aa r X i v : . [ m a t h . SP ] A p r Canonical systems with discrete spectrum
Roman Romanov ∗ Harald Woracek ‡ Abstract:
We study spectral properties of two-dimensional canonical systems y ′ ( t ) = zJH ( t ) y ( t ), t ∈ [ a, b ), where the Hamiltonian H is locally integrable on [ a, b ),positive semidefinite, and Weyl’s limit point case takes place at b . We answer thefollowing questions explicitly in terms of H :Is the spectrum of the associated selfadjoint operator discrete ?If it is discrete, what is its asymptotic distribution ?Here asymptotic distribution means summability and limit superior conditionsrelative to comparison functions growing sufficiently fast. Making an analogy withcomplex analysis, this correponds to convergence class and type w.r.t. proximateorders having order larger than 1. It is a surprising fact that these properties dependonly on the diagonal entries of H .In 1968 L.de Branges posed the following question as a fundamental problem:Which Hamiltonians are the structure Hamiltonian of somede Branges space ?We give a complete and explicit answer. AMS MSC 2010:
Keywords: canonical system, discrete spectrum, eigenvalue distribution, operatorideal, Volterra operator, de Branges space
We study the spectrum of the selfadjoint model operator associated with a two-dimensional canonical system y ′ ( t ) = zJH ( t ) y ( t ) , t ∈ [ a, b ) . (1.1)Here H is the Hamiltonian of the system, −∞ < a < b ≤ ∞ , J is the symplecticmatrix J := (cid:16) −
11 0 (cid:17) , and z ∈ C is the eigenvalue parameter. We assumethroughout that H satisfies ⊲ H ∈ L (cid:0) [ a, c ) , R × (cid:1) , c ∈ ( a, b ), and { t ∈ [ a, b ) : H ( t ) = 0 } has measure 0, ⊲ H ( t ) ≥ t ∈ [ a, b ) a.e. and R ba tr H ( t ) dt = ∞ .Differential equations of this form orginate from Hamiltonian mechanics, andappear frequently in theory and applications. Various kinds of equations canbe rewritten to the form (1.1), and several problems of classical analysis canbe treated with the help of canonical systems. For example we mentionSchr¨odinger operators [ remling:2002 ], Dirac systems [ sakhnovich:2002 ], orthe extrapolation problem of stationary Gaussian processes via Bochners the-orem [ krein.langer:2014 ]. Other instances can be found, e.g., in [ kac:1999 ; kac:1999a ], [ kaltenbaeck.winkler.woracek:bimmel ], [ akhiezer:1961 ], or[ arov.dym:2008 ]. ‡ The second author was supported the project P 30715–N35 of the Austrian Science Fund. gohberg.krein:1967 ; debranges:1968 ]. Recent references are[ remling:2018 ; romanov:1408.6022v1 ].With a Hamiltonian H a Hilbert space L ( H ) is associated, and in L ( H )a selfadjoint operator A [ H ] is given by the differential expression (1.1) and byprescribing the boundary condition (1 , y ( a ) = 0 (in one exceptional situation A [ H ] is a multivalued operator, but this is only a technical difficulty). Thisoperator model behind (1.1) was given its final form in [ kac:1983 ; kac:1984 ].A more accessible reference is [ hassi.snoo.winkler:2000 ], and the relationwith de Branges’ work on Hilbert spaces of entire functions was made explicitin [ winkler:1993 ; winkler:1995 ].In the present paper we answer the following questions: Is the spectrum of A [ H ] discrete ?If σ ( A [ H ] ) is discrete, what is its asymptotic distribution ? The question about asymptotic distribution is understood as the problem offinding the convergence exponent and the upper density of eigenvalues in termsof the Hamiltonian.
Discreteness of the spectrum
In our first theorem we characterise discreteness of σ ( A [ H ] ). Let H = (cid:16) h h h h (cid:17) be a Hamiltonian on [ a, b ) and assume that R ba h ( s ) ds < ∞ . Then σ ( A [ H ] ) is discrete if and only if lim t ր b (cid:16) Z bt h ( s ) ds · Z ta h ( s ) ds (cid:17) = 0 . (1.2) The assumption that R ba h ( s ) ds < ∞ in Theorem 1 . / ∈ σ ess ( A [ H ] ) is that there exists some angle φ ∈ R such that R ba (cos φ, sin φ ) H ( s )(cos φ, sin φ ) ∗ ds < ∞ . Second, applying rotation isomor-phism always allows to reduce to the case that φ = 0. We will give details inSection 5.2. ♦ Let us remark that Theorem 1 . kac.krein:1958 ], of[ remling.scarbrough:1811.07067v1 ], and of [ kac:1995 ]. Structure Hamiltonians of de Branges spaces
Recall that a de Branges space H ( E ) is a reproducing kernel Hilbert space ofentire functions with certain additional properties, whose kernel is generated bya Hermite-Biehler function E . For each de Branges space there exists a uniquemaximal chain of de Branges subspaces H ( E t ), t ≤
0, contained isometrically(on exceptional intervals only contractively) in H ( E ). The generating Hermite-Biehler functions E t satisfy a canonical system on the interval ( −∞ ,
0] withsome Hamiltonian H , and this Hamiltonian is called the structure Hamiltonianof H ( E ). 2.de Branges identified in [ debranges:1961 ] a particular class of Hamilto-nians which are structure Hamiltonians of de Branges spaces. A mild general-isation of de Branges’ theorem can be found in [ linghu:2015 ], and a furtherclass of structure Hamiltonians is identified (in a different language) by the al-ready mentioned work of I.S.Kac and M.G.Krein [ kac.krein:1958 ] and its mildgeneralisation [ remling.scarbrough:1811.07067v1 ]. These classes do by farnot exhaust the set of all structure Hamiltonians. In 1968, after having finalisedhis theory of Hilbert spaces of entire functions, de Branges posed the followingquestion as a fundamental problem, cf. [ debranges:1968 ]: Which Hamiltonians H are the structure Hamiltonian of somede Branges space H ( E ) ? In the following decades there was no significant progress towards a solution ofthis question. One result was claimed by I.S.Kac in 1995; proofs have never beenpublished. He states a sufficient condition and a (different) necessary conditionfor H to be a structure Hamiltonian. Unfortunately, his conditions are difficultto handle.The connection with Theorem 1 . H ( E ), if and only if the oper-ator A [ ˜ H ] associated with the reversed Hamiltonian˜ H ( t ) := (cid:16) − (cid:17) H ( − t ) (cid:16) − (cid:17) , t ∈ [0 , ∞ ) , has discrete spectrum. This can be seen by a simple “juggling with fundamentalsolutions”–argument. A proof based on a different argument was published in[ kac:2007 ], see also [ linghu:2015 ].Hence we obtain from Theorem 1 . Summability properties
We turn to discussing the asymptotic distribution of σ ( A [ H ] ). Consider a Hamil-tonian H with discrete spectrum. Then its spectrum is a (finite or infinite)sequence of simple eigenvalues without finite accumulation point. If σ ( A [ H ] )is finite, any questions about the asymptotic behaviour of the eigenvalues areobsolete. Moreover, under the normalisation that R ba h ( s ) ds < ∞ , the point0 is not an eigenvalue of A [ H ] . Hence, we can think of σ ( A [ H ] ) as a sequence( λ n ) ∞ n =1 of pairwise different real numbers arranged such that0 < | λ | ≤ | λ | ≤ | λ | ≤ . . . (1.3)In our second theorem we characterise summability of the sequence ( λ − n ) ∞ n =1 relative to suitable comparison functions. In particular, this answers the ques-tion whether ( λ − n ) ∞ n =1 ∈ ℓ p when p >
1. The only known result in this directionis [ kaltenbaeck.woracek:hskansys ], which settles the case p = 2; we reobtainthis theorem.As comparison functions we use functions g defined on the ray [0 , ∞ ) andtaking values in (0 , ∞ ) which satisfy: ⊲ The limit ρ g := lim r →∞ log g ( r )log r exists and belongs to (0 , ∞ ).3 The function g is continuously differentiable with g ′ ( r ) >
0, andlim r →∞ r g ′ ( r ) g ( r ) = ρ g .Functions of this kind are known as growth functions; the number ρ g is calledthe order of g . They form a comparison scale which is finer than the scaleof powers r ρ . The history of working with growth scales other than powersprobably starts with the paper [ lindeloef:1905 ], where E.Lindel¨of comparedthe growth of an entire function with functions of the form g ( r ) = r ρ · (cid:0) log r (cid:1) β · (cid:0) log log r (cid:1) β · . . . · (cid:0) log · · · log | {z } m th -iterate r (cid:1) β m for r large . In what follows the reader may think of g ( r ) for simplicity as a concrete functionof this form, or simply as a power r ρ . Let H = (cid:16) h h h h (cid:17) be a Hamiltonian on an interval [ a, b ) suchthat R ba h ( s ) ds < ∞ and that A [ H ] has discrete spectrum. Moreover, assumethat h does not vanish a.e. on any interval ( c, b ) with c ∈ ( a, b ) . Let g be agrowth function with order ρ g > . Then ∞ X n =1 g ( | λ n | ) < ∞ ⇔ Z ba (cid:20) g (cid:18)(cid:16) Z bt h ( s ) ds · Z ta h ( s ) ds (cid:17) − (cid:19)(cid:21) − · h ( t ) dt R bt h ( s ) ds < ∞ . For the same reasons as explained in Remark 1 .
