On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators
aa r X i v : . [ m a t h . SP ] A ug On the similarity of Sturm-Liouvilleoperators with non-Hermitian boundaryconditions to self-adjoint and normaloperators
David Krejˇciˇr´ık a,b , Petr Siegl, a,c,d and Jakub ˇZelezn´y a,c a ) Department of Theoretical Physics, Nuclear Physics Institute, CzechAcademy of Sciences, ˇReˇz, Czech RepublicE-mail: [email protected], [email protected], [email protected] b ) IKERBASQUE, Basque Foundation for Science, Alameda Urquijo,36, 5, 48011 Bilbao, Kingdom of Spain c ) Faculty of Nuclear Sciences and Physical Engineering, Czech Tech-nical University in Prague, Prague, Czech Republic d ) Laboratoire Astroparticule et Cosmologie, Universit´e Paris 7, Paris,France
24 August 2011
Abstract
We consider one-dimensional Schr¨odinger-type operators in a boundedinterval with non-self-adjoint Robin-type boundary conditions. It is wellknown that such operators are generically conjugate to normal operatorsvia a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties ofthe transformations in detail. We show that they can be expressed asthe sum of the identity and an integral Hilbert-Schmidt operator. In thecase of parity and time reversal boundary conditions, we establish closedintegral-type formulae for the similarity transformations, derive the simi-lar self-adjoint operator and also find the associated “charge conjugation”operator, which plays the role of fundamental symmetry in a Krein-spacereformulation of the problem.
Mathematics Subject Classification (2010) :Primary: 34B24, 47B40, 34L10 Secondary: 34L40, 34L05, 81Q12.
Keywords : Sturm-Liouville operators, non-symmetric Robin boundaryconditions, similarity to normal or self-adjoint operators, discrete spectraloperator, complex symmetric operator, PT -symmetry, metric operator, C operator, Hilbert-Schmidt operators Introduction
Let us consider the m-sectorial realization H of the second derivative operator ψ
7→ − ψ ′′ (1.1)in the Hilbert space H := L ( − a, a ), with a >
0, subjected to separated, Robin-type boundary conditions ψ ′ ( ± a ) + c ± ψ ( ± a ) = 0 (1.2)where c ± are arbitrary complex numbers. The operator H is self-adjoint if,and only if, the constants c ± are real. The present paper is concerned withthe existence and properties of similarity transformations of H to a normal orself-adjoint operator in the non-trivial case of non-real c ± .The similarity to the normal (respectively, self-adjoint) operator is under-stood as the existence of a bounded operator Ω with bounded inverse such that h := Ω H Ω − (1.3)is normal (respectively, self-adjoint). We remark that this concept is equivalentto the existence of a topologically equivalent inner product in H with respectto which H is normal (respectively, self-adjoint). In addition to results on thegeneral structure of the similarity transformations, modified inner products, andtransformed operators, we present explicit closed formulae for these objects inspecial cases of boundary conditions.The operators of the type (1.1)–(1.2) have been studied from many aspectsand there exist a large number of known results; we particularly mention theclassical monograph of Dunford and Schwartz [11, Chapter XIX.3]. Recent yearsbrought new motivations and focused attention to some aspects of the problemwhich attracted little attention earlier.As an example, let us mention that one-dimensional Schr¨odinger operatorswith non-Hermitian boundary conditions of the type (1.2) were used as a modelin semiconductor physics by Kaiser, Neidhardt and Rehberg [18]. In their paperthe imaginary parts of the constants c ± are required to have opposite signs suchthat the system is dissipative. The authors find the characteristic functionof the operators, construct its minimal self-adjoint dilation and develop thegeneralized eigenfunction expansion for the dilation. See also [16, 17] for furthergeneralizations. Here the main idea of using non-self-adjointness comes fromembedding a quantum-mechanically described structure into a macroscopic flowand regarding the system as an open one.However, the principal motivation of the present work is the possibility ofgiving a direct quantum-mechanical interpretation of non-Hermitian operatorswhich are similar to self-adjoint ones [29]. The most recent strong impetus tothis point of view comes from the so-called PT -symmetric quantum mechanics.Here the reality of the spectrum of a class of non-Hermitian operators – causedby certain symmetries rather than self-adjointness – suggests their potential2elevance as quantum-mechanical Hamiltonians; see the review articles [4, 27].It has been confirmed during the last years that it is indeed the case providedthat the similarity transformation to a self-adjoint operator can be ensured.However, it is a difficult task.Motivated by the lack of rigorous results, the authors of [21] introduceda simple non-Hermitian PT -symmetric operator of the type (1.1)–(1.2) andwrote down a closed formula for the (square of the) similarity transformation(see also [20, 22]). Let us also mention that the importance of (not only) PT -symmetric version of (1.1)–(1.2) in quantum mechanical scattering has beenrecently established in [15].The present paper can be regarded as a step further. In addition to consider-ing more general situations of larger classes of boundary conditions and similar-ity to normal operators, we provide an alternative and more elegant (integral-kernel) formulae for the similarity transformations in the PT -symmetric sit-uation. Moreover, we also give a remarkably simple formula for the similarself-adjoint operator in this case. Finally, we succeed in finding the so-called C -operator in a closed form, which plays the role of fundamental symmetry ina Krein-space reformulation of the problem.The paper is organised as follows. In Section 2 we give a precise definitionof the operator H , summarize its known properties and recall the general con-cepts of quasi-Hermitian, PT -symmetric, and C -symmetric operators. Our mainresults about the universal structure of the similarity transformations can befound in Section 3. In Section 4 we show how these can be applied to particular( PT -symmetric) classes of boundary conditions and we present some explicitconstructions of the studied objects. In Section 5 we discuss how the results canbe extended to bounded and even second-order perturbations of H . Our finalSection 6 presents a series of concluding remarks. We start with recalling general properties of H and concepts of similarity trans-formations in Hilbert spaces. H The standard norm in our Hilbert space
