Canonical systems with discrete spectrum

TL;DR

This paper investigates the spectral properties of two-dimensional canonical systems with specific Hamiltonians, providing explicit conditions for the discreteness and distribution of the spectrum, and characterizes Hamiltonians related to de Branges spaces.

Contribution

It offers explicit criteria for the discreteness and asymptotic distribution of spectra based solely on Hamiltonian diagonal entries and characterizes Hamiltonians of de Branges spaces.

Findings

Spectrum is discrete under certain conditions on H

Asymptotic distribution depends only on diagonal entries of H

Complete characterization of Hamiltonians for de Branges spaces

Abstract

We study spectral properties of two-dimensional canonical systems $y'(t)=zJH(t)y(t)$, $t\in[a,b)$, where the Hamiltonian $H$ is locally integrable on $[a,b)$, positive semidefinite, and Weyl's limit point case takes place at $b$. We answer the following 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 conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this corresponds to convergence class and type w.r.t.\ proximate orders having order larger than $1$. It is a surprising fact that these properties depend only 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 some\\ de~Branges space ? We give a complete and explicit answer.

Equations

$$y^{\prime }(t)=zJH(t)y(t),t\in [a,b).$$

$$\lim_{t\nearrow b}\Big{(}\int_{t}^{b}h_{1}(s)\mkern 4.0muds\cdot\int_{a}^{t}h_{2}(s)\mkern 4.0muds\Big{)}=0.$$

$$\tilde{H}(t)\mathrel{\mathop{:}}=\Bigl{(}\begin{smallmatrix}1\hskip 0.60275pt&\hskip 0.60275pt0\\[2.15277pt] 0\hskip 0.60275pt&\hskip 0.60275pt-1\end{smallmatrix}\Bigr{)}H(-t)\Bigl{(}\begin{smallmatrix}1\hskip 0.60275pt&\hskip 0.60275pt0\\[2.15277pt] 0\hskip 0.60275pt&\hskip 0.60275pt-1\end{smallmatrix}\Bigr{)},\quad t\in[0,\infty),$$

$$0<|{\lambda }_1|\le |{\lambda }_2|\le |{\lambda }_3|\le \dots$$

$${\mathscr{g}}(r)=r^{\rho}\cdot\big{(}\log r\big{)}^{\beta_{1}}\cdot\big{(}\log\log r\big{)}^{\beta_{2}}\cdot\ldots\cdot\big{(}\underbrace{\log\cdots\log}_{\text{\footnotesize$m$\textsuperscript{th}-iterate}}r\big{)}^{\beta_{m}}\quad\text{for $r$ large}.$$

$$\sum_{n=1}^{\infty}\frac{1}{{\mathscr{g}}(|\lambda_{n}|)}<\infty\ \Leftrightarrow\quad\\ \int_{a}^{b}\bigg{[}{\mathscr{g}}\bigg{(}\Big{(}\int_{t}^{b}h_{1}(s)\mkern 4.0muds\cdot\int_{a}^{t}h_{2}(s)\mkern 4.0muds\Big{)}^{-\frac{1}{2}}\bigg{)}\bigg{]}^{-1}\mkern-5.0mu\cdot\mkern 1.0mu\frac{h_{1}(t)\mkern 4.0mudt}{\int_{t}^{b}h_{1}(s)\mkern 4.0muds}<\infty.$$

$$\underset{n\to \infty }{\lim }\frac{{\lambda }_n^{+}}{n}=\underset{n\to \infty }{\lim }\frac{{\lambda }_n^{-}}{n}=\pi {\int }_a^b\sqrt{\det H(s)}ds,$$

$$\left\{\begin{array}[]{rcl}[a,b]&\to&[0,1]\\ t&\mapsto&\Big{(}\int_{a}^{b}h_{1}(s)\mkern 4.0muds\Big{)}^{-1}\Big{(}\int_{t}^{b}h_{1}(s)\mkern 4.0muds\Big{)}\end{array}\right.$$

$$\omega_{n}\mathrel{\mathop{:}}=2^{-\frac{n}{2}}\bigg{(}\int\limits_{\chi(2^{1-n})}^{\chi(2^{-n})}h_{2}(s)\mkern 4.0muds\bigg{)}^{\frac{1}{2}},\quad n\in{\mathbb{N}}.$$

$$H_{\alpha ;{\alpha }_1,{\alpha }_2}(t):=\left(\begin{array}{cc}1& 0\\ 0& h_2(t)\end{array}\right),t\in [0,1),$$

$$h_{2}(t)\mathrel{\mathop{:}}=\Big{(}\frac{1}{1-t}\Big{)}^{\alpha}\Big{(}1+\log\frac{1}{1-t}\Big{)}^{-\alpha_{1}}\Big{(}1+\log^{+}\log\frac{1}{1-t}\Big{)}^{-\alpha_{2}},\quad t\in[0,1).$$

$$({\alpha }_1>0)\vee ({\alpha }_1=0,{\alpha }_2>0).$$

$$\text{conv.exp. of }\sigma (A_{[H_{2;{\alpha }_1,{\alpha }_2}]})=\{\begin{array}{ll}\infty & ,{\alpha }_1=0,\\ \frac{2}{{\alpha }_1}& ,{\alpha }_1\in (0,2),\\ 1& ,{\alpha }_1\ge 2.\end{array}$$

$$0<\underset{n\to \infty }{\mathrm{limsup}}\frac{n}{|{\lambda }_n{|}^{\frac{2}{{\alpha }_1}}(\log |{\lambda }_n|{)}^{-\frac{{\alpha }_2}{{\alpha }_1}}}<\infty .$$

$${\mathring{H}}_{{\alpha }_1,{\alpha }_2}:=\left(\begin{array}{cc}1& -\sqrt{h_2(t)}\\ -\sqrt{h_2(t)}& h_2(t)\end{array}\right),t\in [0,1),$$

$$\text{conv.exp. of }\sigma (A_{[{\mathring{H}}_{{\alpha }_1,{\alpha }_2}]})=\{\begin{array}{ll}\frac{2}{{\alpha }_1}& ,{\alpha }_1\in (0,4),\\ \frac{1}{2}& ,{\alpha }_1\ge 4.\end{array}$$

$$a_{n}(T)\mathrel{\mathop{:}}=\inf\big{\{}\|T-A\|\colon\mkern 3.0muA\in{\mathcal{B}}({\mathcal{H}}),\dim\operatorname{ran}A<n\big{\}},\quad n\in{\mathbb{N}}.$$