2, the assumptionthat R ba h ( s ) ds < ∞ is just a normalisation and no loss in generality. Alsothe assumption that h cannot vanish a.e. on any interval ( c, b ) is no loss ofgenerality. The reason being that, if h does vanish on an interval of this form,then the Krein-de Branges formula, cf. [ krein:1951 ], [ debranges:1961 ], saysthat lim n →∞ λ + n n = lim n →∞ λ − n n = π Z ba p det H ( s ) ds, (1.4)where λ ± n denote the sequences of positive and negative, respectively, eigenvaluesarranged according to increasing modulus. In particular, the series P ∞ n =1 1 g ( | λ n | ) converges whenever ρ g > ♦ Let us note that, besides the condition for square summability given in[ kaltenbaeck.woracek:hskansys ], Theorem 1 . kac.krein:1958 ] and[ kac:1962 ], and of [ kac:1986 ] for orders between and 1. Limit superior properties
In our third theorem, we characterise lim sup–properties of the sequence( λ n ) ∞ n =1 , again relative to growth functions g with ρ g >
1. While the char-acterisations in Theorems 1 . . H , theconditions occurring in this context are somewhat more complicated. The rea-son for this is intrinsic, and manifests itself in the necessity to pass to thenonincreasing rearrangement of a certain sequence.4 .5 Theorem. Let H = (cid:16) h h h h (cid:17) be a Hamiltonian on an interval [ a, b ) suchthat R ba h ( s ) ds < ∞ and that A [ H ] has discrete spectrum. Moreover, assumethat h does not vanish a.e. on any interval ( c, b ) with c ∈ ( a, b ) . Let g be agrowth function with order ρ g > .Choose a right inverse χ of the nonincreasing surjection ( [ a, b ] → [0 , t (cid:16) R ba h ( s ) ds (cid:17) − (cid:16) R bt h ( s ) ds (cid:17) and let ( ω ∗ n ) n ∈ N be the nonincreasing rearrangement of the sequence ( ω n ) n ∈ N defined as ω n := 2 − n (cid:18) χ (2 − n ) Z χ (2 − n ) h ( s ) ds (cid:19) , n ∈ N . Then (i) lim sup n →∞ n g ( | λ n | ) < ∞ ⇔ lim sup n →∞ n g (( ω ∗ n ) − ) < ∞ , (ii) lim n →∞ n g ( | λ n | ) = 0 ⇔ lim n →∞ n g (( ω ∗ n ) − ) = 0 . Remember here Remark 1 . Outline of the proofs
The proof of Theorems 1 .
1, 1 .
3, and 1 . ➀ The first stage is to pass from eigenvalue distribution to operator theoreticproperties. This is done in a standard way using symmetrically normed operatorideals: discreteness of σ ( A [ H ] ) is equivalent to ( A [ H ] − z ) − being compact,summability properties of σ ( A [ H ] ) are equivalent to ( A [ H ] − z ) − belonging toOrlicz ideals, and lim sup–properties of σ ( A [ H ] ) are equivalent to ( A [ H ] − z ) − belonging to Lorentz spaces. ➁ The reader has certainly observed the – probably surprising – fact that theconditions in our theorems do not involve the off-diagonal entry h of the Hamil-tonian H . The second stage is to prove an Independence Theorem which saysthat membership of resolvents ( A [ H ] − z ) − in operator ideals I is indeed in-dependent of h , provided I possesses a certain additional property. This ad-ditional property is that a weak variant of Matsaev’s Theorem on real parts ofVolterra operators holds in I . ➂ In a work of A.B.Aleksandrov, S.Janson, V.V.Peller, and R.Rochberg, mem-bership in Schatten classes of integral operators whose kernel has a particularform is characterised using a dyadic discretisation method. The third stage isto realise that a minor generalisation of one of their results suffices to prove thementioned weak Matsaev Theorem in the ideal S ∞ of all compact operators.For Orlicz- and Lorentz ideals, it is known that (the full) Matsaev Theoremholds. Thus the Independence Theorem stated in ➁ will apply to all idealsoccurring in ➀ . 5 The final stage is to characterise membership in the mentioned ideals fora diagonal Hamiltonian (meaning that h = 0). This again rests on the dis-cretisation method from [ aleksandrov.janson.peller.rochberg:2002 ], whichyields characterisations of a sequential form (as the one stated in Theorem 1 . ➀ . For the cases of S ∞ and Orlicz ideals, sequentialcharacterisations can be rewritten to a continuous form (as stated in Theorems1 . . S ∞ , while for Orlicz ideals a little moreeffort and passing to dual spaces is needed. The threshold ρ g = 1 The Krein-de Branges formula (1.4) implies that for every Hamiltonian H whosedeterminant does not vanish a.e., the spectrum σ ( A [ H ] ), if discrete, satisfieslim inf n →∞ n | λ n | >
0. On the other hand, σ ( A [ H ] ) can be arbitrarily sparse ifdet H = 0 a.e. It is not difficult to find Hamiltonians H = (cid:16) h h h h (cid:17) whosespectrum is discrete and satisfies lim n →∞ n | λ n | ρ ∈ (0 , ∞ ) for some ρ <
1, butin the same time h h does not vanish a.e., see Example 1 .
7. For each suchHamiltonian and every Schatten-von Neumann ideal S p with ρ < p ≤
1, theIndependence Theorem mentioned in ➁ fails. This shows that our methodnecessarily must break down at (and below) trace class, i.e., growth of speed g ( r ) := r .On a less concrete level, growth of order 1 is a threshold because of (at least)four reasons. ⊲ Orders larger than 1, meaning eigenvalue distriubution more dense than inte-gers, can occur only from the behaviour of tails of H at its singular endpoint b . In fact, for ρ >
1, the spectrum of A [ H ] is discrete with convergence expo-nent ρ if and only if for some c ∈ ( a, b ) the spectrum of A [ H | [ c,b ) ] is discretewith convergence exponent ρ .Contrasting this, orders less than 1 will in general accumulate over the wholeinterval ( a, b ). In fact, it may happen that σ ( A [ H ] ) has convergence exponent1 while for every c ∈ ( a, b ) the spectrum of the tail H | [ c,b ) has convergenceexponent 0. ⊲ Entire functions of bounded type have very specific properties related toexponential type. In complex analysis, orders larger than 1 are usuallyconsidered as more stable than smaller orders. ⊲ The theory of symmetrically normed operator ideals is significantly morecomplicated for ideals close to trace class than for ideals containing someSchatten-von Neumann ideal S p with p >
1. When going even below traceclass, a lot of the theory breaks down completely. ⊲ Rewriting asymptotic conditions on the spectral distribution to conditionson membership in Orlicz- and Lorentz ideals is not anymore possible whencoming close to trace class.Let us now give two examples which illustrate our results. They are simple, andgiven by Hamiltonians related to a string, but, as we hope, still illustrative. Atthis point we only state their spectral properties; the proof is given in Section 5.3.6 .6 Example.