H ≡ L ( − a, a ) is denoted by k · k .The corresponding inner product is denoted by h· , ·i and it is assumed to beantilinear in the first component.We consider the m-sectorial realization H of the operator (1.1) subjectedto the boundary conditions (1.2) as the operator associated on H with thequadratic form t H [ ψ ] := k ψ ′ k + c + | ψ ( a ) | − c − | ψ ( − a ) | , Dom ( t H ) := W , ( − a, a ) . (2.1)3ote that the boundary terms are well defined because of the embedding ofthe Sobolev space W , ( − a, a ) in the space of uniformly continuous functions C [ − a, a ]. An elementary idea of the proof of the embedding can be also usedto show that the boundary terms represent a relatively bounded perturbationof the form associated with the Neumann Laplacian ( i.e. , c ± = 0). Since theNeumann form is clearly non-negative and closed by definition of the Sobolevspace, we know that t H is a closed sectorial form by a standard perturbativeargument [19, Sec. VI.1.6].By the representation theorem [19, Thm. VI.2.1] and an elementary versionof standard elliptic regularity theory, it is easy to see that Hψ = − ψ ′′ , Dom ( H ) = (cid:8) ψ ∈ W , ( − a, a ) : ψ ′ ( ± a ) + c ± ψ ( ± a ) = 0 (cid:9) . (2.2)We refer to [19, Ex. VI.2.16] for more details. The operator definition (2.2)gives a precise meaning to (1.1)–(1.2). This subsection is mainly intended to collect some notation we shall use later.We have already mentioned that the special choice c ± = 0 gives rise tothe Neumann Laplacian − ∆ N on H . The Dirichlet Laplacian − ∆ D on H canbe considered as the other extreme case by formally putting c ± = + ∞ . Itis properly defined as the second derivative operator (1.1) with the operatordomain Dom ( − ∆ D ) := W , ( − a, a ) ∩ W , ( − a, a ).The spectrum of the Dirichlet and Neumann Laplacians in our one-dimen-sional situation is well known: σ ( − ∆ D ) = { k n } ∞ n =1 ,σ ( − ∆ N ) = { k n } ∞ n =0 , with k n := nπ a . The corresponding eigenfunctions are respectively given by χ Dn ( x ) := 1 √ a sin k n ( x + a ) , χ Nn ( x ) := ( √ a if n = 0 , √ a cos k n ( x + a ) if n ≥ . (2.3)To simplify some expressions in the sequel, we extend the notation by χ D := 0.Next we introduce a “momentum” operator p and its adjoint p ∗ : pψ := − i ψ ′ , p ∗ ψ = − i ψ ′ , Dom ( p ) := W , ( − a, a ) , Dom ( p ∗ ) = W , ( − a, a ) . (2.4)The following identities hold:i pχ Dn = k n χ Nn , i p ∗ χ Nn = − k n χ Dn , − ∆ D = p ∗ p, − ∆ N = pp ∗ . (2.5)4he resolvents ( − ∆ D − k ) − , ( − ∆ N − k ) − act as integral operators withsimple kernels (Green’s functions) G kD and G kN , respectively: G kD ( x, y ) = − sin( k ( x + a )) sin( k ( y − a )) k sin(2 ka ) , x < y , G kN ( x, y ) = − cos( k ( x + a )) cos( k ( y − a )) k sin(2 ka ) , x < y , (2.6)with x, y exchanged for x > y . Here the spectral parameter k is supposed tobelong to the resolvent set of the respective operator.For k = 0, the kernel of ( − ∆ D ) − simplifies to G D ( x, y ) = ( x + a )( a − y )2 a , x < y , (2.7)with x, y exchanged for x > y . The resolvent of − ∆ N does not exist for k = 0,of course, but one can still introduce the reduced resolvent (cid:0) − ∆ ⊥ N (cid:1) − of theNeumann Laplacian with respect to the eigenvalue 0 (see [19, Sec. III.6.5] for theconcept of reduced resolvent). From the point of view of the spectral theorem: (cid:0) − ∆ ⊥ N (cid:1) − = ∞ X n =1 k n χ Nn h χ Nn , ·i . (2.8)The corresponding integral kernel G ⊥ N ( x, y ) can be obtained by taking the limit k → G kN ( x, y ) + k − χ N ( x ) χ N ( y ). We find G ⊥ N ( x, y ) = ( x + a ) a + ( y − a ) a − a , x < y , (2.9)with x, y exchanged for x > y .Finally, we introduce operators J ι := ∞ X n =0 C n χ ιn h χ ιn , ·i , ι ∈ { D, N } , (2.10)where C n are positive numbers satisfying0 < m < C n < m < ∞ (2.11)for all n ≥
0, with given positive m , m . The sum in the definition (2.10), aswell as all other analogous expressions in the following, are understood as limitsin the strong sense. H Now we are in a position to recall some general properties of the operator H .5 roposition 2.1 (General known facts) . (i) H is m-sectorial. The adjoint operator H ∗ is obtained by taking the com-plex conjugation of c ± in the boundary conditions (1.2) . (ii) H forms a holomorphic family of operators of type ( B ) with respect to theboundary parameters c ± . (iii) The resolvent of H is a compact operator. (iv) H is a discrete spectral operator. (v) If all eigenvalues are simple, then H is similar to a normal operator. Ifthe spectrum of H is in addition real, then H is similar to a self-adjointoperator. We have already shown that H is m-sectorial as the operator associated withthe closed sectorial form (2.2). The rest of the claim (i) follows by the fact thatthe adjoint operator is associated with the adjoint form t ∗ H ( φ, ψ ) := t H ( ψ, φ )( cf. [19, Thm. VI.2.5]). Property (ii) follows from (2.2) as well if we recall thatthe boundary terms represent a relatively bounded perturbation of the formassociated with the Neumann Laplacian and the relative bound can be madearbitrarily small ( cf. [19, Sec. VII.4.3]). This also proves (iii) as a consequenceof the perturbation result [19, Sec. VI.3.4]. The proof of (iv) is contained in [11,Chapter XIX.3]. Property (v) is a consequence of (iv).The similarity to a normal operator can be equivalently stated as the Rieszbasicity of the eigenvectors of H . This property is shared by all second derivativeoperators with strongly regular boundary conditions, see [26]. Using the notionof spectral operator, this has been investigated in [11] as well.Although the eigenvalues of H are generically simple, degeneracies may ap-pear. However, the only possibility are the eigenvalues of algebraic multiplicitytwo and geometric multiplicity one. In this case, operator H cannot be sim-ilar to a normal one, nevertheless, the eigenvectors together with generalizedeigenvectors still form a Riesz basis.Now we turn to symmetry properties of H . Definition 2.1 ( PT -symmetry) . We say that H is PT -symmetric if [ PT , H ] = 0 , (2.12) where ( P ψ )( x ) := ψ ( − x ) , ( T ψ )( x ) := ψ ( x ) . (2.13)It should be stressed that PT is an antilinear operator. The commutatorrelation (2.12) means precisely that ( PT ) H ⊂ H ( PT ), as usual for the commu-tativity of an unbounded operator with a bounded one ( cf. [19, Sec. III.5.6]). Inthe quantum-mechanical context, P corresponds to the parity inversion (spacereflection), while T is the time reversal operator.6 efinition 2.2 ( S -self-adjointness) . We say that H is S -self-adjoint if the re-lation H = S − H ∗ S holds with a boundedly invertible operator S . We will use this concept in a wide sense, with S being either linear or anti-linear operator. If S is a conjugation operator ( i.e. antilinear involution), thenour definition coincides with the concept of J -self-adjointness [12, Sec. III.5].While Definition 2.2 is quite general, Definition 2.1 makes sense for operatorsin a complex functional Hilbert space only. In our case, we have: Proposition 2.2 (Symmetry properties) . (i) H is T -self-adjoint. (ii) H is P -self-adjoint if, and only if, c − = − c + . (iii) H is PT -symmetric if, and only if, c − = − c + . Property (ii) coincides with the notion of self-adjointness in the Krein spaceequipped with the indefinite inner product h· , P·i . It is also referred to as P -pseudo-Hermiticity in physical literature (see, e.g. , [27]).It follows from Proposition 2.2.(i) that the residual spectrum of H is empty( cf. [7, Corol. 2.1]). Alternatively, it is a consequence of Proposition 2.1.