$$({\alpha }_n{)}_{n=1}^{\infty }\in \mathcal{S}\wedge |{\beta }_n|\le |{\alpha }_n|,n\in \mathbb{N}\Rightarrow ({\beta }_n{)}_{n=1}^{\infty }\in \mathcal{S},$$

$$({\alpha }_n{)}_{n=1}^{\infty }\in \mathcal{S},\sigma \text{ permutation of }\mathbb{N}\Rightarrow ({\alpha }_{\sigma (n)}{)}_{n=1}^{\infty }\in \mathcal{S}.$$

$$T\in \mathfrak{I}\iff (a_n(T){)}_{n=1}^{\infty }\in \mathrm{Seq}(\mathfrak{I}).$$

$$\big{\{}T\in{\mathcal{B}}({\mathcal{H}})\colon\mkern 3.0mu(a_{n}(T))_{n=1}^{\infty}\in{\mathcal{S}}\big{\}},\quad{\mathcal{H}}\text{ Hilbert space}.$$

$$T:\text{mathds}{1}_{t<s}(t,s)\phi (t)\overline{\kappa (s)}$$

$$(Tf)(t):=\phi (t){\int }_t^{b}f(s)\overline{\kappa (s)}ds,t\in (a,b),$$

$$\operatorname{dom}T\mathrel{\mathop{:}}=\Big{\{}f\in L^{2}(a,b)\colon\mkern 3.0mu\Big{(}t\mapsto\varphi(t)\int_{t}^{b}f(s)\overline{\kappa(s)}\mkern 4.0muds\Big{)}\in L^{2}(a,b)\Big{\}}.$$

$$\mathrm{diag}H:=\left(\begin{array}{cc}{h}_1& 0\\ 0& h_2\end{array}\right).\text{vspace}\ast -3mm$$

$$(B_{[H]}f)(t)=-\underset{c\nearrow b}{\lim }{\int }_a^c\left(\begin{array}{cc}0& \text{mathds}{1}_{s<t}(t,s)\\ \text{mathds}{1}_{s>t}(t,s)& 0\end{array}\right)H(s)f(s)ds,$$

$$\operatorname{dom}B_{[{H}]}=\bigg{\{}f\in L^{2}(H)\colon\mkern 3.0mu\parbox{142.26378pt}{\small$\lim_{c\nearrow b}(0,1)\int_{a}^{c}JH(s)f(s)\mkern 4.0muds$ exists,\\ r.h.s.\ of {\rm(\ref{Q125})} belongs to $L^{2}(H)$}\bigg{\}}.$$

$$\Phi :f(t)\mapsto H(t{)}^{\frac{1}{2}}f(t)$$

$$C_{[H]}:-H(t{)}^{\frac{1}{2}}\left(\begin{array}{cc}0& \text{mathds}{1}_{s<t}(t,s)\\ \text{mathds}{1}_{s>t}(t,s)& 0\end{array}\right)H(s{)}^{\frac{1}{2}}$$

$$B_{[H]}={\Phi }^{-1}P{C}_{[H]}\Phi \text{and}{C}_{[H]}=\Phi B_{[H]}{\Phi }^{-1}P.$$

Full text

Canonical systems with discrete spectrum

Roman Romanov   $\ast$  Harald Woracek

‡‡‡The second author was supported the project P 30715–N35 of the Austrian Science Fund.

Abstract: We study spectral properties of two-dimensional canonical systems $y^{\prime}(t)=zJH(t)y(t)$, $t\in[a,b)$, where the Hamiltonian $H$ is locally integrable on $[a,b)$, positive semidefinite, and Weyl’s limit point case takes place at $b$. We answer the following 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 conditions relative to comparison functions growing sufficiently fast. Making an analogy with complex analysis, this correponds to convergence class and type w.r.t. proximate orders having order larger than $1$. It is a surprising fact that these properties depend only 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 some

de Branges space ?

We give a complete and explicit answer.

AMS MSC 2010: 37J05, 34L20, 45P05, 46E22

Keywords: canonical system, discrete spectrum, eigenvalue distribution, operator ideal, Volterra operator, de Branges space

1 Introduction

We study the spectrum of the selfadjoint model operator associated with a two-dimensional canonical system

[TABLE]

Here $H$ is the Hamiltonian of the system, $-\infty<a<b\leq\infty$, $J$ is the symplectic matrix J\mathrel{\mathop{:}}=\Bigl{(}\begin{smallmatrix}0\hskip 0.60275pt&\hskip 0.60275pt-1\\[2.15277pt] 1\hskip 0.60275pt&\hskip 0.60275pt0\end{smallmatrix}\Bigr{)}, and $z\in{\mathbb{C}}$ is the eigenvalue parameter. We assume throughout that $H$ satisfies

$\triangleright$

H\in L^{1}\big{(}[a,c),{\mathbb{R}}^{2\times 2}\big{)}, $c\in(a,b)$, and $\{t\in[a,b)\colon\mkern 3.0muH(t)=0\}$ has measure [math],

$\triangleright$

$H(t)\geq 0$, $t\in[a,b)$ a.e. and $\int_{a}^{b}\operatorname{tr}H(t)\mkern 4.0mudt=\infty$.

Differential equations of this form orginate from Hamiltonian mechanics, and appear frequently in theory and applications. Various kinds of equations can be rewritten to the form (1.1), and several problems of classical analysis can be treated with the help of canonical systems. For example we mention Schrödinger operators [remling:2002], Dirac systems [sakhnovich:2002], or the extrapolation problem of stationary Gaussian processes via Bochners theorem [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 direct and inverse spectral theory of the equation (1.1) was developed in [gohberg.krein:1967, debranges:1968]. Recent references are [remling:2018, romanov:1408.6022v1].

With a Hamiltonian $H$ a Hilbert space $L^{2}(H)$ is associated, and in $L^{2}(H)$ a selfadjoint operator $A_{[{H}]}$ is given by the differential expression (1.1) and by prescribing the boundary condition $(1,0)y(a)=0$ (in one exceptional situation $A_{[{H}]}$ is a multivalued operator, but this is only a technical difficulty). This operator 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 relation with de Branges’ work on Hilbert spaces of entire functions was made explicit in [winkler:1993, winkler:1995].

In the present paper we answer the following questions:

Is the spectrum of $A_{[{H}]}$ discrete ?

If $\sigma(A_{[{H}]})$ is discrete, what is its asymptotic distribution ?

The question about asymptotic distribution is understood as the problem of finding the convergence exponent and the upper density of eigenvalues in terms of the Hamiltonian.

Discreteness of the spectrum

In our first theorem we characterise discreteness of $\sigma(A_{[{H}]})$.