Given α > α , α ∈ R , we consider the Hamiltonian (toavoid bulky notation, we skip indices α, α , α at h ) H α ; α ,α ( t ) := (cid:18) h ( t ) (cid:19) , t ∈ [0 , , where h ( t ) := (cid:16) − t (cid:17) α (cid:16) − t (cid:17) − α (cid:16) + log 11 − t (cid:17) − α , t ∈ [0 , . (1.5)Since α >
1, we have R h ( t ) dt = ∞ .If α >
2, then 0 belongs to the essential spectrum of A [ H α ; α ,α ] , and if α ∈ (1 , n →∞ n | λ n | >
0, in particular, P ∞ n =1 1 | λ n | = ∞ .A behaviour between those extreme situations occurs when α = 2. First,the spectrum of A [ H α ,α ] is discrete, if and only if( α > ∨ ( α = 0 , α > . For such parameter values, the convergence exponent of the spectrum isconv.exp. of σ ( A [ H α ,α ] ) = ∞ , α = 0 , α , α ∈ (0 , , , α ≥ . (1.6)For α ∈ (0 , < lim sup n →∞ n | λ n | α (log | λ n | ) − α α < ∞ . ♦ Given α > α ∈ R consider the Hamiltonian (again indicesat h are skipped)˚ H α ,α := (cid:18) − p h ( t ) − p h ( t ) h ( t ) (cid:19) , t ∈ [0 , , where h is as in (1.5) with α = 2. Then σ ( A [˚ H α ,α ] ) is discrete, and itsconvergence exponent isconv.exp. of σ ( A [˚ H α ,α ] ) = ( α , α ∈ (0 , , , α ≥ . (1.7)The diagonalisation of ˚ H α ,α , i.e., the Hamiltonian obtained by skipping its off-diagonal entries, is H α ,α . Comparing the convergence exponents computedin (1.6) and (1.7), illustrates validity of the Independence Theorem from ➁ aslong as the convergence exponent is not less than 1, and its failure for othervalues. ♦ rganisation of the manuscript The structuring of this article is straightforward. We start off in Section 2 withproving the central Independence Theorem mentioned in ➁ . Section 3 containsthe proof of Theorem 1 .
1, and Section 4 the proofs of Theorems 1 . . R ba h ( s ) ds < ∞ , and providedetails for the Examples 1 . . ➂ and ➃ above, our arguments use a (very) minor gen-eralisation of the AJPR-results. This is established in just the same way asthe results of [ aleksandrov.janson.peller.rochberg:2002 ] with only a fewtechnical additions. For the convenience of the reader we provide full details inAppendix A. Moreover, in Appendix B, we provide detailed proof for some ele-mentary facts being used in the text, and in Appendix C we make the connectionof our Theorem 1 . kac:1995 ]. Let H be a Hilbert space and B ( H ) the set of all bounded linear operators on H .For an operator T ∈ B ( H ) we denote by a n ( T ) the n -th approximation numberof T , i.e., a n ( T ) := inf (cid:8) k T − A k : A ∈ B ( H ) , dim ran A < n (cid:9) , n ∈ N . The Calkin correspondence [ calkin:1941 ] is the map assigning to each T ∈B ( H ) the sequence ( a n ( T )) ∞ n =1 of its approximation numbers.An operator ideal I in H is a two-sided ideal of the algebra B ( H ). Everyproper operator ideal I contains the ideal of all finite rank operators, and,provided H is separable, is contained in the ideal S ∞ of all compact operators.Moreover, every operator ideal contains with an operator T also its adjoint T ∗ .Via the Calkin correspondence, operator ideals can be identified with certainsequence spaces. Recall [ garling:1967 ]: there is a bijection Seq of the set of alloperator ideals of H onto the set of all solid symmetric sequence spaces , suchthat for all T ∈ B ( H ) T ∈ I ⇔ ( a n ( T )) ∞ n =1 ∈ Seq( I ) . For example, the ideal S ∞ of all compact operators corresponds to c , the trivialideal B ( H ) to ℓ ∞ , and the Schatten–von Neumann classes S p to ℓ p . Taking theviewpoint of sequence spaces is natural in (at least) two respects. ⊲ It allows to compare ideals in B ( H ) for different base spaces H . A solidsymmetric sequence space S invokes the family of “same-sized” operatorideals (cid:8) T ∈ B ( H ) : ( a n ( T )) ∞ n =1 ∈ S (cid:9) , H Hilbert space . A linear subspace S of C N is called solid, if( α n ) ∞ n =1 ∈ S ∧ | β n | ≤ | α n | , n ∈ N ⇒ ( β n ) ∞ n =1 ∈ S , and it is called symmetric, if( α n ) ∞ n =1 ∈ S , σ permutation of N ⇒ ( α σ ( n ) ) ∞ n =1 ∈ S . Virtually all examples of operator ideals I which “appear in nature” aredefined by a specifying their sequence space Seq( I ).From now on we do not anymore distinguish between sequence spaces and op-erator ideals, and always speak of an operator ideal.A central role is played by integral operators whose kernel has a very specialform. Let −∞ ≤ a < b ≤ ∞ , and κ, ϕ : ( a, b ) → C be measurable functions suchthat κ ∈ L ( a, b ) and ( a,c ) ϕ ∈ L ( a, b ) for every c ∈ ( a, b ). Then we considerthe (closed, but possibly unbounded) integral operator T in L ( a, b ) with kernel T : tt ( t, s ) 0 (cid:19) H ( s ) f ( s ) ds, (2.2) on the domain dom B [ H ] = (cid:26) f ∈ L ( H ) : lim c ր b (0 , R ca JH ( s ) f ( s ) ds exists,r.h.s. of (2.2) belongs to L ( H ) (cid:27) . (2.3) ❑ Denote by L ( Idt ) the L -space of 2-vector valued functions on ( a, b ) with re-spect to the matrix measure (cid:16) (cid:17) dt . The functionΦ : f ( t ) H ( t ) f ( t )maps the model space L ( H ) isometrically onto some closed subspace of L ( Idt ).This holds since, by its definition, L ( H ) is a closed subspace of the L -space of2-vector valued functions on ( a, b ) with respect to the matrix measure H ( t ) dt .Let C [ H ] be the (closed, but possibly unbounded) integral operator on L ( Idt ) with kernel C [ H ] : − H ( t ) (cid:18) s
5, the operators B [ H ] and C [ H ] are togetherbounded or unbounded, and if they are bounded their approximation numberscoincide. Thus, for every operator ideal I , we have B [ H ] ∈ I ⇔ C [ H ] ∈ I . The following simple computation is a key step to the proof of Theorem 2 . Let H be a Hamiltonian on [ a, b ) with R ba h ( s ) ds < ∞ . Denote H ( t ) = (cid:18) v ( t ) v ( t ) v ( t ) v ( t ) (cid:19) , t ∈ [ a, b ) , and let T ij , ( i, j ) ∈ { , } × { , } , be the integral operators in L ( a, b ) withkernel T ij : ts ( t, s ) v ( t ) v ( s ) ts ( t, s ) v ( t ) v ( s ) ts ( t, s ) v ( t ) v ( s ) ts ( t, s ) v ( t ) v ( s ) The adjoint T ∗ ij is the integral operator with kernel T ∗ ij : t>s ( t, s ) v j ( t ) v i ( s )and the assertion follows. ❑ Let H = (cid:16) h h (cid:17) be a diagonal Hamiltonian, and let S be theintegral operator in L ( a, b ) with kernel S : t
Lemma 2 . C [ H ] = − (cid:18) S ∗ S (cid:19) . (2.5) ❑ For a bounded function ψ , we denote by M ψ the multiplication operator with ψ on L ( a, b ): ( M ψ f )( t ) := ψ ( t ) f ( t ) , k M ψ k = k ψ k ∞ . Proof of Theorem . . It holds that h = v + v and h = v + v , and hence0 ≤ v ≤ p h , ≤ v ≤ p h , | v | ≤ min { p h , p h } . Thus the functions (quotients are understood as 0 if their denominator vanishes) v √ h , v √ h , v √ h , v √ h , ψ := v √ h + v , ψ := v √ h + v , are all bounded. We have T = M v / √ h S M v / √ h , T = M v / √ h S M v / √ h ,T = M v / √ h S M v / √ h , T = M v / √ h S M v / √ h ,S = M ψ T M ψ + M ψ T + T M ψ + T , where the last relation holds since, by a short computation, p h j = (cid:16) v p h j + v j (cid:17) · v + v j , j = 1 , .