(iii),which in addition implies that the spectrum of H is purely discrete.We denote the (countable) set of eigenvalues of H by { λ n } ∞ n =0 and the corre-sponding set of eigenfunctions by { ψ n } ∞ n =0 . Similarly, let { λ n } ∞ n =0 and { φ n } ∞ n =0 be the set of eigenvalues and eigenfunctions of the adjoint operator H ∗ . That is Hψ n = λ n ψ n , H ∗ φ n = λ n φ n . (2.14)Eigenfunctions ψ n and φ m corresponding to different eigenvalues, i.e. λ n = λ m ,are clearly orthogonal. Solving the eigenvalue equation for H in terms of sineand cosine functions, it is straightforward to reduce the boundary value problemto an algebraic one. Proposition 2.3 (Spectrum) . The eigenvalues λ n = l n of H are solutions ofthe implicit equation sin(2 al )( c − c + + l ) + ( c − − c + ) l cos(2 al ) = 0 . (2.15) The corresponding eigenfunctions of H and H ∗ respectively read ψ n ( x ) = A n √ a (cid:18) cos( l n ( x + a )) − c − l n sin( l n ( x + a )) (cid:19) ,φ n ( x ) = 1 √ a (cid:18) cos( l n ( x + a )) − c − l n sin( l n ( x + a )) (cid:19) . (2.16) If all eigenvalues are simple, ψ n can be normalized through the coefficients A n in such a way that h ψ n , φ m i = δ nm . H has been described more explicitly for the PT -symmetriccase. First of all, as a consequence of the symmetry, we know that the spectrumis symmetric with respect to the real axis. In the following proposition wesummarize more precise results obtained in [21, 22]. Proposition 2.4 ( PT -symmetric spectrum) . Let c ± = i α ± β , with α, β ∈ R .1. If β = 0 then all eigenvalues of H are real, λ = α , λ n = k n , n ∈ N . (2.17) The corresponding eigenfunctions of H and H ∗ respectively read ψ ( x ) = A e − i α ( x + a ) , ψ n ( x ) = A n (cid:18) χ Nn ( x ) − i αk n χ Dn ( x ) (cid:19) ,φ ( x ) = 1 √ a e i α ( x + a ) , φ n ( x ) = χ Nn ( x ) + i αk n χ Dn ( x ) . (2.18) If α = k n for every n ∈ N , then all the eigenvalues are simple and choosing A := αe αa √ a sin(2 αa ) , A n := k n k n − α , (2.19) we have the biorthonormal relations h ψ n , φ m i = δ nm .2. If β > , then all the eigenvalues of H are real and simple.3. If β < , then all the eigenvalues are either real or there is one pair ofcomplex conjugated eigenvalues with real part located in the neighborhoodof α + β .In any case, the eigenvalue equation (2.15) can be rewritten as ( l − α − β ) sin(2 al ) − βl cos(2 al ) = 0 . (2.20) We recall the concept of metric operator (or quasi-Hermitian operators intro-duced in [9]), widely used in PT -symmetric literature. Definition 2.3 (Metric operator and quasi-Hermiticity) . Bounded positive operator Θ with bounded inverse is called a metric operator for H , if H is Θ -self-adjoint. H is then called quasi-Hermitian. It is obvious that the quasi-Hermitian operator H is self-adjoint with respectto the modified inner product h· , ·i Θ := h· , Θ ·i . It is also not difficult to showthat the metric operator exists if, and only if, H is similar to a self-adjoint A is positive if h f, Af i > f ∈ H , f = 0. H has purely discrete spectrum, the metric operatorcan be obtained as Θ = ∞ X n =0 C n φ n h φ n , ·i , (2.21)where φ n are eigenfunctions of H ∗ and C n are real constants satisfying (2.11).The expression (2.21) illustrates a non-uniqueness of the metric operatorcaused by the arbitrariness of C n . The latter can be actually viewed as amodification of the normalization of functions φ n . Choosing different sequences { C n } ∞ n =0 , we obtain all metric operators for H , cf. [31, 33].It is important to stress that if we define an operator Θ by (2.21), we findthat such Θ is bounded, positive, and with bounded inverse whenever { φ n } ∞ n =0 is a Riesz basis. Thus, by virtue of Proposition 2.1.(v), such a Θ exists if,and only if, all eigenvalues of H are simple. However, the Θ-self-adjointnessof H is satisfied if, and only if, the spectrum of H is real. Otherwise, onlyΘ H Θ − H ∗ = H ∗ Θ H Θ − holds, cf. [33], which is equivalent to the fact that H is similar to a normal operator.In the following, the operator Θ is always defined by (2.21) regardless if itis a metric operator for H in view of Definition 2.3.It should be also noted that Θ, as a positive operator, can be always decom-posed to Θ = Ω ∗ Ω . (2.22)One example of such Ω is obviously √ Θ. We shall take the advantage of somedifferent decompositions of the type (2.22) later. It follows easily from Defini-tion 2.3 that the similar operator h defined by (1.3) with Ω given by (2.22) isself-adjoint if Θ is a metric operator for H . If all eigenvalues of H are simplebut no longer entirely real, h is (only) a normal operator. C operator For PT -symmetric operators, the notion of C operator was introduced in [6] andformalized in [2]. It was observed in [25] and in many works after that paper thatKrein spaces provide suitable framework for studying PT -symmetric operators.Indeed, PT -symmetric operators which are at the same time P -self-adjoint arein fact self-adjoint in the Krein space equipped with the indefinite inner product h· , P·i . Recall that our operator H is P -self-adjoint if, and only if, it is PT -symmetric ( cf. Proposition 2.2).
Definition 2.4 ( C operator) . Assume that H is P -self-adjoint ( cf. Proposi-tion 2.2). We say that H possesses the property of C -symmetry, if there existsa bounded linear operator C such that [ H, C ] = 0 , C = I, and PC is a metricoperator for H . Thus, from the point of view of metric operators, we can find the C operatoras C := P Θ for Θ satisfying ( P Θ) = I . Hence C -symmetry allows us tonaturally choose a metric operator. Besides a possible physical interpretation9f C discussed in [5, 4], it appears naturally in the Krein spaces framework aspointed out in [23, 24] as a fundamental symmetry of the Krein space ( H , h· , P·i )with an underlying Hilbert space ( H , h· , PC·i ). In this section we provide general properties of the metric operator Θ definedin (2.21) and its decompositions Ω from (2.22).Let { ψ n } ∞ n =0 and { φ n } ∞ n =0 denote the set of eigenvectors of H and H ∗ ,respectively. We assume that ψ n and φ n form Riesz bases and that they arenormalized in such a way that h ψ n , φ m i = δ mn . In view of Propositions 2.1, 2.3,we know that this is satisfied if all the eigenvalues of H are simple, which is ageneric situation.Let { e n } ∞ n =0 be any orthonormal basis of H . If all eigenvalues of H aresimple, we introduce an operator Ω byΩ := ∞ X n =0 e n h φ n , ·i . (3.1)Clearly, Ω : ψ n e n .Ω is defined only if all eigenvalues are simple, however, sometimes it is pos-sible to extend it by continuity, see examples in Section 4. Nonetheless, suchΩ is typically not invertible and the dimension of the kernel corresponds to thesize of Jordan blocks appearing in the spectrum of H .Basic properties of Ω are summarized in the following. Lemma 3.1.
Let all eigenvalues of H be simple. Then Ω is a bounded operatorwith bounded inverse given by Ω − = ∞ X n =0 ψ n h e n , ·i , (3.2)i.e. Ω − : e n ψ n . The adjoint of Ω reads Ω ∗ = ∞ X n =0 φ n h e n , ·i . (3.3)i.e. Ω ∗ : e n φ n and Ω ∗ Ω = Θ , where Θ is defined in (2.21) with C n = 1 . Furthermore, we show how the operator Ω can be realized.
Theorem 3.2.