1.1 Theorem**.**

\thlabel

Q102 Let H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pth_{3}\\[2.15277pt] h_{3}\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)} be a Hamiltonian on $[a,b)$ and assume that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$. Then $\sigma(A_{[{H}]})$ is discrete if and only if

[TABLE]

1.2 Remark*.*

\thlabel

Q103 The assumption that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$ in \threfQ102 is made for normalisation and is no loss in generality. First, a necessary condition that $0\notin\sigma_{\operatorname{ess}}(A_{[{H}]})$ is that there exists some angle $\phi\in{\mathbb{R}}$ such that $\int_{a}^{b}(\cos\phi,\sin\phi)H(s)(\cos\phi,\sin\phi)^{*}\mkern 4.0muds<\infty$. Second, applying rotation isomorphism always allows to reduce to the case that $\phi=0$. We will give details in Section 5.2.

∎

Let us remark that \threfQ102 yields a new proof of the discreteness criterion for strings given by I.S.Kac and M.G.Krein in [kac.krein:1958, Theorema 4,5], of [remling.scarbrough:1811.07067v1, Theorem 1.4], and of [kac:1995, Theorem 1].

Structure Hamiltonians of de Branges spaces

Recall that a de Branges space ${\mathcal{H}}(E)$ is a reproducing kernel Hilbert space of entire functions with certain additional properties, whose kernel is generated by a Hermite-Biehler function $E$. For each de Branges space there exists a unique maximal chain of de Branges subspaces ${\mathcal{H}}(E_{t})$, $t\leq 0$, contained isometrically (on exceptional intervals only contractively) in ${\mathcal{H}}(E)$. The generating Hermite-Biehler functions $E_{t}$ satisfy a canonical system on the interval $(-\infty,0]$ with some Hamiltonian $H$, and this Hamiltonian is called the structure Hamiltonian of ${\mathcal{H}}(E)$.

L.de Branges identified in [debranges:1961, Theorem IV] a particular class of Hamiltonians which are structure Hamiltonians of de Branges spaces. A mild generalisation of de Branges’ theorem can be found in [linghu:2015, Theorem 4.11], and a further class of structure Hamiltonians is identified (in a different language) by the already mentioned work of I.S.Kac and M.G.Krein [kac.krein:1958] and its mild generalisation [remling.scarbrough:1811.07067v1]. These classes do by far not exhaust the set of all structure Hamiltonians. In 1968, after having finalised his theory of Hilbert spaces of entire functions, de Branges posed the following question as a fundamental problem, cf. [debranges:1968, p.140]:

*Which Hamiltonians $H$ are the structure Hamiltonian of some

de Branges space $\mathcal{H}(E)$ ?*

In the following decades there was no significant progress towards a solution of this question. One result was claimed by I.S.Kac in 1995; proofs have never been published. He states a sufficient condition and a (different) necessary condition for $H$ to be a structure Hamiltonian. Unfortunately, his conditions are difficult to handle.

The connection with \threfQ102 is the following: a Hamiltonian is the structure Hamiltonian of some de Branges space ${\mathcal{H}}(E)$, if and only if the operator $A_{[{\tilde{H}}]}$ associated with the reversed Hamiltonian

[TABLE]

has discrete spectrum. This can be seen by a simple “juggling with fundamental solutions”–argument. A proof based on a different argument was published in [kac:2007], see also [linghu:2015, Theorem 2.3].

Hence we obtain from \threfQ102 a complete and explicit answer to de Branges’ question.

Summability properties

We turn to discussing the asymptotic distribution of $\sigma(A_{[{H}]})$. Consider a Hamiltonian $H$ with discrete spectrum. Then its spectrum is a (finite or infinite) sequence of simple eigenvalues without finite accumulation point. If $\sigma(A_{[{H}]})$ is finite, any questions about the asymptotic behaviour of the eigenvalues are obsolete. Moreover, under the normalisation that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$, the point [math] is not an eigenvalue of $A_{[{H}]}$. Hence, we can think of $\sigma(A_{[{H}]})$ as a sequence $(\lambda_{n})_{n=1}^{\infty}$ of pairwise different real numbers arranged such that

[TABLE]

In our second theorem we characterise summability of the sequence $(\lambda_{n}^{-1})_{n=1}^{\infty}$ relative to suitable comparison functions. In particular, this answers the question whether $(\lambda_{n}^{-1})_{n=1}^{\infty}\in\ell^{p}$ when $p>1$. The only known result in this direction is [kaltenbaeck.woracek:hskansys, Theorem 2.4], which settles the case $p=2$; we reobtain this theorem.

As comparison functions we use functions ${\mathscr{g}}$ defined on the ray $[0,\infty)$ and taking values in $(0,\infty)$ which satisfy:

$\triangleright$

The limit $\rho_{{\mathscr{g}}}\mathrel{\mathop{:}}=\lim_{r\to\infty}\frac{\log{\mathscr{g}}(r)}{\log r}$ exists and belongs to $(0,\infty)$.

$\triangleright$

The function ${\mathscr{g}}$ is continuously differentiable with ${\mathscr{g}}^{\prime}(r)>0$, and $\lim_{r\to\infty}\frac{r{\mathscr{g}}^{\prime}(r)}{{\mathscr{g}}(r)}=\rho_{\mathscr{g}}$.

Functions of this kind are known as growth functions; the number $\rho_{{\mathscr{g}}}$ is called the order of ${\mathscr{g}}$. They form a comparison scale which is finer than the scale of powers $r^{\rho}$. The history of working with growth scales other than powers probably starts with the paper [lindeloef:1905], where E.Lindelöf compared the growth of an entire function with functions of the form

[TABLE]

In what follows the reader may think of ${\mathscr{g}}(r)$ for simplicity as a concrete function of this form, or simply as a power $r^{\rho}$.

1.3 Theorem**.**

\thlabel

Q104 Let H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pth_{3}\\[2.15277pt] h_{3}\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)} be a Hamiltonian on an interval $[a,b)$ such that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$ and that $A_{[{H}]}$ has discrete spectrum. Moreover, assume that $h_{1}$ does not vanish a.e. on any interval $(c,b)$ with $c\in(a,b)$. Let ${\mathscr{g}}$ be a growth function with order $\rho_{\mathscr{g}}>1$. Then

[TABLE]

1.4 Remark*.*

\thlabel

Q105 For the same reasons as explained in \threfQ103, the assumption that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$ is just a normalisation and no loss in generality. Also the assumption that $h_{1}$ cannot vanish a.e. on any interval $(c,b)$ is no loss of generality. The reason being that, if $h_{1}$ does vanish on an interval of this form, then the Krein-de Branges formula, cf. [krein:1951, p.369 (english translation)], [debranges:1961, Theorem X], says that

[TABLE]

where $\lambda_{n}^{\pm}$ denote the sequences of positive and negative, respectively, eigenvalues arranged according to increasing modulus. In particular, the series $\sum_{n=1}^{\infty}\frac{1}{{\mathscr{g}}(|\lambda_{n}|)}$ converges whenever $\rho_{\mathscr{g}}>1$.