11e see that S ∈ I ⇔ T , T , T , T ∈ I . From this it readily follows that B [diag H ] ∈ I implies B [ H ] ∈ I .Assume that the weak Matsaev Theorem holds for I . If B [ H ] ∈ I , thenRe T , Re T , Re T + Re T ∈ I . The operator Re T is one-dimensional, hence certainly belongs to I , and itfollows that also Re T ∈ I . We conclude that T , T , T , and T , all belongto I . From this S ∈ I , and in turn B [diag H ] ∈ I . ❑ Theorem 1 . aleksandrov.janson.peller.rochberg:2002 ], namely Theorem 3 . −∞ ≤ a < b ≤ ∞ ,let κ, ϕ : ( a, b ) → C be measurable functions such that κ ∈ L ( a, b ) and ( a,c ) ϕ ∈ L ( a, b ) for every c ∈ ( a, b ). Then the function t
7→ k ( t,b ) κ k is anonincreasing surjection of [ a, b ] onto [0 , k κ k ]. Hence, we can choose an in-creasing sequence c := a < c < c < . . . < b such that k ( c n ,b ) κ k = 2 − n k κ k , n ∈ N . Note that this requirement is equivalent to k ( c n − ,c n ) κ k = (cid:16) (cid:17) n k κ k , n ∈ N . (3.1)Having chosen c n , we denote J n := ( c n − , c n ) , ω n := k J n κ k · k J n ϕ k , n ∈ N . (3.2)Explicitly, by (3.1), ω n = k κ k · − n (cid:16) Z c n c n − | ϕ ( s ) | ds (cid:17) , n ∈ N . Let −∞ ≤ a < b ≤ ∞ , let κ, ϕ : ( a, b ) → C be measurablefunctions with κ ∈ L ( a, b ) and ( a,c ) ϕ ∈ L ( a, b ) , c ∈ ( a, b ) , and consider theintegral operator T on L ( a, b ) with kernel (2.1) . Then T is compact Re T is compact lim n →∞ ω n = 0 where ω n are as in (3.2) . ❑ The proof of Theorem 3 . aleksandrov.janson.peller.rochberg:2002 ]. We therefore skip details fromthe main text; the reader can find the fully elaborated argument in Appendix A.Rewriting the sequential condition occuring from Theorem 3 . . .2 Lemma. Letting notation be as in Theorem . , we have lim n →∞ ω n = 0 ⇔ lim t ր b k ( a,t ) ϕ kk ( t,b ) κ k = 0 . ❑ Now Theorem 1 . Proof of Theorem . . Theorem 3 . I = c of all compact operators (remember that wedo not distinguish between a concrete operator ideal and its sequence space).Hence the Independence Theorem applies, and together with Corollary 2 . . H we find B [ H ] ∈ S ∞ ⇔ lim n →∞ ω n = 0 , (3.3)where ω n is buildt with κ ( t ) := p h ( t ), ϕ ( t ) := p h ( t ). By Lemma 3 . n →∞ ω n = 0 ⇔ lim t ր b (cid:16) Z bt h ( s ) ds · Z ta h ( s ) ds (cid:17) = 0 , and the proof of Theorem 1 . ❑ Using the connection between Krein strings and diagonal canonicalsystems elaborated in [ kaltenbaeck.winkler.woracek:bimmel ], the presentTheorem 1 . kac.krein:1958 ] fora string to have discrete spectrum. ♦ A symmetrically normed ideal I is a (two-sided) operator ideal which is endowedwith a norm k . k I , such that ⊲ ( I , k . k I ) is complete, ⊲ k AT B k I ≤ k A k · k T k I · k B k , T ∈ I , A, B ∈ B ( H ) , H Hilbert space, ⊲ k T k I = k T k for T with dim ran T = 1.Basic examples of symmetrically normed ideal are the Schatten-von Neumannideals S p , 1 ≤ p ≤ ∞ .Our standard reference about symmetrically normed ideals is[ gohberg.krein:1965 ]; another classical reference is [ schatten:1970 ].Recall that an operator T is called a Volterra operator, if it is compact and σ ( T ) = { } . Let I be a symmetrically normed ideal which is properly con-tained in S ∞ . We say that Matsaev’s Theorem holds in I , if the followingstatement is true. ⊲ Let H be a Hilbert space, and let T be a Volterra operator in H . ThenRe T ∈ I implies T ∈ I . ♦ .
1, we obtain the followingfact (which also justifies our terminology introduced in Definition 2 . Let I ( S ∞ be a symmetrically normed ideal. If Matsaev’sTheorem holds in I , then also the weak Matsaev Theorem holds in I . ❑ Consequently, for every proper symmetrically normed ideal in which Matsaev’sTheorem holds, the Independence Theorem applies.The characterisations stated in Theorems 1 . . aleksandrov.janson.peller.rochberg:2002 ] (proof details of this result aregiven in Appendix A).We use the notation introduced in Section 3, in particular recall (3.1) and(3.2). Moreover, recall that an operator ideal I is called fully symmetric, if foreach two nonincreasing sequences of nonnegative numbers ( α n ) ∞ n =1 and ( β n ) ∞ n =1 it holds that (cid:16) ( α n ) ∞ n =1 ∈ I ∧ ∀ n ∈ N . n X k =1 β k ≤ n X k =1 α k (cid:17) = ⇒ ( β n ) ∞ n =1 ∈ I Let −∞ ≤ a < b ≤ ∞ , let κ, ϕ : ( a, b ) → C be measurablefunctions with κ ∈ L ( a, b ) and ( a,c ) ϕ ∈ L ( a, b ) , c ∈ ( a, b ) , and consider theintegral operator T on L ( a, b ) with kernel (2.1) . Moreover, let I ( S ∞ be anoperator ideal. ⊲ If I is fully symmetric, then Re T ∈ I implies ( ω n ) ∞ n =1 ∈ I . ⊲ If I is symmetrically normed and Matsaev’s Theorem holds in I , then ( ω n ) ∞ n =1 ∈ I implies T ∈ I . ❑ To obtain the proof of Theorem 1 .
3, we use a particular class of symmetricallynormed ideals.
Let M : [0 , ∞ ) → [0 , ∞ ) be a continuous increasing and convexfunction with M (0) = 0, lim t →∞ M ( t ) = ∞ , and M ( t ) > t >
0. Assumethat lim sup t ց M (2 t ) M ( t ) < ∞ , and that M is normalised by M (1) = 1. The Orliczspace S J M K is the symmetrically normed ideal S J M K := n ( α n ) ∞ n =1 ∈ c : ∞ X n =1 M ( | α n | ) < ∞ o , k ( α n ) ∞ n =1 k S J M K := inf n β > ∞ X n =1 M (cid:16) | α n | β (cid:17) ≤ o . Orlicz ideals are separable; in fact the unit vectors e j := ( δ nj ) ∞ n =1 , j ∈ N , form anunconditional basis in S J M K . For a systematic treatment of this type of sequencespaces we refer to [ maligranda:1989 ] and [ lindenstrauss.tzafriri:1977 ]. ♦ .5 Remark. Given a growth function g with order ρ g >
1, set M ( t ) := 1 g ( t ) . In general, M will not be convex. However, based on[ bingham.goldie.teugels:1989 ], [ lelong.gruman:1986 ], we always findan equivalent growth function g , i.e., one with lim t →∞ g ( t ) g ( t ) = 1, such thatthe corresponding M satisfies all requirements made in Example 4 .