Let all eigenvalues of H be simple. Ω can be expressed as Ω = U + L, (3.4) where U := P ∞ n =0 e n h χ Nn , ·i , i.e. U : χ Nn e n , is a unitary operator, and L is aHilbert-Schmidt operator. roof. At first we remark that it suffices to prove that Ω = I + ˜ L for e n := χ Nn , where ˜ L is Hilbert-Schmidt. More precisely, if we compose U from theclaim and I + ˜ L , we obtain Ω in (3.4) since L = U ˜ L is Hilbert-Schmidt too.Thus, we consider this choice of e n in the following. Furthermore, we put a := π/ a using the isometry V : L ( − π/ , π/ → L ( − a, a ) defined by ψ ( x ) p π a ψ ( πx a ).The asymptotic analysis of eigenvalues of H in [11, proof of Lem. XIX.3.10]shows that l n = n + c + − c − πn + O ( n − ) ,λ n ≡ l n = k n + 2( c + − c − ) π + O ( n − ) , (3.5)and | Im ( l n ) | is uniformly bounded in n . These formulae are valid except for afinite number N of eigenvalues.We set ε n := l n − k n = l n − n . Using elementary trigonometric identities,we rewrite the eigenfunctions φ n as follows φ n ( x ) = χ Nn ( x ) cos ( ε n ( x + a )) − χ Dn ( x ) sin ( ε n ( x + a )) − c − l n (cid:2) χ Dn ( x ) cos( ε n ( x + a )) + χ Nn ( x ) sin ( ε n ( x + a )) (cid:3) . (3.6)We further rewrite the cosine and sine functions in this expression ascos ( ε n ( x + a )) = 1 + ε n cos ( ε n ( x + a )) − ε n =: 1 + ε n c n ( x ) , sin ( ε n ( x + a )) = ε n sin ( ε n ( x + a )) ε n =: ε n s n ( x ) . (3.7)Note that k c n k and k s n k are uniformly bounded in n because of the propertiesof ε n . The building block χ Nn h φ n , ·i of Ω then becomes χ Nn h φ n , ·i = χ Nn h χ Nn , ·i + ε n χ Nn h χ Nn c n , ·i − ε n χ Nn h χ Dn s n , ·i− c − l n (cid:0) χ Nn h χ Dn , ·i + ε n χ Nn h χ Dn c n , ·i + ε n χ Nn h χ Nn s n , ·i (cid:1) . (3.8)Taking the sum of χ Nn h φ n , ·i as in (3.1), we obviously get Ω = I + ˜ L .It remains to show that the Hilbert-Schmidt norm k ˜ L k HS of ˜ L is finite. Wewill understand ˜ L as a sum ˜ L = ˜ L N + ˜ L ∞ , where˜ L N := N − X n =0 χ Nn h ˜ φ n , ·i , ˜ L ∞ := ∞ X n = N χ Nn h ˜ φ n , ·i , (3.9)and ˜ φ n := φ n − χ Nn . ˜ L N is a finite rank operator, hence it is automaticallyHilbert-Schmidt and it suffices to consider ˜ L ∞ in the rest of the proof. We11stimate explicitly only one term in the expression for k ˜ L ∞ k , the rest followsin a similar way: ∞ X p =0 * ∞ X n = N ε n χ Nn (cid:10) χ Dn s n , χ Np (cid:11) , ∞ X m = N ε m χ Nm (cid:10) χ Dm s m , χ Np (cid:11)+ = ∞ X p =0 ∞ X n = N | ε n | (cid:12)(cid:12)(cid:10) χ Dn s n , χ Np (cid:11)(cid:12)(cid:12) ≤ a ∞ X n = N | ε n | k s n k < ∞ . (3.10)Here the first inequality follows by the Bessel inequality (after interchanging theorder of summation, which is justified) and by estimating χ Dn by its supremumnorm. The asymptotic behavior of ε n and the uniform boundedness of k s n k areused in the last step. Corollary 3.3.
Let all eigenvalues of H be simple. Then Θ := Ω ∗ Ω = I + K (3.11) coincides with Θ defined in (2.21) with C n = 1 . Here K is a Hilbert-Schmidtoperator that can be realized as an integral operator with a kernel belonging to L (( − a, a ) × ( − a, a )) .Proof. The claim follows from Theorem 3.2 and the well-known facts thatHilbert-Schmidt operators are *-both-sided ideal in the space of bounded oper-ators and can be realized as integral ones, see [28, Thm.VI.23].
Remark . Slight modification of the definition of Ω and the proof of The-orem 3.2 yields the analogous result for operators Θ defined in (2.21) witharbitrary C n . It suffices to consider f n := C n e n instead of e n . The resultingform is Θ = J N + ˜ K, (3.12)where J N is defined in (2.10) and ˜ K is again a Hilbert-Schmidt operator. J N it-self, however, can be a sum of a bounded and a Hilbert-Schmidt operator, aswe shall see in examples. Proposition 3.4.
Let S be an open connected set in C such that for all ( c − , c + ) ∈ S all eigenvalues of H are simple. Then Ω and thereby Θ are boundedholomorphic families in S with respect to parameters c ± .Proof. We verify the criterion stated in [19, Sec. VII.1.1]. We have provedalready that Ω is bounded. It remains to show that h f, Ω g i is holomorphic forevery f, g from a fundamental set of H that we choose as the orthonormal basis { e n } ∞ n =0 . h e m , Ω e n i = h φ m , e n i is holomorphic because φ m is an eigenfunctionof the operator H ∗ , which can be viewed as a holomorphic family of operatorsof type ( B ) with respect to the parameters c ± . Corollary 3.5.
Assume the hypothesis of Proposition 3.4. Then h := Ω H Ω − is a holomorphic family of operators in S with respect to parameters c ± . H is a holomorphic family of type ( B ), i.e. it is naturallydefined via quadratic forms with the domain W , ( − a, a ) independent of theparameters c ± , h is expected to possess a similar property. To prove it, wehave to particularly show that the associated quadratic forms correspondingto different values of c ± have the same domain, which is not guaranteed byCorollary 3.5. To this end we analyse the quadratic form associated to h , wherewe set e n := χ Nn in the definition of Ω. Theorem 3.6.