∎

Let us note that, besides the condition for square summability given in [kaltenbaeck.woracek:hskansys], \threfQ104 also yields new proofs of the results on the genus of the spectrum of a string given in [kac.krein:1958, p.139f] and [kac:1962, Theorema 1,2], and of [kac:1986] for orders between $\frac{1}{2}$ and $1$.

Limit superior properties

In our third theorem, we characterise $\limsup$–properties of the sequence $(\lambda_{n})_{n=1}^{\infty}$, again relative to growth functions ${\mathscr{g}}$ with $\rho_{\mathscr{g}}>1$. While the characterisations in \threfQ102,Q104 are perfectly explicit in terms of $H$, the conditions occurring in this context are somewhat more complicated. The reason for this is intrinsic, and manifests itself in the necessity to pass to the nonincreasing rearrangement of a certain sequence.

1.5 Theorem**.**

\thlabel

Q106 Let H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pth_{3}\\[2.15277pt] h_{3}\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)} be a Hamiltonian on an interval $[a,b)$ such that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$ and that $A_{[{H}]}$ has discrete spectrum. Moreover, assume that $h_{1}$ does not vanish a.e. on any interval $(c,b)$ with $c\in(a,b)$. Let ${\mathscr{g}}$ be a growth function with order $\rho_{\mathscr{g}}>1$.

Choose a right inverse $\chi$ of the nonincreasing surjection

[TABLE]

and let $(\omega_{n}^{*})_{n\in{\mathbb{N}}}$ be the nonincreasing rearrangement of the sequence $(\omega_{n})_{n\in{\mathbb{N}}}$ defined as

[TABLE]

Then

1. (i)

${\displaystyle\limsup_{n\to\infty}\frac{n}{{\mathscr{g}}(|\lambda_{n}|)}<\infty\ \Leftrightarrow\ \limsup_{n\to\infty}\frac{n}{{\mathscr{g}}((\omega_{n}^{*})^{-1})}<\infty}$, 2. (ii)

${\displaystyle\lim_{n\to\infty}\frac{n}{{\mathscr{g}}(|\lambda_{n}|)}=0\ \Leftrightarrow\ \lim_{n\to\infty}\frac{n}{{\mathscr{g}}((\omega_{n}^{*})^{-1})}=0}$.

Remember here \threfQ105.

Outline of the proofs

The proof of \threfQ102,Q104,Q106 proceeds through four stages. StepsCount191

The first stage is to pass from eigenvalue distribution to operator theoretic properties. This is done in a standard way using symmetrically normed operator ideals: discreteness of $\sigma(A_{[{H}]})$ is equivalent to $(A_{[{H}]}-z)^{-1}$ being compact, summability properties of $\sigma(A_{[{H}]})$ are equivalent to $(A_{[{H}]}-z)^{-1}$ belonging to Orlicz ideals, and $\limsup$–properties of $\sigma(A_{[{H}]})$ are equivalent to $(A_{[{H}]}-z)^{-1}$ belonging to Lorentz spaces.

The reader has certainly observed the – probably surprising – fact that the conditions in our theorems do not involve the off-diagonal entry $h_{3}$ of the Hamiltonian $H$. The second stage is to prove an Independence Theorem which says that membership of resolvents $(A_{[{H}]}-z)^{-1}$ in operator ideals ${{\mathfrak{I}}}$ is indeed independent of $h_{3}$, provided ${{\mathfrak{I}}}$ possesses a certain additional property. This additional property is that a weak variant of Matsaev’s Theorem on real parts of Volterra operators holds in ${{\mathfrak{I}}}$.

In a work of A.B.Aleksandrov, S.Janson, V.V.Peller, and R.Rochberg, membership in Schatten classes of integral operators whose kernel has a particular form is characterised using a dyadic discretisation method. The third stage is to realise that a minor generalisation of one of their results suffices to prove the mentioned weak Matsaev Theorem in the ideal ${{\mathfrak{S}}}_{\infty}$ of all compact operators. For Orlicz- and Lorentz ideals, it is known that (the full) Matsaev Theorem holds. Thus the Independence Theorem stated in ➁ will apply to all ideals occurring in ➀.

The final stage is to characterise membership in the mentioned ideals for a diagonal Hamiltonian (meaning that $h_{3}=0$). This again rests on the discretisation method from [aleksandrov.janson.peller.rochberg:2002], which yields characterisations of a sequential form (as the one stated in \threfQ106) for all ideals occurring in ➀. For the cases of ${{\mathfrak{S}}}_{\infty}$ and Orlicz ideals, sequential characterisations can be rewritten to a continuous form (as stated in \threfQ102,Q104). This is nearly obvious for ${{\mathfrak{S}}}_{\infty}$, while for Orlicz ideals a little more effort and passing to dual spaces is needed.

The threshold $\bm{\rho_{\mathscr{g}}=1}$

The Krein-de Branges formula (1.4) implies that for every Hamiltonian $H$ whose determinant does not vanish a.e., the spectrum $\sigma(A_{[{H}]})$, if discrete, satisfies $\liminf_{n\to\infty}\frac{n}{|\lambda_{n}|}>0$. On the other hand, $\sigma(A_{[{H}]})$ can be arbitrarily sparse if $\det H=0$ a.e. It is not difficult to find Hamiltonians H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pth_{3}\\[2.15277pt] h_{3}\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)} whose spectrum is discrete and satisfies $\lim_{n\to\infty}\frac{n}{|\lambda_{n}|^{\rho}}\in(0,\infty)$ for some $\rho<1$, but in the same time $h_{1}h_{2}$ does not vanish a.e., see \threfQ132. For each such Hamiltonian and every Schatten-von Neumann ideal ${{\mathfrak{S}}}_{p}$ with $\rho<p\leq 1$, the Independence Theorem mentioned in ➁ fails. This shows that our method necessarily must break down at (and below) trace class, i.e., growth of speed ${\mathscr{g}}(r)\mathrel{\mathop{:}}=r$.

On a less concrete level, growth of order $1$ is a threshold because of (at least) four reasons.

