 + bce iφ φ ( π, π]

12 System on a line - interaction at x = 0 Symmetries PT -symmetry: PT Hψ = HPT ψ, ψ Dom(H) P-pseudo-Hermiticity: H = PHP T -self-adjointness: H = T HT T -complex conjugation, P-parity Spectrum residual part is empty σ r (H) = 2008 Borisov, Krejčiřík continuous spectrum σ c (H) = [0, ) b = 0, c = 0 point spectrum - at most two eigenvalues real if bc sin 2 φ cos 2 φ or bc sin 2 φ cos 2 and cos φ Albeverio, Fei, Kurasov

13 Special case b = 0, c = 0 Boundary conditions ψ(0+) = e iφ ψ(0 ) ψ (0+) = e iφ ψ (0 ) Special cases φ = 0 - self-adjoint operator, no interaction ψ(0+) = ψ(0 ), ψ (0+) = ψ (0 ) φ = ±π/2 - continuous spectrum [0, ), no eigenvalues, quasi-hermitian φ = ±π/2 - surprising case ψ(0+) = ±iψ(0 ), ψ (0+) = iψ (0 )

14 Special case b = 0, c = 0, φ = ±π/2 Quasi-Hermiticity H φ is quasi-hermitian: ΘH φ = H φθ, Θ, Θ 1 B(H), Θ > 0 Θ = I i sin φp sign P (P sign f) (x) = signxf(x), P-parity similarity to s-a operator ρ = Θ = cos φ i sin φ 2 I P 2 cos φ sign P 2 Metric operator Θ spectrum - only two eigenvalues 1 sin φ, 1 + sin φ Θ > 0, Θ 1 B(H) ΘH φ = H φθ is valid Θ is not invertible if φ = ±π/2!

15 Special case b = 0, c = 0, φ = ±π/2 Quasi-Hermiticity H φ is quasi-hermitian: ΘH φ = H φθ, Θ, Θ 1 B(H), Θ > 0 Θ = I i sin φp sign P (P sign f) (x) = signxf(x), P-parity similarity to s-a operator ρ = Θ = cos φ i sin φ 2 I P 2 cos φ sign P 2 Metric operator Θ spectrum - only two eigenvalues 1 sin φ, 1 + sin φ Θ > 0, Θ 1 B(H) ΘH φ = H φθ is valid Θ is not invertible if φ = ±π/2!

16 Surprising case b = 0, c = 0, φ = π/2 Properties of H π/2 ψ(0+) = iψ(0 ), ψ (0+) = iψ (0 ) H π/2 is PT -symmetric, P-pseudo-Hermitian, T -self-adjoint Hπ/2 = H π/2, H π/2 is closed ΘHπ/2 = H π/2θ Θ = I ip sign P, Θ 0, Θ is not invertible! Spectrum 2005 Albeverio and Kuzhel residual spectrum is empty continuous spectrum [0, ) point spectrum C [0, )

17 Surprising case b = 0, c = 0, φ = π/2 Properties of H π/2 ψ(0+) = iψ(0 ), ψ (0+) = iψ (0 ) H π/2 is PT -symmetric, P-pseudo-Hermitian, T -self-adjoint Hπ/2 = H π/2, H π/2 is closed ΘHπ/2 = H π/2θ Θ = I ip sign P, Θ 0, Θ is not invertible! Spectrum 2005 Albeverio and Kuzhel residual spectrum is empty continuous spectrum [0, ) point spectrum C [0, )

18 Surprising case b = 0, c = 0, φ = π/2 Eigenfunctions of H π/2 ψ k (x) = ζ k (x) = { { e kx, x < 0, ie kx, x > 0, e ikx, x < 0, ie ikx, x > 0. φ k (x) = { e kx, x < 0, ie kx, x > 0, ψ k L 2 (R) for Re k > 0, φ k L 2 (R) for Re k < 0, ζ k L 2 (R) for Re k = 0 and Im k > 0

19 Models on finite interval Models on a finite interval ( l, l) L 2 ( l, l), H = d2 dx 2 Dom(H) = AC 1 ( l, l) 2 interactions - at x = 0 and x = ±l - 2 BC at x = 0 - PT -symmetric interaction b = 0, c = 0 at x = ±l - general PT -symmetric interactions

20 Compact resolvent guaranteed? Theorem (Kato) Let T 1, T 2 C (H) have non-empty resolvent sets. Let T 1, T 2 be extensions of a common operator T 0, with order of extension for T 1 being finite. Then T 1 has compact resolvent if and only if T 2 has compact resolvent.

21 Two PT -symmetric interactions PT -symmetric interactions ψ(0+) = e iφ1 ψ(0 ) ψ (0+) = e iφ1 ψ (0 ) ( ) ( ) ψ(l) ψ( l) ψ = B (l) ψ ( l) ( 1 + b2 c 2 e iφ2 b 2 B = 1 + b2 c 2 e iφ2 c 2 )

22 Two PT -symmetric interactions Spectrum discrete (λ = k 2 ) if φ 1 = ±π/2, φ 2 = ±π/2 φ 1 = ±π/2, φ 2 = ±π/2 and b 2 = 0 or c 2 = 0 ( (b2 cos φ 1 k 2 ) ) c 2 sin 2kl + 2k 1 + b2 c 2 cos φ 2 cos 2kl + ( ) +2k 1 + b2 c 2 sin φ 1 sin φ 2 1 = 0. empty if φ 1 = ±π/2 and 1 + b 2 c 2 sin φ 2 1 = 0 entire C if φ 1 = ±π/2 and 1 + b 2 c 2 sin φ 2 1 = 0 b 2 = c 2 = 0 empty if φ 1 = ±π/2 and φ 2 = ±π/2 entire C if φ 1 = φ 2 = ±π/2

23 Conclusions Conclusions Antilinear symmetry is not equivalent to pseudo-hermiticity Equivalent subclases are not determined PT -symmetry, pseudo-hermiticity, C-self-adjointness do not guarantee non-empty spectrum, countable point spectrum, spectral decomposition Examples of PT -symmetric point interactions line R - σ = C, σ c = [0, ), σ p = C [0, ) finite interval ( l, l) - σ = versus σ = σ p = C

24 References 2002 Albeverio, Fei, Kurasov, LMP, Point Interactions: PT-Hermiticity and Reality of the Spectrum 2002 Solombrino, Weak pseudo-hermiticity and antilinear commutant, quant-ph/ Scolarici, Solombrino, On the pseudo-hermitian nondiagonalizable Hamiltonians, quant-ph/ Albeverio, Kuzhel, LMP, One-dimensional Schrdinger operators with P-symmetric zero-range potentials 2008 Siegl, JPA, Supersymmetric quasi-hermitian Hamiltonians with point interactions on a loop 2009 Albeverio, Gunther, Kuzhel, JPA, J-self-adjoint operators with C-symmetries: an extension theory approach 2009 Siegl, PRAMANA, The non-equivalence of pseudo-hermiticity and presence of antilinear symmetry 2009 Siegl, Surprising spectra of PT-symmetric point interactions, arxiv: v Kuzhel, Shapovalova, Vavrykovych, On J-self-adjoint extensions of the Phillips symmetric operator, arxiv: v1