Let all eigenvalues of H be simple and let e n := χ Nn in (3.1) .Then Ω = I + L and Ω − = I + M , where L , M are Hilbert-Schmidt opera-tors. Ω , Ω ∗ , Ω − , (Ω − ) ∗ are bounded operators on W , ( − a, a ) and W , ( − a, a ) .Furthermore, the following estimates hold for all φ ∈ W , ( − a, a ) and arbitrary δ > : k ( L ∗ φ ) ′ k ≤ C (cid:0) δ k φ ′ k + δ − k φ k (cid:1) , k ( M φ ) ′ k ≤ C (cid:0) δ k φ ′ k + δ − k φ k (cid:1) , (3.13) with C being constants not dependent on δ and φ .Proof. We set a := π/ M is Hilbert-Schmidtsince I = ΩΩ − = I + L + M + LM and L is Hilbert-Schmidt.We consider Ω ∗ at first. Following the proof of Theorem 3.2, L ∗ can bewritten as L ∗ f = ∞ X k =0 ˜ φ k h χ Nk , f i , (3.14)where ˜ φ k := φ k − χ Nk and f ∈ H . We show that L ∗ is bounded on W , ( − a, a ).We estimate the Hilbert-Schmidt norm of L ∗ on W , ( − a, a ) with help of theorthonormal basis f n := χ Nn / √ n . In fact, it suffices to estimate: ∞ X n =0 h ( L ∗ f n ) ′ , ( L ∗ f n ) ′ i = ∞ X n =0
11 + n k ˜ φ ′ n k (3.15)where (recall (3.6) and (3.7))˜ φ ′ n = − nε n χ Dn c n − ε n χ Nn s n − nε n χ Nn s n − ε n χ Dn (1 + ε n c n )+ c − (cid:2) χ Nn (1 + ε n c n ) − ε n χ Dn s n (cid:3) . (3.16)Using the asymptotic properties of ε n and the uniform boundedness of c n , s n (see (3.5) and (3.7), respectively) together with the normalization of χ ιn , weconclude that k ˜ φ ′ n k ≤ C uniformly in n . Therefore (3.15) is finite.Using the same technique, we can show that the Hilbert-Schmidt norm of L ∗ in W , ( − a, a ) is finite. To this end we select the basis χ Nn / √ n + n , therest is based on k ˜ φ ′′ n k = O ( n ) as n → ∞ .Let us now establish the inequalities (3.13). Consider φ ∈ W , ( − a, a ), itsbasis decomposition φ = P ∞ n =0 α n χ Nn , and the identity ∞ X n =0 | nα n | = k φ ′ k . (3.17)13ence, k ( L ∗ φ ) ′ k = ∞ X m,n =0 α m α n h ˜ φ ′ m , ˜ φ ′ n i , (3.18)and having the explicit form of ˜ φ ′ n , see (3.16), we have to estimate several terms.We show the technique only for one term, the estimate of remaining terms isanalogous. First, using the uniform boundedness of k c n k , k s n k , the asymptotics ε n = O ( n − ) and the uniform boundedness of k χ Nn k ∞ , it is easy to see that ∞ X m,n =0 m n | α m || α n || ε m || ε n ||h χ Nm s m , χ Nn s n i| ≤ C ∞ X n =1 | α n | ! holds with some positive constant C . It remains to estimate the l -norm of α n by the l -norms of α n and nα n (which equal k φ k and k φ ′ k , respectively). Thisis rather algebraic: ∞ X n =1 | α n | ! = ∞ X n =1 (cid:0) | α n | n (cid:1) b | α n | − b n − b ! ≤ ∞ X n =1 | α n | n ! b ∞ X n =1 | α n | ! − b ∞ X n =1 n − b ! ≤ C b k φ ′ k b k φ k − b ) ≤ C b (cid:16) b δ k φ ′ k + (1 − b ) δ − b − b k φ k (cid:17) , with any b, δ ∈ (0 , b is chosen in such a way that 2 b >
1, so that the sum of n − b (denotedby C b ) converges. If we put b = 2 /
3, we obtain the inequality in the claim.One can show, using the asymptotics (3.5), that it follows from the normal-ization requirement h φ n , ψ n i = 1 that A n , the normalization constants of ψ n ,see (2.16), satisfy A n = 1 + O ( n − ). Then the claims for Ω − and M can bederived in the same manner.To justify that Ω and (Ω − ) ∗ are bounded on W , ( − a, a ) and W , ( − a, a ),it suffices to realize that Ω − and Ω ∗ are invertible because they are invertible in L ( − a, a ) and the inverse is bounded because of the form identity plus compactoperator on considered Sobolev spaces. Corollary 3.7.
Assume the hypotheses of Theorem 3.6. Then h := Ω H Ω − isa holomorphic family of operators of type ( B ) with respect to c ± . The associatedquadratic form t h , in the sense of the representation theorem [19, Thm. VI.2.1], eads t h [ ψ ] = k ψ ′ k + h ( L ∗ ψ ) ′ , ψ ′ i + h ψ ′ , ( M ψ ) ′ i + h ( L ∗ ψ ) ′ , ( M ψ ) ′ i + c + h(cid:0) ψ ( a ) + ( L ∗ ψ )( a ) (cid:1)(cid:0) ψ ( a ) + ( M ψ )( a ) (cid:1)i − c − h(cid:0) ψ ( − a ) + ( L ∗ ψ )( − a ) (cid:1)(cid:0) ψ ( − a ) + ( M ψ )( − a ) (cid:1)i , Dom ( t h ) = W , ( − a, a ) . (3.19) Proof.
The form t h defined in (3.19) is sectorial and closed due to the perturba-tion result [19, Thm. VI.1.33], regarding u [ ψ ] := t h [ ψ ] − k ψ ′ k as a perturbationof t [ ψ ] := k ψ ′ k . Indeed, the inequalities (3.13) applied on u [ ψ ] yield that u is t -bounded with t -bound 0. Therefore, due to the first representation theorem[19, Thm. VI.2.1], there is a unique m-sectorial operator associated with t h . Letus denote it by ˜ h . Our objective is to show that ˜ h = h .Using the definition of h by the similarity transformation, i.e. h = Ω H Ω − ,and the fact that H is associated to t H , we know that the domain of h arefunctions u such that, firstly, Ω − u ∈ W , ( − a, a ) and, secondly, there exists w ∈ L ( − a, a ) such that t H (Ω ∗ v, Ω − u ) = ( v, w ) (3.20)for all v such that Ω ∗ v ∈ W , ( − a, a ). However, by Theorem 3.6, Ω, Ω ∗ , Ω − ,(Ω ∗ ) − are bounded on W , ( − a, a ) and it is easy to check that the identity t H (Ω ∗ v, Ω − u ) = t h ( v, u ) (3.21)holds for all u, v ∈ W , ( − a, a ). Consequently, the operators ˜ h and h indeedcoincide. Remark . We remark that the boundedness of Ω, Ω ∗ , Ω − and (Ω − ) ∗ in W , ( − a, a ) was not used in the proof Corollary 3.7. Nevertheless, this propertyis useful if we analyse the domain of h directly from the relation h = Ω H Ω − . Itfollows that Dom ( h ) consists of functions ψ from W , ( − a, a ) satisfying bound-ary conditions (Ω − ψ ) ′ ( ± a ) + c ± (Ω − ψ )( ± a ) = 0. P T -symmetric cases
We present closed formulae of operators Θ, Ω and h corresponding to H withspecial PT -symmetric choice of boundary conditions, c ± := i α, with α ∈ R .This case has already been studied in a similar context in [21, 20], where thefirst formulae of the metric Θ were given. We substantially generalize theseresults here.We essentially rely on the original idea of [20] to “use the spectral theorembackward” to sum up the infinite series appearing in the definition of Θ in (2.21).The attempts to find Ω as the square root of Θ using the holomorphic andself-adjoint calculus are contained in [35, 34], however, only approximations15f the resulting similar self-adjoint operator h were found there. The mainnovelty of the present approach comes from the more general factorization (2.22)with (3.1), which enables us to obtain exact results. Formulae contained in thissection are obtained by tedious although straightforward calculations that wedo not present entirely.Finally, we present the metric operator for H with general PT -symmetricboundary conditions, c ± := i α ± β . In this case, the eigenvalues are no longerexplicitly known, nevertheless, the experience from previous examples and for-mulation of partial differential equation together with a set of “boundary con-ditions” for the kernel of the integral operator provide the correct result. We start with the following fundamental result.
Proposition 4.1.