$\triangleright$

Orders larger than $1$, meaning eigenvalue distriubution more dense than integers, can occur only from the behaviour of tails of $H$ at its singular endpoint $b$. In fact, for $\rho>1$, the spectrum of $A_{[{H}]}$ is discrete with convergence exponent $\rho$ if and only if for some $c\in(a,b)$ the spectrum of $A_{[{H|_{[c,b)}}]}$ is discrete with convergence exponent $\rho$.

Contrasting this, orders less than $1$ will in general accumulate over the whole interval $(a,b)$. In fact, it may happen that $\sigma(A_{[{H}]})$ has convergence exponent $1$ while for every $c\in(a,b)$ the spectrum of the tail $H|_{[c,b)}$ has convergence exponent [math].

$\triangleright$

Entire functions of bounded type have very specific properties related to exponential type. In complex analysis, orders larger than $1$ are usually considered as more stable than smaller orders.

$\triangleright$

The theory of symmetrically normed operator ideals is significantly more complicated for ideals close to trace class than for ideals containing some Schatten-von Neumann ideal ${{\mathfrak{S}}}_{p}$ with $p>1$. When going even below trace class, a lot of the theory breaks down completely.

$\triangleright$

Rewriting asymptotic conditions on the spectral distribution to conditions on membership in Orlicz- and Lorentz ideals is not anymore possible when coming close to trace class.

Let us now give two examples which illustrate our results. They are simple, and given by Hamiltonians related to a string, but, as we hope, still illustrative. At this point we only state their spectral properties; the proof is given in Section 5.3.

1.6 Example*.*

\thlabel

Q109 Given $\alpha>1$ and $\alpha_{1},\alpha_{2}\in{\mathbb{R}}$, we consider the Hamiltonian (to avoid bulky notation, we skip indices $\alpha,\alpha_{1},\alpha_{2}$ at $h_{2}$)

[TABLE]

where

[TABLE]

Since $\alpha>1$, we have $\int_{0}^{1}h_{2}(t)\mkern 4.0mudt=\infty$.

If $\alpha>2$, then [math] belongs to the essential spectrum of $A_{[{H_{\alpha;\alpha_{1},\alpha_{2}}}]}$, and if $\alpha\in(1,2)$, then the spectrum is discrete with convergence exponent $1$ but $\liminf_{n\to\infty}\frac{n}{|\lambda_{n}|}>0$, in particular, $\sum_{n=1}^{\infty}\frac{1}{|\lambda_{n}|}=\infty$.

A behaviour between those extreme situations occurs when $\alpha=2$. First, the spectrum of $A_{[{H_{2;\alpha_{1},\alpha_{2}}}]}$ is discrete, if and only if

[TABLE]

For such parameter values, the convergence exponent of the spectrum is

[TABLE]

For $\alpha_{1}\in(0,2)$, we have a more refined $\limsup$-property relative to a comparison function which is not a power:

[TABLE]

∎

1.7 Example*.*

\thlabel

Q132 Given $\alpha_{1}>0$ and $\alpha_{2}\in{\mathbb{R}}$ consider the Hamiltonian (again indices at $h_{2}$ are skipped)

[TABLE]

where $h_{2}$ is as in (1.5) with $\alpha=2$. Then $\sigma(A_{[{\mathring{H}_{\alpha_{1},\alpha_{2}}}]})$ is discrete, and its convergence exponent is

[TABLE]

The diagonalisation of $\mathring{H}_{\alpha_{1},\alpha_{2}}$, i.e., the Hamiltonian obtained by skipping its off-diagonal entries, is $H_{2;\alpha_{1},\alpha_{2}}$. Comparing the convergence exponents computed in (1.6) and (1.7), illustrates validity of the Independence Theorem from ➁ as long as the convergence exponent is not less than $1$, and its failure for other values.

∎

Organisation of the manuscript

The structuring of this article is straightforward. We start off in Section 2 with proving the central Independence Theorem mentioned in ➁. Section 3 contains the proof of \threfQ102, and Section 4 the proofs of \threfQ104,Q106. Finally, in Section 5, we give a fourth theorem in the spirit of the above three theorems, disuss the normalisation condition $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$, and provide details for the \threfQ109,Q132.

As mentioned in ➂ and ➃ above, our arguments use a (very) minor generalisation of the AJPR-results. This is established in just the same way as the results of [aleksandrov.janson.peller.rochberg:2002] with only a few technical additions. For the convenience of the reader we provide full details in Appendix A. Moreover, in Appendix B, we provide detailed proof for some elementary facts being used in the text, and in Appendix C we make the connection of our \threfQ102 with [kac:1995, Theorem 1].

2 The Independence Theorem

Let ${\mathcal{H}}$ be a Hilbert space and ${\mathcal{B}}({\mathcal{H}})$ the set of all bounded linear operators on ${\mathcal{H}}$. For an operator $T\in{\mathcal{B}}({\mathcal{H}})$ we denote by $a_{n}(T)$ the $n$-th approximation number of $T$, i.e.,

[TABLE]

The Calkin correspondence [calkin:1941] is the map assigning to each $T\in{\mathcal{B}}({\mathcal{H}})$ the sequence $(a_{n}(T))_{n=1}^{\infty}$ of its approximation numbers.

An operator ideal ${{\mathfrak{I}}}$ in ${\mathcal{H}}$ is a two-sided ideal of the algebra ${\mathcal{B}}({\mathcal{H}})$. Every proper operator ideal ${{\mathfrak{I}}}$ contains the ideal of all finite rank operators, and, provided ${\mathcal{H}}$ is separable, is contained in the ideal ${{\mathfrak{S}}}_{\infty}$ 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 certain sequence spaces. Recall [garling:1967, Theorem 1]: there is a bijection Seq of the set of all operator ideals of ${\mathcal{H}}$ onto the set of all solid symmetric sequence spaces111A linear subspace ${\mathcal{S}}$ of ${\mathbb{C}}^{{\mathbb{N}}}$ is called solid, if

$(\alpha_{n})_{n=1}^{\infty}\in{\mathcal{S}}\wedge|\beta_{n}|\leq|\alpha_{n}|,n\in{\mathbb{N}}\ \Rightarrow\ (\beta_{n})_{n=1}^{\infty}\in{\mathcal{S}},$

and it is called symmetric, if

$(\alpha_{n})_{n=1}^{\infty}\in{\mathcal{S}},\sigma\text{ permutation of }{\mathbb{N}}\ \Rightarrow\ (\alpha_{\sigma(n)})_{n=1}^{\infty}\in{\mathcal{S}}.$

, such that for all $T\in{\mathcal{B}}({\mathcal{H}})$

[TABLE]

For example, the ideal ${{\mathfrak{S}}}_{\infty}$ of all compact operators corresponds to $c_{0}$, the trivial ideal ${\mathcal{B}}(H)$ to $\ell^{\infty}$, and the Schatten–von Neumann classes ${{\mathfrak{S}}}_{p}$ to $\ell^{p}$. Taking the viewpoint of sequence spaces is natural in (at least) two respects.

