Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces

Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces

Metric operators, generalized hermiticity and lattices of Hilbert spaces Jean-Pierre Antoine Institut de Recherche en Math´ematiqueet Physique (IRMP) Universit´ecatholique de Louvain, Louvain-la-Neuve, Belgium (Joint work with Camillo Trapani) 15th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics (PHHQP 15) Universit`adi Palermo, May 2015 Jean-Pierre Antoine Metric operators, generalized hermiticity and . 1/28 MOTIVATION Non-self-adjoint operators with real spectrum appear in different contexts : PT-symmetric quantum mechanics (Bender et al.) Pseudo-Hermitian quantum mechanics (Mostafazadeh) Three-Hilbert-Space formulation of quantum mechanics (Znojil) Nonlinear pseudo-bosons (Bagarello) Nonlinear supersymmetry ... under various names : pseudo-Hermitian, quasi-Hermitian, cryptohermitian operators Generic structure : A∗G = GA, i.e., A∗ is similar to A via G : H!K, a metric operator, i.e. G > 0, thus invertible, with (possibly unbounded) inverse G −1 Aim of this talk : to study the problem of operator similarity under a metric operator in the framework of partial inner product spaces (pip-spaces) Jean-Pierre Antoine Metric operators, generalized hermiticity and . 2/28 I. Metric operators and generalized hermiticity Jean-Pierre Antoine Metric operators, generalized hermiticity and . 3/28 Metric operators - 1 Metric operator : strictly positive self-adjoint operator G > 0 : i.e., hGξjξi > 0; 8 ξ 2 D(G), and hGξjξi = 0 , ξ = 0 ) G invertible, G −1 densely defined in H, not necessarily bounded G −1 bounded ) G −1 is a metric operator Let G; G1, G2 be metric operators. Then (1) if G1; G2 are bounded, G1 + G2 is a bounded metric operator (2) λG is a metric operator if λ > 0 1=2 t (3) G ; G (t 2 R) are metric operators Take G bounded and G −1 unbounded and define 1=2 1=2 1=2 hξjηiG := hG ξjG ηi;ξ;η 2 D(G ) : positive definite inner product 1=2 on H with corresponding norm kξkG = kG ξk Completion ) new Hilbert space H(G) and H ⊆ H(G) Conjugate dual space H(G)× of H(G) ' subspace of H and H(G)× ≡ H(G −1) = D(G −1=2) with inner product −1=2 −1=2 hξjηiG −1 = hG ξjG ηi ) Triplet (RHS) H(G −1) ,!H ,!H(G) (,! = continuous embedding with dense range) G −1 bounded )H(G −1) = H(G) = H with norms equivalent (but different) to the norm of H Jean-Pierre Antoine Metric operators, generalized hermiticity and . 4/28 Metric operators - 2 In the triplet H(G −1) ⊂ H ⊂ H(G) (*) G −1=2= unitary operator H(G −1) !H and H!H(G) G 1=2= unitary operator H!H(G −1) and H(G) !H Triplet (*) = central part of the infinite scale of Hilbert spaces built on −1=2 −n=2 the powers of G , VI := fHn; n 2 Zg, where Hn = D(G ); n 2 N, × with a norm equivalent to the graph norm, and H−n = Hn ::: ⊂ H2 ⊂ H1 ⊂ H ⊂ H−1 ⊂ H−2 ⊂ ::: Question : what are the end spaces of the scale ? −1=2 \ −1=2 [ H1(G ) := Hn; H−∞(G ) := Hn: n2Z n2Z By quadratic interpolation, build continuous scale Ht ; 0 6 t 6 1, between −t=2 −t=2 H1 and H, where Ht = D(G ), with norm kξkt = kG ξk × Define H−t = Ht and iterate ) full continuous scale V := fH ; t 2 g : -space eI t R pip Jean-Pierre Antoine Metric operators, generalized hermiticity and . 5/28 Similar operators Easy properties : Given (A; D(A))= linear operator in H : D(A) dense in H) D(A) is dense in H(G) A closed in H(G) ) closed in H Definitions (1) Let H; K be Hilbert spaces, A; B densely defined linear operators in H; resp. K. A bounded operator T : H!K is called a bounded intertwining operator for A and B if (io1) T : D(A) ! D(B); (io2) BT ξ = TAξ; 8ξ 2 D(A). (2) A and B are similar( A ∼ B) if there exists a bounded intertwining operator T for A and B with bounded inverse T −1 : K!H, intertwining for B and A (∼ is an equivalence relation) (3) A and B are unitarily equivalent( A ∼u B) if A ∼ B and T : H!K is unitary Properties Let A ∼ B. Then TD(A) = D(B) A closed , B closed A−1 exists , B−1 exists; then, B−1 ∼ A−1 Jean-Pierre Antoine Metric operators, generalized hermiticity and . 6/28 Similar operators : spectral considerations Decomposition of spectrum of a closed operator A: Resolvent set −1 ρ(A) := fλ 2 C :(A − λI ) one-to-one and (A − λI ) is boundedg Spectrum σ(A) := C n ρ(A) Point spectrum σp(A) := fλ 2 C :(A − λI ) not one-to-oneg = feigenvalues of Ag Continuous spectrum σc (A) := fλ 2 C :(A − λI ) one-to-one, dense range 6= Hg ) (A − λI )−1 densely defined, unbounded Residual spectrum σr (A) := fλ 2 C :(A − λI ) one-to-one, range not denseg, ) (A − λI )−1 not densely defined ∗ Note σr (A) = σp(A∗) = fλ : λ 2 σp(A )g σ(A) = σp(A) [ σc (A) [ σr (A) (Dunford-Schwartz) Jean-Pierre Antoine Metric operators, generalized hermiticity and . 