Let c ± := i α, with α ∈ R . Then the operator Θ definedin (2.21) has the form Θ = J N + C θ + J N θ + J D θ , (4.1) where J ι , with ι ∈ { D, N } , are defined in (2.10) , C > , and θ i are integraloperators with kernels: θ ( x, y ) := i a e i α ( x − y ) sin (cid:16) α x − y ) (cid:17) ,θ ( x, y ) := i α a (cid:2) y − a sgn( y − x ) (cid:3) ,θ ( x, y ) := α a (cid:0) a − xy (cid:1) − i α a x − i α (cid:2) − i α ( y − x ) (cid:3) sgn( y − x ) . (4.2)Θ is the metric operator for H , see Definition 2.3, if, and only if, α = k n forevery n ∈ N .Proof. Using the explicit form (2.18) of functions φ n and the definition (2.21)of Θ, we obtainΘ = ∞ X n =0 C n χ Nn h χ Nn , ·i + C (cid:0) φ h φ , ·i − χ N h χ N , ·i (cid:1) + α ∞ X n =1 C n k n χ Dn h χ Dn , ·i + i α ∞ X n =1 C n k n χ Dn h χ Nn , ·i − i α ∞ X n =1 C n k n χ Nn h χ Dn , ·i . (4.3)Employing the operators J ι and p, p ∗ introduced in (2.10) and (2.4), respectively,16nd relations (2.5) we obtain:Θ = J N ∞ X n =0 χ Nn h χ Nn , ·i + C (cid:0) φ h φ , ·i − χ N h χ N , ·i (cid:1) + αJ N p ∞ X n =1 k n χ Dn h χ Dn , ·i + J D α ∞ X n =1 k n χ Dn h χ Dn , ·i + αp ∗ ∞ X n =1 k n χ Nn h χ Nn , ·i ! . (4.4)It follows from the functional calculus for self-adjoint operators that (4.4) canbe written asΘ = J N + C (cid:0) φ h φ , ·i − χ N h χ N , ·i (cid:1) + αJ N p ( − ∆ D ) − + J D (cid:2) α ( − ∆ D ) − + α p ∗ ( − ∆ ⊥ N ) − (cid:3) . (4.5)By inserting the explicit integral kernels of the resolvents, see Section 2.2, weobtain the formula (4.1) with (4.2).To ensure that such Θ represents as metric operator, we recall that thespectrum of H is always real, see Proposition 2.4. Moreover, it is simple if, andonly if, the condition in the last claim is satisfied. Remark . The formula (4.1) can be rewritten in terms of the operator J N only. Indeed, it is possible to show that J D = p ∗ J N p ( − ∆ D ) − . (4.6)The final result is thenΘ = J N + C θ + J N θ + p ∗ J N θ , (4.7)where θ := p ( − ∆ D ) − θ is an integral operator with kernel θ ( x, y ) = α a (cid:18) y (3 − i αy ) + 3 x (1 − i αy ) + 2 a (cid:2) α (3 x − y ) (cid:3)(cid:19) − α (cid:18) − i α ( y − x ) (cid:19) ( y − x ) sgn( y − x ) . (4.8)Note that the expression (4.8) is a result of a rather lengthy computation.Any metric operator for H in Proposition 4.1 can be obtained by determin-ing J N for given constants C n . Thus we managed to transform the problem ofconstructing the metric operators for non-self-adjoint operator H to the prob-lem of constructing the metric operators J N for the Neumann Laplacian − ∆ N .This significantly simplifies the problem, since − ∆ N is self-adjoint and its met-ric operators are bounded, positive operators with bounded inverse commutingwith − ∆ N . For instance, any bounded, uniformly positive function of − ∆ N J N canbe approximated in the strong sense by a polynomial of I + λ ( − ∆ N − λ ) − ,with λ ∈ ρ ( − ∆ N ).We consider two choices of constants C n in the following and we find finalformulae for the corresponding metric operators. Let C n := 1 for every n ≥
0. Then J N = J D = I and the metric operator Θreads Θ = I + K , where K is an integral operator with the kernel K ( x, y ) = i a e i α ( x − y ) sin (cid:16) α x − y ) (cid:17) + i α a (cid:0) | y − x | − a (cid:1) sgn( y − x )+ α a (cid:0) a − xy − a | y − x | (cid:1) . (4.9)Formula (4.9) represents a remarkably elegant form for the metric operatorfound firstly in [21, 20]. C operator Another choice of C n is motivated by the concept of C operator, see Defini-tion 2.4. We want to find such Θ that C = I , where C = P Θ. Since H is P -self-adjoint, we have P φ n = D n ψ n with some numbers D n . Assuming thenon-degeneracy condition α = k n for every n ≥
0, an explicit calculation showsthat D = sin(2 αa )2 αa , D n = ( − n k n − α k n , n ∈ N . (4.10)The condition ( P Θ) = I then restricts C n from (2.21) to C = 2 | α | a | sin(2 αa ) | , C n = k n | k n − α | , n ∈ N . (4.11)In order to simplify the formulae, we consider only α ∈ (0 , k ) in the following. Remark . As mentioned below (2.21), any choice of C n can be interpretedas a sort of normalisation of φ n . It is therefore interesting to notice that (4.11)results into the symmetric normalization of φ n and ψ n when h φ n , ψ n i = 1 isrequired: ψ ( x ) = r α sin(2 αa ) e i αa e − i αx , ψ n ( x ) = k n p k n − α (cid:18) χ Nn ( x ) − i αk n χ Dn ( x ) (cid:19) ,φ ( x ) = r α sin(2 αa ) e i αa e i αx , φ n ( x ) = k n p k n − α (cid:18) χ Nn ( x ) + i αk n χ Dn ( x ) (cid:19) . These expressions should be compared with the normalization of (2.18)–(2.19),standardly used in the present paper. The symmetric form of the “presentnormalization” indicates that the choice (4.11) will lead to a simpler form of Θthan (4.9). 18sing (4.11) in the series (2.10), the operators J ι can be determined by thefunctional calculus: J N = ∞ X n =0 k n k n − α χ Nn h χ Nn , ·i + C χ N h χ N , ·i = ( − ∆ N )( − ∆ N − α ) − + C χ N h χ N , ·i = I + α ( − ∆ N − α ) − + C χ N h χ N , ·i ,J D = ∞ X n =1 k n k n − α χ Dn h χ Dn , ·i = ( − ∆ D )( − ∆ D − α ) − = I + α ( − ∆ D − α ) − . (4.12)A direct (but very tedious) way how to derive the metric Θ for the choice (4.11)is to express the resolvents of the Dirichlet and Neumann Laplacians from theultimate expressions in (4.12) by means of the Green’s functions (2.6) and com-pose them with the operators θ i in (4.1).However, a more clever way how to proceed is to come back to the operatorform (4.5) and perform first some algebraic manipulations with the interme-diate expressions appearing in (4.12). First, we clearly have J D ( − ∆ D ) − =( − ∆ D − α ) − . Second, employing (2.4) and the identity ( − ∆ N )( − ∆ ⊥ N ) − = I − χ N h χ N , ·i , we check (cid:2) J D p ∗ ( − ∆ ⊥ N ) − (cid:3) ∗ = p ( − ∆ D − α ) − , (cid:2) J N p ( − ∆ D ) − (cid:3) ∗ = p ∗ ( − ∆ N − α ) − . Finally, again using (2.4), we verify the intertwining relation [ p ( − ∆ D − α ) − ] ∗ = p ∗ ( − ∆ N − α ) − . Summing up, with our choice (4.11), formula (4.5) simplifiesto Θ = I + C φ h φ , ·i + α ( − ∆ N − α ) − + α ( − ∆ D − α ) − + α p ( − ∆ D − α ) − + α p ∗ ( − ∆ N − α ) − . (4.13)Now it is easy to substitute (2.6) and after elementary manipulations to concludewith Θ = I + K , where K is an integral operator with the kernel K ( x, y ) = α e − i α ( y − x ) (cid:2) tan( αa ) − i sgn( y − x ) (cid:3) . (4.14)The operator C can be found easily by composing P and Θ. We finally arriveat the formula C = P + L , where L is an integral operator with the kernel L ( x, y ) = α e − i α ( y + x ) (cid:2) tan( αa ) − i sgn( y + x ) (cid:3) . (4.15) Next we present an example of operator Ω, defined in (3.1) with e n := χ Nn , thatwill be used to find the similar self-adjoint operator h from (1.3). We recall19hat the similarity transformation Ω is invertible if all the eigenvalues of H aresimple, which is ensured by the condition α = k n for every n ∈ N . We willactually search for the quadratic form associated to h for which we have theresult in Corollary 3.7.We follow the analogous strategy to obtain formula for Ω as in the proof ofProposition 4.1. The definition of Ω with e n := χ Nn leads to the sum:Ω = χ N h φ , ·i + ∞ X n =1 χ Nn h χ Nn , ·i − i α ∞ X n =1 k n χ Nn h χ Dn , ·i = I + χ N h φ , ·i − χ N h χ N , ·i + αp ∞ X n =1 k n χ Dn h χ Dn , ·i = I + χ N h φ , ·i − χ N h χ N , ·i + αp ( − ∆ D ) − , (4.16)where we have used identities (2.5). In the same manner, we obtain the resultfor the inverse Ω − :Ω − = ψ h χ N , ·i + ∞ X n =1 k n k n − α χ Nn h χ Nn , ·i − i α ∞ X n =1 k n k n − α χ Dn h χ Nn , ·i = I + ψ h χ N , ·i + α ( − ∆ N − α ) − − αp ∗ ( − ∆ N − α ) − . (4.17)The operators L , M appearing in the expressions for Ω = I + L and Ω − = I + M are, as expected, integral operators with the kernels L , M that can be easilyobtained using formulae for the Neumann and Dirichlet resolvents (2.6)–(2.7): L ( x, y ) = i α a (cid:2) y − a sgn( y − x ) (cid:3) + 12 a (cid:16) e − i α ( y + a ) − (cid:17) , M ( x, y ) = αe i α ( a − x ) sin(2 αa ) − α e − i α ( x − y ) (cid:2) cot(2 αa ) − i sgn( y − x ) (cid:3) − αe − i α ( x + y ) αa ) . (4.18)To find the similar self-adjoint operator (1.3), we start from the quadraticform (3.19). Inserting (4.18) into the latter and performing several integrationsby parts with noticing that LM = − L − M and ( M ψ ) ′ = − i αM ψ − i αψ resultsin: t h [ ψ ] = k ψ ′ k + α |h χ N , ψ i| . (4.19)The corresponding operator h reads: hψ = − ψ ′′ + α χ N h χ N , ψ i , Dom ( h ) = (cid:8) ψ ∈ W , ( − a, a ) : ψ ′ ( ± a ) = 0 (cid:9) . (4.20)We remark that h is a rank one perturbation of the Neumann Laplacian. Theeigenfunctions of h are χ Nn with χ N corresponding to the eigenvalue α .It is interesting to compare the spectra of H and h for α = k n , i.e. in thepoints where the spectra are degenerate and similarity transformation breaks20own because the operator Ω is not invertible. k n is an eigenvalue with thealgebraic multiplicity two for both H and h . However, the geometric multiplicitydiffers: it is one for H and two for h .The form of h also explains the origin of the peculiar α -dependence of theeigenvalues of H (which are all constant except for λ ( α ) = α ). In fact, it isthe nature of the rank one perturbation to leave all the Neumann eigenvaluesuntouched except for the lowest one that is driven to the α behavior. Finally, we consider the general PT -symmetric boundary conditions c ± := i α ± β , with α, β ∈ R . We start with formal considerations. The Θ-self-adjointnessof H can be expressed in the following way. We take the advantage of therealization of Θ = I + K , which we insert into Θ Hψ = H ∗ Θ ψ , ψ ∈ Dom ( H ). Aformal interchange of differentiation with integration and integration by partsyield following problem that we can understand in distributional sense:( ∂ x − ∂ y ) K ( x, y ) = 0 , (4.21) ∂ y K ( x, ± a ) + (i α ± β ) K ( x, ± a ) = 0 . (4.22)Moreover, Θ ψ must belong to Dom ( H ∗ ), from which we have a condition ∂ x K ( ± a, y ) + ( − i α ± β ) K ( ± a, y ) = 2i αδ ( y ∓ a ) . (4.23)Here δ denotes the Dirac delta function.Already presented examples of Θ for β = 0 satisfy these requirements, par-ticularly K solves the wave equation (4.21). The kernel (4.14), correspondingto the simpler form of presented metric operators, is a function of x − y only.Inspired by this, we find the solution of the wave equation K ( x, y ) = e i α ( x − y ) − β | x − y | (cid:2) c + i α sgn( x − y ) (cid:3) , c ∈ R , (4.24)that satisfies the “boundary conditions” (4.22) and (4.23) as well. The oneparametric family of solutions (4.24) of (4.21)–(4.23) demonstrates the knownnon-uniqueness of solutions to this problem. We also remark that c can be takenas α or a dependent as well.The positivity of Θ is ensured if the norm of K is smaller than 1. This canbe estimated by the Hilbert-Schmidt norm of K which is explicitly computable: k K k = ( c + α ) 4 aβ + e − aβ − β . (4.25)Consequently, the positivity of Θ can be achieved by several ways, e.g. , if a issmall; or if β is positive and large; or | c | and | α | are small. In any of the regimes,the formal manipulations above are justified.21 Bounded perturbations
In this section we show that results of Section 3 remain valid if we consider abounded perturbation V of H .Firstly we remark that the perturbation result [11, Thm. XIX 2.7] guar-antees that H + V remains a discrete spectral operator. That is, if all theeigenvalues of H + V are simple, then the metric operator Θ exists. We showthat the claim of Theorem 3.2 is valid for H + V as well. The rest of the resultsfrom Section 3 then follows straightforwardly.Our approach is to use analytic perturbation theory for the operator h :=Ω H Ω − that is perturbed by a bounded operator Ω V Ω − . We denote by ξ n , η n the eigenfunctions of H + V and H ∗ + V ∗ , respectively. Let e n be elements ofany orthonormal basis in H . Theorem 5.1.