$\triangleright$

It allows to compare ideals in ${\mathcal{B}}({\mathcal{H}})$ for different base spaces ${\mathcal{H}}$. A solid symmetric sequence space ${\mathcal{S}}$ invokes the family of “same-sized” operator ideals

[TABLE]

$\triangleright$

Virtually all examples of operator ideals ${{\mathfrak{I}}}$ which “appear in nature” are defined by a specifying their sequence space ${\rm Seq}({{{\mathfrak{I}}}})$.

From now on we do not anymore distinguish between sequence spaces and operator ideals, and always speak of an operator ideal.

A central role is played by integral operators whose kernel has a very special form. Let $-\infty\leq a<b\leq\infty$, and $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ be measurable functions such that $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$ for every $c\in(a,b)$. Then we consider the (closed, but possibly unbounded) integral operator $T$ in $L^{2}(a,b)$ with kernel

[TABLE]

Explicitly, this is the operator acting as

[TABLE]

on its natural maximal domain

[TABLE]

In the next definition we single out a crucial property which an operator ideal ${{\mathfrak{I}}}$ may or may not have.

2.1 Definition**.**

\thlabel

Q111 Let ${{\mathfrak{I}}}$ be an operator ideal. We say that the weak Matsaev Theorem holds in ${{\mathfrak{I}}}$, if the following statement is true.

$\triangleright$

Let $-\infty\leq a<b\leq\infty$, let $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ be measurable functions such that $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$ for every $c\in(a,b)$, and let $T$ be the integral operator with kernel (2.1). Then $\operatorname{Re}T\in{{\mathfrak{I}}}$ implies $T\in{{\mathfrak{I}}}$.

∎

We are going to compare a Hamiltonian $H$ with its diagonal part.

2.2 Definition**.**

\thlabel

Q112 Let $H$ be a Hamiltonian and write H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pth_{3}\\[2.15277pt] h_{3}\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)}. Then we denote the corresponding diagonal Hamiltonian as

[TABLE]

∎

2.3 Theorem** (Independence Theorem).**

\thlabel

Q113 Let $H$ be a Hamiltonian defined on an interval $[a,b)$, and assume that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$. Moreover, let ${{\mathfrak{I}}}$ be an operator ideal.

Then $A_{[{\operatorname{diag}H}]}\in{{\mathfrak{I}}}$ implies that $A_{[{H}]}\in{{\mathfrak{I}}}$. If the weak Matsaev Theorem holds in ${{\mathfrak{I}}}$, also the converse holds.

Before giving the proof, we have to recall some known facts about the model operator $A_{[{H}]}$. The first lemma is folklore; one possible reference is [kaltenbaeck.woracek:hskansys] where it appears implicitly.

2.4 Lemma**.**

\thlabel

Q114 Under the assumption that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$, the operator $A_{[{H}]}$ is injective and its inverse $B_{[{H}]}\mathrel{\mathop{:}}=A_{[{H}]}^{-1}$ acts as

[TABLE]

on the domain

[TABLE]

∎

Denote by $L^{2}(Idt)$ the $L^{2}$-space of $2$-vector valued functions on $(a,b)$ with respect to the matrix measure \Bigl{(}\begin{smallmatrix}1\hskip 0.60275pt&\hskip 0.60275pt0\\[2.15277pt] 0\hskip 0.60275pt&\hskip 0.60275pt1\end{smallmatrix}\Bigr{)}\mkern 4.0mudt. The function

[TABLE]

maps the model space $L^{2}(H)$ isometrically onto some closed subspace of $L^{2}(Idt)$. This holds since, by its definition, $L^{2}(H)$ is a closed subspace of the $L^{2}$-space of $2$-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 $L^{2}(Idt)$ with kernel

[TABLE]

and the natural maximal domain.

The next lemma says that the operator $B_{[{H}]}$ can be transformed into $C_{[{H}]}$, and was shown in [kaltenbaeck.woracek:hskansys, Proof of Lemma 2.2].

2.5 Lemma**.**

\thlabel

Q116 Assume that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$, and denote by $P$ the orthogonal projection of $L^{2}(Idt)$ onto $\operatorname{ran}\Phi$. Then

[TABLE]

∎

As a consequence of \threfQ116, the operators $B_{[{H}]}$ and $C_{[{H}]}$ are together bounded or unbounded, and if they are bounded their approximation numbers coincide. Thus, for every operator ideal ${{\mathfrak{I}}}$, we have

[TABLE]

The following simple computation is a key step to the proof of \threfQ113.

2.6 Lemma**.**

\thlabel

Q117 Let $H$ be a Hamiltonian on $[a,b)$ with $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$. Denote

[TABLE]

and let $T_{ij}$, $(i,j)\in\{2,3\}\times\{1,3\}$, be the integral operators in $L^{2}(a,b)$ with kernel

[TABLE]

Then

[TABLE]

Proof.

Multiplying out the kernel (2.4) of the integral operator $C_{[{H}]}$ gives

[TABLE]

The adjoint $T_{ij}^{*}$ is the integral operator with kernel

[TABLE]

and the assertion follows. ∎

2.7 Corollary**.**

\thlabel

Q119 Let H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pt0\\[2.15277pt] 0\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)} be a diagonal Hamiltonian, and let $S_{21}$ be the integral operator in $L^{2}(a,b)$ with kernel

[TABLE]

Then for every operator ideal ${{\mathfrak{I}}}$ we have

[TABLE]

Proof.

\thref

Q117 gives

[TABLE]

∎

For a bounded function $\psi$, we denote by $M_{\psi}$ the multiplication operator with $\psi$ on $L^{2}(a,b)$:

[TABLE]

Proof of \threfQ113.

It holds that $h_{1}=v_{1}^{2}+v_{3}^{2}$ and $h_{2}=v_{2}^{2}+v_{3}^{2}$, and hence

[TABLE]

Thus the functions (quotients are understood as [math] if their denominator vanishes)

[TABLE]

are all bounded. We have

[TABLE]

where the last relation holds since, by a short computation,

[TABLE]

We see that

[TABLE]

From this it readily follows that $B_{[{\operatorname{diag}H}]}\in{{\mathfrak{I}}}$ implies $B_{[{H}]}\in{{\mathfrak{I}}}$.