7/28 Similar operators : spectral considerations Properties of similar operators A; B closed and A ∼ B with the bounded intertwining operator T . Then ρ(A) = ρ(B) σp(A) = σp(B) and eigenvectors of A; B are in bijection, with same eigenvalues and same multiplicity : ξ 2 D(A) eigenvector of A ) T ξ eigenvector of B η 2 D(B) eigenvector of B ) T −1η eigenvector of A σc (A) = σc (B) and σr (A) = σr (B) A self-adjoint ) B has real spectrum and σr (B) = ; Similarity : often too strong! A is quasi-similar to B (A a B) if there exists a bounded intertwining operator T for A and B which is invertible, with inverse T −1 densely defined (but not necessarily bounded). Note : one can always suppose that T is a metric operator () `metrically (quasi-)similar operators') NB. Great confusion about terminology in the literature ! Jean-Pierre Antoine Metric operators, generalized hermiticity and . 8/28 Quasi-similar operators : spectral considerations - 1 Comparison of spectra Let A; B be closed and densely defined, and assume A a B with the bounded intertwining operator T . Then: (i) σp(A) ⊆ σp(B) and, for every λ 2 σp(A), one has mA(λ) 6 mB (λ) (ii) σr (B) ⊆ σr (A) (iii) If TD(A) = D(B), then σp(B) = σp(A) −1 (iv) If T is bounded and TD(A) is a core for B, then σp(B) ⊆ σ(A) (v) T −1 everywhere defined and bounded ) ρ(A) n σp(B) ⊆ ρ(B) and ρ(B) n σr (A) ⊆ ρ(A) (vi) Assume that T −1 is everywhere defined and bounded and TD(A) is a core for B. Then σp(A) ⊆ σp(B) ⊆ σ(B) ⊆ σ(A): Remark :This situation is important for applications : it gives some information on σ(B) once σ(A) is known. For instance : A has a pure point spectrum ) B is isospectral to A A self-adjoint and A a B via an intertwining operator T with bounded inverse T −1 ) B has real spectrum Jean-Pierre Antoine Metric operators, generalized hermiticity and . 9/28 Quasi-similar operators : spectral considerations - 2 Examples: 2 (1) In H = L (R; dx), take Q of multiplication by x, on the dense domain Z 2 2 2 D(Q) = f 2 L (R): x jf (x)j dx < 1 R 2 P' = j'ih'j ≡ ' ⊗ ', for ' 2 L (R), with k'k = 1 2 2 −1 A'f = h(I + Q )f j'i(I + Q ) '; ' 2 D(A') ) Then 2 −1 P' a A' with the bounded intertwining operator T = (I + Q ) P' is everywhere defined and bounded, but the operator A' is closable iff ' 2 D(Q2). 2 If ' 2 D(Q ), A' is bounded and everywhere defined, and σ(A') = σ(P') = f0; 1g 2 (2) In H = L (R; dx), define (Af )(x) = f 0(x) − 2x f (x); f 2 D(A) = W 1;2( ) 1+x2 R 0 1;2 (Bf )(x) = f (x); f 2 D(B) = W (R) ) A a B with the bounded intertwining operator T = (I + Q2)−1 σ(A) = σ(B) σp(A) = ;, σr (A) = f0g, but σ(B) = σc (B) = iR Jean-Pierre Antoine Metric operators, generalized hermiticity and . 10/28 Quasi-similarity with unbounded intertwining operators A closed (densely defined) operator T : H!K is called an intertwining operator for A and B if (io0) D(TA) = D(A) ⊂ D(T ); (io1) T : D(A) ! D(B); (io2) BT ξ = TAξ; 8 ξ 2 D(A). First part of condition (io0): ξ 2 D(A) implies Aξ 2 D(T ). A is quasi-similar to B, A a B, if there exists a (possibly unbounded) intertwining operator T for A and B which is invertible, with inverse T −1 densely defined ∗ ∗ ∗ Note : A a B 6) B a A , since (io0) may fail for B ) This is a very singular situation ! Indeed if A a B w.r. to T and ρ(A) 6= ;, then T is bounded A and B are mutually quasi-similar if we have both A a B and B a A, which we denote by A a` B a` is an equivalence relation A a` B ) A∗ a` B∗ Jean-Pierre Antoine Metric operators, generalized hermiticity and . 11/28 Quasi-Hermitian operators - 1 Dieudonn´e'sdefinition :A bounded operator A is called quasi-Hermitian if there exists a metric operator G such that GA = A∗G New definitions : A closed operator A is called quasi-Hermitian if there exists a metric operator G such that D(A) ⊂ D(G) and hAξjGηi = hGξjAηi; ξ; η 2 D(A) A is strictly quasi-Hermitian if it is quasi-Hermitian and AD(A) ⊂ D(G) , D(GA) = D(A) Then A strictly quasi-Hermitian , A a A∗ (1) Bounded quasi-Hermitian operators : If A is bounded, the following statements are equivalent (i) A is quasi-Hermitian (ii) There exists a bounded metric operator G, with bounded inverse, such that GA (= A∗G) is self-adjoint (iii) A is metrically similar to a self-adjoint operator K (2) Unbounded quasi-Hermitian operators : If G is bounded, then A is quasi-Hermitian , GA is symmetric in H Let G be bounded.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    28 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us