Let all the eigenvalues of H be simple and let V be a boundedoperator. If all eigenvalues of H + V are simple, then Ω V = P ∞ n =0 e n h η n , ·i , i.e. Ω V : ξ n e n , can be expressed as Ω V = U + L, (5.1) where U is a unitary operator and L is a Hilbert-Schmidt operator.Proof. As in the proof of Theorem 3.2, without loss of generality, we restrictourselves to e n := χ Nn and we show that Ω V = I + L with L being Hilbert-Schmidt. We consider the normal operator h := Ω H Ω − and we perturb it by v := Ω V Ω − . More specifically, we construct h ( ε ) := h + ε v forming a holomor-phic family of type ( A ) with respect to the parameter ε . We denote by µ n ( ε ) ,µ n ( ε ) the eigenvalues and by ˜ ξ n ( ε ), ˜ η n ( ε ) the corresponding eigenfunctions of h ( ε ) and of h ( ε ) ∗ respectively. h (0), h (0) ∗ are normal, therefore the eigenfunc-tions ˜ ξ n (0) and ˜ η n (0) form orthonormal bases. In fact, with our choice of e n ,˜ ξ n (0) = ˜ η n (0) = χ Nn .We construct operator ˜Ω : ˜ ξ n (1) χ Nn and we will show that ˜Ω = I + ˜ L ,where ˜ L is Hilbert-Schmidt. Ω V is the composition of Ω and ˜Ω and the claimthen follows easily using of the fact that Hilbert-Schmidt operators are a *-both-sided ideal.The distance of µ n (0) and µ n (1) can be at most k v k . Since we know theasymptotics of µ n (0) = λ n , see (3.5), it is clear that there exists N such thatfor all n > N , | µ n +1 (1) − µ n (1) | > n holds. Moreover, for such n the radius ofconvergence of perturbation series for eigenvalues and eigenfunctions is largerthan 1. Thus, we have ˜ η n ( ε ) = χ Nn + ∞ X j =1 ˜ η ( j ) n ε j . (5.2)We estimate the norms of ˜ η ( j ) n using the analytic perturbation theory: k ˜ η ( j ) n k ≤ π I Γ n (cid:13)(cid:13) ( h (0) ∗ − E ) − ( v ∗ ( h (0) ∗ − E ) − ) j χ Nn (cid:13)(cid:13) d E ≤ π I Γ n j +1 k v k j n j +1 d E ≤ c j n j , (5.3)22here Γ n is a circle around µ n (0) of radius n/ c does notdepend on n . We define N as such that N ≥ N and c/N < ∗ = P ∞ n =0 ˜ η n (1) h χ Nn , ·i can be written as ˜Ω ∗ = I + ˜ L ∗ N + ˜ L ∗∞ , where˜ L ∗ N := N − X n =0 (˜ η n (1) − χ Nn ) h χ Nn , ·i , ˜ L ∗∞ := ∞ X n = N ∞ X j =1 ˜ η ( j ) n h χ Nn , ·i , (5.4)and ˜ L ∗ N and ˜ L ∗∞ are Hilbert-Schmidt. The decomposition of ˜Ω ∗ follows im-mediately if we consider the expansions (5.2) for n > N and rewrite ˜ η n (1) = χ Nn + (˜ η n (1) − χ Nn ) for n ≤ N . ˜ L ∗ N is a finite rank operator therefore it isobviously Hilbert-Schmidt. ˜ L ∗∞ is bounded and the defining sum is absolutelyconvergent since ∞ X n = N ∞ X j =2 k ˜ η ( j ) n k|h χ Nn , ψ i| ≤ k ψ k ∞ X n = N ∞ X j =2 (cid:16) cn (cid:17) j ≤ k ψ k ∞ X n = N c n − nc , ∞ X n = N k ˜ η (1) n k|h χ Nn , ψ i| ≤ c vuut ∞ X n = N n vuut ∞ X n = N |h χ Nn , ψ i| ≤ c k ψ k vuut ∞ X n = N n . (5.5)Finally we estimate the Hilbert-Schmidt norm of ˜ L ∗∞ : ∞ X p =0 * ∞ X m = N ∞ X i =1 ˜ η ( i ) m h χ Nm , χ Np i , ∞ X n = N ∞ X j =1 ˜ η ( j ) n h χ Nn , χ Np i + ≤ ∞ X p = N ∞ X i =1 (cid:18) cp (cid:19) i ∞ X j =1 (cid:18) cp (cid:19) j ≤ ∞ X p = N (cid:18) cp − c (cid:19) < ∞ . (5.6)This concludes the proof of the theorem. Remark . Let us conclude this sectionby a remark on how to extend the previous result on bounded perturbations V for the operator H in the general form Hψ := − ( ρψ ′ ) ′ + V ψ on L ( − a, a ) , subject to the boundary conditions ρ ( ± a ) ψ ′ ( ± a ) + c ± ψ ( ± a ) = 0 . (5.7)Assuming merely that ρ is a bounded and uniformly positive function, i.e. ,there exists a positive constant C such that C − ≤ ρ ( x ) ≤ C for all x ∈ ( − a, a ),the operator can be defined ( cf. [8, Corol. 4.4.3]) as an m-sectorial operatorassociated with a closed sectorial form with domain W , ( − a, a ). If, in addition,we assume that ρ ∈ W , ∞ ( − a, a ), then it is possible to check that the domainof H consists of functions ψ from the Sobolev space W , ( − a, a ) satisfying (5.7).23ow, let us strengthen the regularity hypothesis to ρ ∈ W , ∞ ( − a, a ) and in-troduce the unitary (Liouville) transformation U : L ( − a, a ) → L ( f ( − a ) , f ( a ))by U − φ := ρ − / φ ◦ f , where f ( x ) := Z x dξ p ρ ( ξ ) . Then it is straightforward to check that the unitarily equivalent operator ˜ H := U H U − on L ( f ( − a ) , f ( a )) satisfies˜ Hφ = − φ ′′ + ˜ V φ + W φ,
Dom ( ˜ H ) = (cid:8) φ ∈ W , (cid:0) f ( − a ) , f ( a ) (cid:1) : φ ′ ( ± f ( a )) + ˜ c ± φ ( ± f ( a )) = 0 (cid:9) , where ˜ V := U V U − and˜ c ± := c ± ρ ( ± a ) / − ρ ′ ( ± a ) ρ ( ± a ) / , W := (cid:18) ρ ′′ − ρ ′ ρ (cid:19) ◦ f − . In this way, we have transformed the second-order perturbation represented by ρ into a bounded potential W and modified boundary conditions. Theorem 5.1applies to ˜ H and, as a consequence of the unitary transform U , to H as well. In this article, we investigated the structure of similarity transformations Ω andmetric operators Θ for Sturm-Liouville operators with separated, Robin-typeboundary conditions. The main result is that Ω and Θ can be expressed as asum of the identity and an integral Hilbert-Schmidt operator.We would like to emphasize that this not always the case for other typesof operators, see, e.g. , [2, 32, 24, 14], where Θ is a sum of the identity anda bounded non-compact operator. The latter is a composition of the parityand the multiplication by sign function. Moreover, corresponding similaritytransformations map (non-self-adjoint) point interactions to (self-adjoint) pointinteractions, which is not typically the case for operators studied here. Thisis illustrated in the example of PT -symmetric boundary conditions where theequivalent self-adjoint operator is not a point interaction but rather a rank oneperturbation of the Neumann Laplacian.In this work we considered the separated boundary conditions only. Nonethe-less, the analogous results are expected to be valid for all strongly regular bound-ary conditions.As the proofs of the results show, the crucial property is the asymptotics ofeigenvalues, i.e. separation distance of eigenvalues tends to infinity, that is usedfor the proof of the existence of similarity transformations [11]. Recent results onbasis properties for perturbations of harmonic oscillator type operators [1, 30, 3]give a possibility to investigate the structure of similarity transformation in thesecases as well. Another step is to extend the results e.g. on Hill operators, where24 criterion on being spectral operator of scalar type has been obtained in [13]and recently extended in [10].On the other hand, the structure of similarity transformations for operatorswith continuous spectrum as well as for multidimensional Schr¨odinger operatorsis almost unexplored and constitutes thus a challenging open problem.We illustrated the results by an example of PT -symmetric boundary condi-tions, where we found all the studied objects in a closed formula form, which ishardly the case in more general situations. However, in general, we may searchfor approximations of Ω or Θ, typically applying the analytic perturbation the-ory to find perturbation series for eigenvalues and eigenfunctions of H to certainorder k . For instance, we perturb the parameters c ± in boundary conditionsby small ε . As a result we find an approximation h app of the similar operator h with resolvents satisfying k ( h − z ) − − ( h app − z ) − k ≤ Cε k . An extensivediscussion and example of such construction can be found in [34]. The sameremark is appropriate for small perturbations by bounded operator discussed inSection 5. Acknowledgement
D.K. acknowledges the hospitality of the Deusto Public Library in Bilbao.This work has been partially supported by the Czech Ministry of Education,Youth, and Sports within the project LC06002 and by the GACR grant No.P203/11/0701. P.S. appreciates the support by GACR grant No. 202/08/H072and by the Grant Agency of the Czech Technical University in Prague, grantNo. SGS OHK4-010/10. J.ˇZ. appreciates the support by the Czech Ministry ofEducation, Youth, and Sports within the project LC527.
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