Assume that the weak Matsaev Theorem holds for ${{\mathfrak{I}}}$. If $B_{[{H}]}\in{{\mathfrak{I}}}$, then

[TABLE]

The operator $\operatorname{Re}T_{33}$ is one-dimensional, hence certainly belongs to ${{\mathfrak{I}}}$, and it follows that also $\operatorname{Re}T_{21}\in{{\mathfrak{I}}}$. We conclude that $T_{23}$, $T_{31}$, $T_{21}$, and $T_{33}$, all belong to ${{\mathfrak{I}}}$. From this $S_{21}\in{{\mathfrak{I}}}$, and in turn $B_{[{\operatorname{diag}H}]}\in{{\mathfrak{I}}}$. ∎

3 Discreteness of the spectrum

\thref

Q102 is shown using a minor extension of [aleksandrov.janson.peller.rochberg:2002, Theorem 3.2], namely \threfQ120 below. In order to formulate this result, we need some notation. Let $-\infty\leq a<b\leq\infty$, let $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ be measurable functions such that $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$ for every $c\in(a,b)$. Then the function $t\mapsto\|\mathds{1}_{(t,b)}\kappa\|^{2}$ is a nonincreasing surjection of $[a,b]$ onto $[0,\|\kappa\|^{2}]$. Hence, we can choose an increasing sequence $c_{0}\mathrel{\mathop{:}}=a<c_{1}<c_{2}<\ldots<b$ such that $\|\mathds{1}_{(c_{n},b)}\kappa\|^{2}=2^{-n}\|\kappa\|^{2}$, $n\in{\mathbb{N}}$. Note that this requirement is equivalent to

[TABLE]

Having chosen $c_{n}$, we denote

[TABLE]

Explicitly, by (3.1),

[TABLE]

3.1 Theorem**.**

\thlabel

Q120 Let $-\infty\leq a<b\leq\infty$, let $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ be measurable functions with $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$, $c\in(a,b)$, and consider the integral operator $T$ on $L^{2}(a,b)$ with kernel (2.1). Then

$T$* is compact ***

*** $\operatorname{Re}T$ is compact*

$\lim_{n\to\infty}\omega_{n}=0$

where $\omega_{n}$ are as in (3.2). ∎

The proof of \threfQ120 is nearly verbatim the same as the argument given in [aleksandrov.janson.peller.rochberg:2002]. We therefore skip details from the main text; the reader can find the fully elaborated argument in Appendix A.

Rewriting the sequential condition occuring from \threfQ120 to a continuous one as stated in \threfQ102 is elementary; details are deferred to Appendix B.

3.2 Lemma**.**

\thlabel

Q122 Letting notation be as in \threfQ120, we have

[TABLE]

∎

Now \threfQ102 follows easily.

Proof of \threfQ102.

\thref

Q120 implies that the weak Matsaev Theorem holds in the operator ideal ${{\mathfrak{I}}}=c_{0}$ of all compact operators (remember that we do not distinguish between a concrete operator ideal and its sequence space). Hence the Independence Theorem applies, and together with \threfQ119 and \threfQ120 applied to $\operatorname{diag}H$ we find

[TABLE]

where $\omega_{n}$ is buildt with $\kappa(t)\mathrel{\mathop{:}}=\sqrt{h_{1}(t)}$, $\varphi(t)\mathrel{\mathop{:}}=\sqrt{h_{2}(t)}$. By \threfQ122

[TABLE]

and the proof of \threfQ102 is complete. ∎

3.3 Remark*.*

\thlabel

Q123 Using the connection between Krein strings and diagonal canonical systems elaborated in [kaltenbaeck.winkler.woracek:bimmel], the present \threfQ102 yields a new proof of the classical criterion [kac.krein:1958, Theorema 4,5] for a string to have discrete spectrum.

∎

4 Summability and limit superior conditions

A symmetrically normed ideal ${{\mathfrak{I}}}$ is a (two-sided) operator ideal which is endowed with a norm $\|.\|_{{{\mathfrak{I}}}}$, such that

$\triangleright$

$({{\mathfrak{I}}},\|.\|_{{{\mathfrak{I}}}})$ is complete,

$\triangleright$

${\displaystyle\|ATB\|_{{{\mathfrak{I}}}}\leq\|A\|\cdot\|T\|_{{{\mathfrak{I}}}}\cdot\|B\|,\quad T\in{{\mathfrak{I}}},\ A,B\in{\mathcal{B}}({\mathcal{H}}),\ {\mathcal{H}}\text{ Hilbert space}}$,

$\triangleright$

$\|T\|_{{{\mathfrak{I}}}}=\|T\|$ for $T$ with $\dim\operatorname{ran}T=1$.

Basic examples of symmetrically normed ideal are the Schatten-von Neumann ideals ${{\mathfrak{S}}}_{p}$, $1\leq p\leq\infty$.

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 $\sigma(T)=\{0\}$.

4.1 Definition**.**

\thlabel

Q150 Let ${{\mathfrak{I}}}$ be a symmetrically normed ideal which is properly contained in ${{\mathfrak{S}}}_{\infty}$. We say that Matsaev’s Theorem holds in ${{\mathfrak{I}}}$, if the following statement is true.

$\triangleright$

Let ${\mathcal{H}}$ be a Hilbert space, and let $T$ be a Volterra operator in ${\mathcal{H}}$. Then $\operatorname{Re}T\in{{\mathfrak{I}}}$ implies $T\in{{\mathfrak{I}}}$.

∎

Notice that an integral operator whose kernel has the form (2.1) has no nonzero eigenvalues. As a consequence of this and \threfQ120, we obtain the following fact (which also justifies our terminology introduced in \threfQ111).

4.2 Corollary**.**

\thlabel

Q151 Let ${{\mathfrak{I}}}\subsetneq{{\mathfrak{S}}}_{\infty}$ be a symmetrically normed ideal. If Matsaev’s Theorem holds in ${{\mathfrak{I}}}$, then also the weak Matsaev Theorem holds in ${{\mathfrak{I}}}$. ∎

Consequently, for every proper symmetrically normed ideal in which Matsaev’s Theorem holds, the Independence Theorem applies.