4. Then S J M K = n ( α n ) ∞ n =1 ∈ c : ∞ X n =1 g (cid:0) | α n | − (cid:1) < ∞ o , (4.1)and we may say that g induces an Orlicz ideal. ♦ Rewriting the sequential condition occuring from Theorem 3 . Letting notation be as above, we have ∞ X n =1 g ( ω − n ) < ∞ ⇔ Z ba h g (cid:16)(cid:0) k ( a,t ) ϕ kk ( t,b ) κ k (cid:1) − (cid:17)i − · | κ ( t ) | dt k ( t,b ) κ k < ∞ . ❑ Proof of Theorem . . The left and right sides of the equivalence asserted inTheorem 1 . g to an equivalent one. Hence, we may assume without loss ofgenerality that g gives rise to an Orlicz space S J M K as in (4.1).Since ρ g >
1, we can apply [ gohberg.krein:1967 ] and conclude that Mat-saev’s Theorem holds in S J M K . By Corollary 4 . S J M K , and hence the Independence Theorem applies. Clearly S J M K isfully symmetric, and Theorem 2 . . . B [ H ] ∈ S J M K ⇔ ∞ X n =1 g ( ω − n ) < ∞ , (4.2)where ω n is buildt with κ ( t ) := p h ( t ), ϕ ( t ) := p h ( t ). By Lemma 4 . ∞ X n =1 g ( ω − n ) < ∞ ⇔ Z ba (cid:20) g (cid:18)(cid:16) Z bt h ( s ) ds · Z ta h ( s ) ds (cid:17) − (cid:19)(cid:21) − · h ( t ) dt R bt h ( s ) ds < ∞ , and the proof of Theorem 1 . ❑ To obtain the proof of Theorem 1 .
5, we use another particular class of symmetri-cally normed ideals. In the following we denote for a sequence ( α n ) ∞ n =1 ∈ c , by( α ∗ n ) ∞ n =1 its nonnegative nonincreasing rearrangement, i.e., the sequence madeup of the elements | α n | arranged nonincreasingly.15 .7 Example. Let ( π n ) ∞ n =1 be a nonincreasing positive sequence with π = 1,lim n →∞ π n = 0, and P ∞ n =0 π n = ∞ . The Lorentz space S [ π ] is the symmetri-cally normed ideal S [ π ] := n ( α n ) ∞ n =1 ∈ c : sup n ∈ N (cid:16) n X k =1 α ∗ k . n X k =1 π k (cid:17) < ∞ o , k ( α n ) ∞ n =1 k S [ π ] := sup n ∈ N (cid:16) n X k =1 α ∗ k . n X k =1 π k (cid:17) . Lorentz spaces may or may not be separable, and we denote by S ◦ [ π ] the sep-arable part of S [ π ] . I.e., S ◦ [ π ] is the closure in S [ π ] of the linear subspaceof all finite rank operators. For this type of sequence spaces we refer to[ gohberg.krein:1965 ] or [ lord.sukochev.zanin:2013 ]. ♦ This type of symmetrically normed ideals correspond to the consideration oflimit superior conditions. Let g be a growth function with ρ g >
1, and set π n := g − ( n ) where g − is the inverse function of g . Then S [ π ] = (cid:8) ( α n ) ∞ n =1 : lim sup n →∞ α ∗ n g − ( n ) < ∞ (cid:9) , S ◦ [ π ] = (cid:8) ( α n ) ∞ n =1 : lim n →∞ α ∗ n g − ( n ) = 0 (cid:9) , cf. [ gohberg.krein:1965 ]. Proof of Theorem . . Since ρ g >
1, we can apply [ gohberg.krein:1967 ]and conclude that Matsaev’s Theorem holds in S [ π ] and in S ◦ [ π ] . By Corol-lary 4 . S J M K , and hence the Indepen-dence Theorem applies. Moreover, S [ π ] and S ◦ [ π ] are fully symmetric, cf.[ gohberg.krein:1965 ]. Now Theorem 2 . . . B [ H ] ∈ S [ π ] ⇔ lim sup n →∞ ω ∗ n g − ( n ) < ∞ ,B [ H ] ∈ S ◦ [ π ] ⇔ lim n →∞ ω ∗ n g − ( n ) = 0 , (4.3)where ω n is buildt with κ ( t ) := p h ( t ), ϕ ( t ) := p h ( t ).Matching notation shows that these numbers ω n are just the same as thenumbers written in Theorem 1 .
5. By a property of growth functions, it holdsthat lim sup n →∞ ω ∗ n g − ( n ) < ∞ ⇔ lim sup n →∞ n g (( ω ∗ n ) − ) < ∞ , lim n →∞ ω ∗ n g − ( n ) = 0 ⇔ lim n →∞ n g (( ω ∗ n ) − ) = 0 , and the proof is complete. ❑ Bounded invertibility, normalisation, and ex-amples
The present method also yields a condition for the model operator A [ H ] to beboundedly invertible. Let H = (cid:16) h h h h (cid:17) be a Hamiltonian on [ a, b ) and assume that R ba h ( s ) ds < ∞ . Then / ∈ σ ( A [ H ] ) if and only if lim sup t ր b (cid:16) Z bt h ( s ) ds · Z ta h ( s ) ds (cid:17) < ∞ . Remember here Remark 1 . remling.scarbrough:1811.07067v1 ].Again we use the notation introduced in Section 3, in particular recall (3.1)and (3.2). The proof of Theorem 5 . aleksandrov.janson.peller.rochberg:2002 ](proof details are given in Appendix A). Let −∞ ≤ a < b ≤ ∞ , let κ, ϕ : ( a, b ) → C be measurablefunctions with κ ∈ L ( a, b ) and ( a,c ) ϕ ∈ L ( a, b ) , c ∈ ( a, b ) , and consider theintegral operator T on L ( a, b ) with kernel (2.1) . Then T is bounded Re T is bounded sup n ∈ N ω n < ∞ where ω n are as in (3.2) . ❑ Rewriting the sequential condition occuring from Theorem 5 . Letting notation be as above, we have sup n →∞ ω n < ∞ ⇔ lim sup t ր b k ( a,t ) ϕ kk ( t,b ) κ k < ∞ . ❑ Proof of Theorem . . Theorem 5 . I = ℓ ∞ of all bounded operators. Hence the Inde-pendence Theorem applies, and together with Corollary 2 . . H we find B [ H ] ∈ B ( L ( H )) ⇔ lim sup n →∞ ω n < ∞ , (5.1)where ω n is buildt with κ ( t ) := p h ( t ), ϕ ( t ) := p h ( t ). By Lemma 5 . n →∞ ω n < ∞ ⇔ lim sup t ր b (cid:16) Z bt h ( s ) ds · Z ta h ( s ) ds (cid:17) < ∞ , and the proof of Theorem 5 . ❑ .2 The normalisation RRR ba h ( s ) ds < ∞ In this section we provide the arguments announced in Remark 1 .
2. Denoteby T min ( H ) and T max ( H ) the minimal and maximal operators induced by theequation (1.1), cf. [ hassi.snoo.winkler:2000 ]. First observe that, in the casesof present interest, the space L ( H ) always contains some constant. Assume that / ∈ σ ess ( A [ H ] ) . Then there exists φ ∈ R such that (cid:18) cos φ sin φ (cid:19) ∈ L ( H ) . (5.2)This follows from [ remling:2018 ] (use t = 0); for the convenience of the readerwe recall the argument. Proof of Lemma . . Since 0 / ∈ σ ess ( A [ H ] ), 0 is a point of regular type for T min ( H ). Thus there exists a selfadjoint extension ˜ A of T min ( H ) such that0 ∈ σ p ( ˜ A ) (see, e.g., [ gorbachuk.gorbachuk:1997 ]), and it follows thatker T max ( H ) = { } . This kernel, however, consists of all constant functionsin L ( H ). ❑ To achieve the normalisation R ba h ( s ) ds < ∞ , equivalently, φ = 0 in (5.2), oneuses rotation isomorphisms. Let α ∈ R , and denote N α := (cid:18) cos α sin α − sin α cos α (cid:19) . (i) For a Hamiltonian H defined on some interval [ a, b ), we set( (cid:9) α H )( t ) := N α H ( t ) N − α , t ∈ [ a, b ) . (ii) For a 2-vector valued function defined on some interval [ a, b ), we set( ω α f )( t ) := N α f ( t ) , t ∈ [ a, b ) . ♦ The following facts hold (see, e.g., [ kaltenbaeck.woracek:p5db ]): ⊲ (cid:9) α H is a Hamiltonian, ⊲ ω α induces an isometric isomorphism of L ( H ) onto L ( (cid:9) α H ), ⊲ T min ( (cid:9) α H ) ◦ ω α = ω α ◦ T min ( H ).Consequently, the Hamiltonians H and (cid:9) α H will share all operator theoreticproperties. ♦ In the context of diagonalisation, making a normalising rotation is inevitable.The reason being the following fact.
