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Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces

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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.
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