For a Hamiltonian H there exists at most one angle α modulo π such that L (diag (cid:9) α H ) contains a nonzero constant. roof. Let α ∈ [0 , π ), write ˜ H = (cid:18) ˜ h ˜ h ˜ h ˜ h (cid:19) := (cid:9) α H, and assume that φ ∈ R with ξ φ := (cid:18) cos φ sin φ (cid:19) ∈ L (diag ˜ H ) . Since ξ ∗ φ (cid:0) diag ˜ H (cid:1) ξ φ = ˜ h cos φ + ˜ h sin φ, and since H (and with it ˜ H and diag ˜ H ) is in limit point case, we conclude that φ ∈ { , π } . Thus, either (cid:0) (cid:1) or (cid:0) (cid:1) belongs to L (diag ˜ H ), and hence to L ( ˜ H ).From this we obtain that either ξ α or ξ α + π belongs to L ( H ). There exists atmost one angle φ modulo π such that ξ φ ∈ L ( H ), and therefore α is uniquelydetermined modulo π . ❑ . and . For Hamiltonians of a particularly simple form the conditions given in Theorems1 .
1, 1 .
3, and 1 . , h ( t ) = 1 a.e., and where h ( t ) grows sufficientlyregularly towards the singular endpoint 1 in the following sense. (i) A function ψ : [1 , ∞ ) → (0 , ∞ ) is called regularly varying with index ρ ψ ∈ R , if it is measurable and ∀ k > . lim x →∞ ψ ( kx ) ψ ( x ) = k ρ ψ . (ii) We call a function ϕ : [0 , → (0 , ∞ ) regularly varying at 1 with index ρ ∈ R , if the function ψ ( x ) := ϕ (cid:16) x − x (cid:17) : [1 , ∞ ) → (0 , ∞ )is regularly varying with index ρ . ♦ Let ϕ : [0 , → (0 , ∞ ) be continuous and regularly varying at with index ρ > , and set κ ( t ) := 1 , t ∈ [0 , . Then the numbers ω n constructedin (3.2) satisfy ω n ≍ − n ϕ (cid:0) − − n (cid:1) . Proof.
We have k ( t,b ) κ k = b − t , and hence the sequence ( c n ) ∞ n =0 is given as c n = 1 − − n , n = 0 , , , . . . . We write α n ≍ β n , if there exist c , c > c α n ≤ β n ≤ c α n , n ∈ N . ϕ is continuous, we find t n ∈ J n with ω n = 2 − n (cid:16) Z J n | ϕ ( s ) | ds (cid:17) = 2 − n ( c n − c n − ) ϕ ( t n ) = 2 − n ϕ ( t n ) . Set ψ ( x ) := ϕ (cid:16) x − x (cid:17) , x n := 11 − c n , y n := 11 − t n . Since x n − x n = , we have y n = k n x n with k n ∈ [ , bingham.goldie.teugels:1989 ], we obtain that (cid:16) (cid:17) ρ ≤ ψ ( y n ) ψ ( x n ) ≤ (cid:16) (cid:17) ρ , n sufficiently large . Passing back to ϕ, c n , t n , this yields ϕ ( t n ) ≍ ϕ ( c n ). ❑ Proof of Example . . We apply Lemma 5 . ϕ ( t ) := p h ( t ).This is justified, since the corresponding function ψ is ψ ( x ) = x α (1 + log x ) − α (1 + log + log x ) − α , and hence is regularly varying with index α . Therefore the numbers ω n whichdecide about the behaviour of the operator S satisfy ω n ≍ − n ϕ (1 − − n )= 2 − n · n α (1 + log 2 n ) − α (1 + log + log 2 n ) − α ≍ n ( α − n − α (log n ) − α . Using this relation, the stated spectral properties of H α ; α ,α follow immediatelyfrom the sequential characterisations given in (4.3), (3.3), (4.2), and (5.1). Letus go through the cases. ⊲ First of all the Krein-de Branges formula implieslim inf n →∞ n | λ n | ≥ Z p h ( s ) ds > . (5.3) ⊲ If ( α >
2) or ( α = 2 , α <
0) or ( α = 2 , α = 0 , α < n →∞ ω n = ∞ , and hence 0 belongs to the essential spectrum. ⊲ If ( α = 2 , α = α = 0), then ω n ≍
1, and hence the spectrum is not discrete,but bounded invertibility takes place. ⊲ If ( α <
2) or ( α = 2 , α >
0) or ( α = 2 , α = 0 , α > n →∞ ω n =0, and hence the spectrum is discrete. ⊲ If ( α = 2 , α > ω n ) ∞ n =1 equals α ,while in the case ( α = 2 , α = 0 , α > ω n ) ∞ n =1 is infinite. From this and (5.3) it follows that (for α = 2)conv.exp. of ( | λ n | ) ∞ n =1 = ∞ , α = 0 , α > , α , α ∈ (0 , , α ∈ R , , α ≥ , α ∈ R . If ( α = 2 , α ∈ (0 , , α ∈ R ) and g ( r ) := r α (log r ) γ , then g (cid:0) ω n (cid:1) ≍ g (cid:0) n α (log n ) α (cid:1) = (cid:2) n α (log n ) α (cid:3) α (cid:2) log (cid:0) n α (log n ) α (cid:1)(cid:3) γ ≍ n (log n ) α α + γ . This shows that for γ = − α α we have n · g ( ω n ) − ≍
1. Since the sequence( ω n ) ∞ n =1 is comparable to a monotone sequence, it follows that0 < lim sup n →∞ n g (( ω ∗ n ) − ) < ∞ . ❑ Proof of Example . . With a simple trick properties of ˚ H α ,α can be obtainedfrom Example 1 .
6. To explain this, we start in the reverse direction. Considerthe Hamiltonian H α ,α , and set˚ H α ,α ( t ) := (cid:18) − m ( t ) − m ( t ) m ( t ) (cid:19) , t ∈ [0 , , where m ( t ) := Z t h ( s ) ds. Moreover, let q be the Weyl-coefficient of H α ,α and ˚ q the one of ˚ H α ,α .Then, by [ kaltenbaeck.winkler.woracek:bimmel ], we have q ( z ) = 1 z ˚ q ( z ) . Thus the spectra of A [˚ H α ,α ] and A [ H α ,α ] are together discrete or not. Ifthese spectra are discrete, then the convergence exponent of σ ( A [ H α ,α ] ) istwice the convergence exponent of σ ( A [˚ H α ,α ] ). This yieldsconv.exp. of σ ( A [˚ H α ,α ] ) = ( α , α ∈ (0 , , , α ≥ . Integrating by parts gives lim t ր m ( t ) h ( t )(1 − t ) = 1 . The function ( h ( t )(1 − t )) is again of the form (1.5) with α = 2, but with theparameters 2 α and 2 α instead of α and α . Thus the spectrum of A [˚ H α ,α ] has the same asymptotic behaviour as the spectrum of A [˚ H α , α ] . ❑ ppendix A. Proof of AJPR-type theorems In this appendix we give a detailed proof of Theorems 3 .
1, 4 .
3, and 5 . a, b ), andmeasurable functions κ, ϕ : ( a, b ) → C with κ ∈ L ( a, b ) and ( a,c ) ϕ ∈ L ( a, b ), c ∈ ( a, b ), we consider the possibly unbounded integral operator T on L ( a, b )acting as ( T f )( t ) := ϕ ( t ) Z bt f ( s ) κ ( s ) ds, t ∈ ( a, b ) , on its natural maximal domain. Let c := a < c < c < . . . < b is a sequencewith k ( c n − ,c n ) κ k = (cid:16) (cid:17) n k κ k , n ∈ N , (A.1)and we denote J n := ( c n − , c n ) , ω n := k J n κ k · k J n ϕ k , n ∈ N . Moreover, let I be an operator ideal (remember that we do not distinguishnotationally between an operator ideal and its sequence space). The task is toprove the implications in the triangle T ∈ I Re T ∈ I ( ω n ) ∞ n =1 ∈ I trivial (A.2)in the following situations.(i) For Theorem 5 . I = ℓ ∞ .(ii) For Theorem 3 . I = c .(iii) For Theorem 4 . I ( S ∞ , assuming that I is fully symmetric for thedownwards implication, and assuming that I is symmetrically normedand Matsaev’s Theorem holds in I for the upwards implication. Proof of “ Re T ∈ I ⇒ ( ω n ) ∞ n =1 ∈ I ”. Let I be ℓ ∞ , c , or a fully symmetric operator ideal, and assume that Re T ∈ I .Moreover, denote by P n the orthogonal projection P n f := J n f of L ( a, b ) ontoits subspace L ( J n ).Since I is of one of the stated forms, we obtain that P ∞ n =1 P n (Re T ) P n +1 ∈ I .For I = ℓ ∞ or I = c , this is obvious. For I being fully symmetric, it is aconsequence of [ gohberg.krein:1965 ] . This is actually a variant of [ gohberg.krein:1965 ] which is easy to obtain in the presentsituation since all spaces L ( J n ) have the same Hilbert space dimension. Choose unitaryoperators U n : L ( J n ) → L ( J n +1 ), let S : L ( a, b ) → L ( a, b ) be the block shift S := U U
0. .. .. . : L ( J ) ⊕ L ( J ) ⊕ ... → L ( J ) ⊕ L ( J ) ⊕ ... , and apply [ gohberg.krein:1965 ] to the operator (Re T ) S . P n T P n +1 = ( , J n +1 κ ) J n ϕ . The adjoint of T is the integral oper-ator with kernel T ∗ : s Proof of “ ( ω n ) ∞ n =1 ∈ I ⇒ T ∈ I ”. Let I be ℓ ∞ , c , or a symmetrically normed ideal in which Matsaev’s Theo-rem holds, and assume that ( ω n ) ∞ n =1 ∈ I . Note that in every case ( ω n ) ∞ n =1 isbounded. ➀ The crucial point is to handle the diagonal cell sum P ∞ n =1 P n T P n . Our aimis to show that this series converges to an operator in I .The summand P n T P n is the integral operator in L ( a, b ) with kernel P n T P n : t In this section we give detailed proofs of Lemmas 3 . 2, 4 . 6, and 5 . 3. Recall therelevant notation: We are given a finite or infinite interval ( a, b ), and measurablefunctions κ, ϕ : ( a, b ) → C with κ ∈ L ( a, b ) and ( a,c ) ϕ ∈ L ( a, b ), c ∈ ( a, b ).Further, c := a < c < c < . . . < b is a sequence with k ( c n ,b ) κ k = (cid:16) (cid:17) n k κ k , equivalently, k ( c n − ,c n ) κ k = (cid:16) (cid:17) n k κ k , and J n := ( c n − , c n ) and ω n := k J n κ k · k J n ϕ k . Moreover, denoteΩ( t ) := k ( t,b ) κ k · k ( a,t ) ϕ k , t ∈ ( a, b ) . The proof of Lemma 3 . . Proof of Lemmas . and . . A sequence ( α n ) ∞ n =1 of nonnegative numbers isbounded (tends to 0), if and only if the sequence (cid:0) − n P nk =1 k α k (cid:1) ∞ n =1 isbounded (tends to 0, respectively). Applying this with α n := 2 − n Z c n c n − | ϕ ( s ) | ds, n ∈ N , yields the assertion. ❑ The proof of Lemma 4 . g with ρ g > g on a finite interval does not change the truthvalue of the left side of the asserted equivalence. Hence, we may assume w.l.o.g.that the function M ( t ) := g ( t − ) − has all properties required in Example 4 . . 5, and additionally that M ( t ) ≍ t ρ g , t ≥ 1. Note that, since ρ g > 1, lim t ց M ( t ) t = lim r →∞ r g ( r ) = 0 , lim t →∞ M ( t ) t = ∞ . Using the language of [ maligranda:1989 ] this means that M belongs to theclass N . B.1 Lemma. Set I := N and let q ∈ (0 , . (i) Set J := { ( n, k ) ∈ I × I : k ≤ n } and let S J M K denote the Orlicz space ofsequences indexed by I or by J depending on the context. For a sequence ( α n ) n ∈ I define sequences ( β ( n,k ) ) ( n,k ) ∈ J and ( β ′ ( n,k ) ) ( n,k ) ∈ J as β n,k := α n q n − k , β ′ ( n,k ) := α k q n − k , ( n, k ) ∈ J. If ( α n ) n ∈ I ∈ S J M K , then ( β ( n,k ) ) ( n,k ) ∈ J , ( β ′ ( n,k ) ) ( n,k ) ∈ J ∈ S J M K . Thereexists a constant C > such that max (cid:8) k ( β ( n,k ) ) ( n,k ) ∈ J k S J M K , k ( β ′ ( n,k ) ) ( n,k ) ∈ J k S J M K (cid:9) ≤ C k ( α n ) n ∈ I k S J M K , ( α n ) n ∈ I ∈ S J M K . (ii) Consider a sequence ( α n ) n ∈ I with ( q n α n ) n ∈ I ∈ S J M K , and define a se-quence ( β n ) n ∈ I as β n := X k ∈ Ik ≤ n α k , n ∈ I. Then ( q n β n ) n ∈ I ∈ S J M K . There exists a constant C > such that k ( q n β n ) n ∈ I k S J M K ≤ C k ( q n α n ) n ∈ I k S J M K , ( q n α n ) n ∈ I ∈ S J M K . Proof. For the proof of item (i) let ( α n ) n ∈ I ∈ S J M K with k ( α n ) n ∈ I k S J M K ≤ | α n | ≤ n ∈ I . Since ρ g > 1, we have C := sup <γ,t ≤ M ( γt ) γM ( t ) < ∞ , cf. [ bingham.goldie.teugels:1989 ]. Note that C ≥ 1. Thus we can estimate X ( n,k ) ∈ J M ( | β n,k | ) = X ( n,k ) ∈ J M ( | α n | q n − k ) ≤ X ( n,k ) ∈ J Cq n − k M ( | α n | )= C X n ∈ I (cid:16) X k ∈ Ik ≤ n q n − k | {z } ≤ − q (cid:17) M ( | α n | ) ≤ C − q X n ∈ I M ( | α n | ) ≤ C − q . Since | β n,k | ≤ | α n | ≤ C − q ≥ 1, it follows that X ( n,k ) ∈ J M (cid:16) | β n,k | C / (1 − q ) (cid:17) ≤ X ( n,k ) ∈ J C − qC M ( | β n,k | ) ≤ . k ( β ( n,k ) ) ( n,k ) ∈ J k S J M K ≤ C − q .The sequence ( β ′ ( n,k ) ) ( n,k ) ∈ J is handled in the same way. Namely X ( n,k ) ∈ J M ( β ′ n,k ) = X ( n,k ) ∈ J M ( | α k | q n − k ) ≤ X ( n,k ) ∈ J Cq n − k M ( | α k | )= C X k ∈ I (cid:16) X n ∈ In ≥ k q n − k | {z } = − q (cid:17) M ( | α k | ) = C − q X k ∈ I M ( | α k | ) ≤ C − q , from which we again obtain that k ( β ′ ( n,k ) ) ( n,k ) ∈ J k S J M K ≤ C − q .The proof of (ii) is based on dualising. For each γ > t ց M ( γt ) M ( t ) = lim r →∞ g ( γr ) g ( r ) = γ ρ g , cf. [ bingham.goldie.teugels:1989 ]. Hence the Orlicz indices of M at 0, cf.[ maligranda:1989 ], are both equal to ρ g . Let M ∗ be the Orlicz functioncomplementary to M , cf. [ maligranda:1989 ]. Then both Orlicz indices of M ∗ are equal to ρ g ρ g − , cf. [ maligranda:1989 ]. From [ maligranda:1989 ] we obtain C ∗ := sup <γ,t ≤ M ∗ ( γt ) γM ∗ ( t ) < ∞ . Now let ( σ n ) n ∈ I ∈ S J M ∗ K . Then we can use (i), the H¨older inequality[ maligranda:1989 ], and the relation [ maligranda:1989 ] between Amemiya-and Luxemburg norms, to estimate (cid:12)(cid:12)(cid:12) X n ∈ I σ n · q n β n (cid:12)(cid:12)(cid:12) ≤ X n ∈ I | σ n | q n X k ∈ Ik ≤ n | α k | = X ( n,k ) ∈ J | σ n | ( √ q ) n − k · (cid:0) q k | α k | (cid:1) ( √ q ) n − k ≤ (cid:13)(cid:13)(cid:0) σ n · ( √ q ) n − k (cid:1) ( n,k ) ∈ J (cid:13)(cid:13) S J M ∗ K (cid:13)(cid:13)(cid:0) q k α k · ( √ q ) n − k (cid:1) ( n,k ) ∈ J (cid:13)(cid:13) S J M K ≤ C ( C ∗ ) (1 − √ q ) k ( σ n ) n ∈ I k S J M ∗ K k ( q n α n ) n ∈ I k S J M K . By [ maligranda:1989 ] it follows that ( q n β n ) n ∈ I ∈ S J M K and k ( q n β n ) n ∈ I k S J M K ≤ C ( C ∗ ) (1 − √ q ) k ( q n α n ) n ∈ I k S J M K . ❑ Proof of Lemma . . For t ∈ J n it holds thatΩ( t ) ≥ k ( a,c n − ) ϕ kk ( c n ,b ) κ k ≥ k J n − ϕ kk J n +1 κ k = ω n − , ∞ X n =1 M (cid:16) ω n (cid:17) = ∞ X n =1 (cid:20) M (cid:16) ω n (cid:17) · Z J n +1 | κ ( t ) | dt k ( t,b ) κ k (cid:21) ≤ ∞ X n =1 (cid:20) Z J n +1 M (Ω( t )) · | κ ( t ) | dt k ( t,b ) κ k (cid:21) ≤ Z ba g (Ω( t ) − ) · | κ ( t ) | dt k ( t,b ) κ k . This shows that the implication “ ⇐ ” holds.Conversely, we have for t ∈ J n Ω( t ) ≤ k ( a,c n ) ϕ kk ( c n − ,b ) κ k ≤ (cid:16) n X k =1 k J k ϕ k (cid:17) · (cid:16) √ (cid:17) n − k κ k . Assume that ( ω n ) ∞ n =1 ∈ S J M K . Since ω n = k κ k (cid:0) √ (cid:1) n · k J n ϕ k , we can applyLemma B. α n := k J n ϕ k , n ∈ N . This shows that (cid:16)(cid:16) √ (cid:17) n n X k =1 k J k ϕ k (cid:17) ∞ n =1 ∈ S J M K , and we obtain Z ba M (Ω( t )) · | κ ( t ) | dt k ( t,b ) κ k = ∞ X n =1 Z J n +1 M (Ω( t )) · | κ ( t ) | dµ ( t ) k ( t,b ) κ k L ( µ ) ≤ log 2 ∞ X n =1 M (cid:16) k κ k (cid:16) √ (cid:17) n − n X k =1 k J k ϕ k (cid:17) < ∞ . ❑ Appendix C. I.S.Kac’s compactness theorem Let us recall [ kac:1995 ], which is stated as the main result in this paper. Un-fortunately, Kac’s original proofs are not available, and we do not know anysource where proofs are given. C.1 Theorem ([ kac:1995 ]) . Let H = (cid:16) h h h h (cid:17) be a Hamiltonian on [0 , ∞ ) which is normalised such that tr H ( t ) = 1 a.e., and denote m j ( t ) := R t h j ( t ) dt .For K > set A + K := (cid:26) λ ∈ R \{ } : lim sup t →∞ (cid:18) ∞ Z t h ( s ) e λm ( s ) ds · t Z h ( s ) e − λm ( s ) ds (cid:19) ≤ Kλ (cid:27) ,B + K := (cid:26) λ ∈ R \{ } : lim sup t →∞ (cid:18) ∞ Z t h ( s ) e − λm ( s ) ds · t Z h ( s ) e λm ( s ) ds (cid:19) ≤ Kλ (cid:27) . Then (i) ⇒ (ii) ⇒ (iii) , where (cid:16) [ K< A + K ∪ [ K< B + K (cid:17) = + ∞ , inf (cid:16) [ K< A + K ∪ [ K< B + K (cid:17) = −∞ . (ii) σ ( A [ H ] ) is discrete with R ∞ h ( s ) ds < ∞ or R ∞ h ( s ) ds < ∞ . (iii) A +1 ∪ B +1 = R \ { } . In the following theorem we make the connection to our present work. C.2 Theorem. Let notation be as in Theorem C. . Then the following itemsare equivalent. (i) There exists K > such that inf A + K = −∞ and sup A + K = + ∞ . (ii) σ ( A [ H ] ) is discrete with R ba h ( s ) ds < ∞ . (iii) \ K> A + K = R \ { } .The analogous statement holds when A + K is replaced by B + K and h by h . Concerning the normalisation in item (ii) remember again Remark 1 . 4. More-over, note that the implication “(iii) ⇒ (i)” is trivial. Proof of “ (i) ⇒ (ii) ”. Choose c ∈ (0 , ∞ ) such that h does not vanish almosteverywhere on [0 , c ], and consider the function F : [ c, ∞ ) × R → [0 , ∞ ] definedas F ( t, λ ) := Z ∞ t h ( s ) e λm ( s ) ds · Z t h ( s ) e − λm ( s ) ds. ➀ We show that for every finite interval [ µ , µ ] ⊆ R it holds that F ( t, λ ) ≤ max (cid:8) F ( t, µ ) , F ( t, µ ) , F ( t, µ ) , F ( t, µ ) , F ( t, µ ) F ( t, µ ) (cid:9) . (C.1)Let λ ∈ ( µ , µ ), and write λ = vµ + (1 − v ) µ with v ∈ (0 , ∞ Z t h ( s ) e λm ( s ) ds = ∞ Z t (cid:16) h ( s ) e µ m ( s ) (cid:17) v · (cid:16) h ( s ) e µ m ( s ) (cid:17) − v ds ≤ (cid:16) ∞ Z t h ( s ) e µ m ( s ) ds (cid:17) v · (cid:16) ∞ Z t h ( s ) e µ m ( s ) ds (cid:17) − v , and similarly t Z h ( s ) e − λm ( s ) ds ≤ (cid:16) t Z h ( s ) e − µ m ( s ) ds (cid:17) v · (cid:16) t Z h ( s ) e − µ m ( s ) ds (cid:17) − v . Multiplying these inequalites, and using the elementary inequality x v y − v ≤ max (cid:8) x, y, x , y , xy (cid:9) , x, y > , v ∈ (0 , , which is seen by distinguishing cases whether x, y are ≤ ≥ v ≤ or v ≥ , we obtain (C.1). This normalisation does not appear in [ kac:1995 ]. However, without it the statement isfalse. Assume now that (i) holds. Let 0 < ε < 1, and choose µ ∈ A + K ∩ ( −∞ ,