The characterisations stated in \threfQ104,Q106 will be shown using another AJPR-type theorem which is a variant of [aleksandrov.janson.peller.rochberg:2002, Theorem 3.3] (proof details of this result are given 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 ${{\mathfrak{I}}}$ is called fully symmetric, if for each two nonincreasing sequences of nonnegative numbers $(\alpha_{n})_{n=1}^{\infty}$ and $(\beta_{n})_{n=1}^{\infty}$ it holds that

[TABLE]

4.3 Theorem**.**

\thlabel

Q152 Let $-\infty\leq a<b\leq\infty$, let $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ be measurable functions with $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$, $c\in(a,b)$, and consider the integral operator $T$ on $L^{2}(a,b)$ with kernel (2.1). Moreover, let ${{\mathfrak{I}}}\subsetneq{{\mathfrak{S}}}_{\infty}$ be an operator ideal.

$\triangleright$

If ${{\mathfrak{I}}}$ is fully symmetric, then $\operatorname{Re}T\in{{\mathfrak{I}}}$ implies $(\omega_{n})_{n=1}^{\infty}\in{{\mathfrak{I}}}$.

$\triangleright$

If ${{\mathfrak{I}}}$ is symmetrically normed and Matsaev’s Theorem holds in ${{\mathfrak{I}}}$, then $(\omega_{n})_{n=1}^{\infty}\in{{\mathfrak{I}}}$ implies $T\in{{\mathfrak{I}}}$.

∎

To obtain the proof of \threfQ104, we use a particular class of symmetrically normed ideals.

4.4 Example*.*

\thlabel

Q154 Let $M:[0,\infty)\to[0,\infty)$ be a continuous increasing and convex function with $M(0)=0$, $\lim_{t\to\infty}M(t)=\infty$, and $M(t)>0$, $t>0$. Assume that $\limsup_{t\searrow 0}\frac{M(2t)}{M(t)}<\infty$, and that $M$ is normalised by $M(1)=1$. The Orlicz space ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$ is the symmetrically normed ideal

[TABLE]

Orlicz ideals are separable; in fact the unit vectors $e_{j}\mathrel{\mathop{:}}=(\delta_{nj})_{n=1}^{\infty}$, $j\in{\mathbb{N}}$, form an unconditional basis in ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$. For a systematic treatment of this type of sequence spaces we refer to [maligranda:1989] and [lindenstrauss.tzafriri:1977, Section 4.a].

∎

4.5 Remark*.*

\thlabel

Q156 Given a growth function ${\mathscr{g}}$ with order $\rho_{{\mathscr{g}}}>1$, set

[TABLE]

In general, $M$ will not be convex. However, based on [bingham.goldie.teugels:1989, Theorem 1.3.3], [lelong.gruman:1986, Proposition 1.22], we always find an equivalent growth function ${\mathscr{g}}_{1}$, i.e., one with $\lim_{t\to\infty}\frac{{\mathscr{g}}_{1}(t)}{{\mathscr{g}}(t)}=1$, such that the corresponding $M_{1}$ satisfies all requirements made in \threfQ154. Then

[TABLE]

and we may say that ${\mathscr{g}}$ induces an Orlicz ideal.

∎

Rewriting the sequential condition occuring from \threfQ120 to a continuous one requires some technique about Orlicz spaces. Details are given in Appendix B.

4.6 Lemma**.**

\thlabel

Q155 Letting notation be as above, we have

[TABLE]

∎

Proof of \threfQ104.

The left and right sides of the equivalence asserted in \threfQ104 do not change their truth value when we pass from the given growth function ${\mathscr{g}}$ to an equivalent one. Hence, we may assume without loss of generality that ${\mathscr{g}}$ gives rise to an Orlicz space ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$ as in (4.1).

Since $\rho_{{\mathscr{g}}}>1$, we can apply [gohberg.krein:1967, Footnote 12,p.139] and conclude that Matsaev’s Theorem holds in ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$. By \threfQ151 the weak Matsaev Theorem holds in ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$, and hence the Independence Theorem applies. Clearly ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$ is fully symmetric, and \threfQ113 combined with \threfQ119 and \threfQ152 yields

[TABLE]

where $\omega_{n}$ is buildt with $\kappa(t)\mathrel{\mathop{:}}=\sqrt{h_{1}(t)}$, $\varphi(t)\mathrel{\mathop{:}}=\sqrt{h_{2}(t)}$. By \threfQ155

[TABLE]

and the proof of \threfQ104 is complete. ∎

To obtain the proof of \threfQ106, we use another particular class of symmetrically normed ideals. In the following we denote for a sequence $(\alpha_{n})_{n=1}^{\infty}\in c_{0}$, by $(\alpha_{n}^{*})_{n=1}^{\infty}$ its nonnegative nonincreasing rearrangement, i.e., the sequence made up of the elements $|\alpha_{n}|$ arranged nonincreasingly.

4.7 Example*.*

\thlabel

Q153 Let $(\pi_{n})_{n=1}^{\infty}$ be a nonincreasing positive sequence with $\pi_{1}=1$, $\lim_{n\to\infty}\pi_{n}=0$, and $\sum_{n=0}^{\infty}\pi_{n}=\infty$. The Lorentz space ${{\mathfrak{S}}}_{[{\pi}]}$ is the symmetrically normed ideal

[TABLE]

Lorentz spaces may or may not be separable, and we denote by ${{\mathfrak{S}}}_{[{\pi}]}^{\bm{\circ}}$ the separable part of ${{\mathfrak{S}}}_{[{\pi}]}$. I.e., ${{\mathfrak{S}}}_{[{\pi}]}^{\bm{\circ}}$ is the closure in ${{\mathfrak{S}}}_{[{\pi}]}$ of the linear subspace of all finite rank operators. For this type of sequence spaces we refer to [gohberg.krein:1965, Theorem III.14.1] or [lord.sukochev.zanin:2013, Example 1.2.7].

∎

This type of symmetrically normed ideals correspond to the consideration of limit superior conditions. Let ${\mathscr{g}}$ be a growth function with $\rho_{{\mathscr{g}}}>1$, and set $\pi_{n}\mathrel{\mathop{:}}=\frac{1}{{\mathscr{g}}^{-1}(n)}$ where ${\mathscr{g}}^{-1}$ is the inverse function of ${\mathscr{g}}$. Then

[TABLE]

cf. [gohberg.krein:1965, Theorem III.14.2].

Proof of \threfQ106.

Since $\rho_{{\mathscr{g}}}>1$, we can apply [gohberg.krein:1967, Theorem III.9.1] and conclude that Matsaev’s Theorem holds in ${{\mathfrak{S}}}_{[{\pi}]}$ and in ${{\mathfrak{S}}}_{[{\pi}]}^{\bm{\circ}}$. By \threfQ151 the weak Matsaev Theorem holds in ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$, and hence the Independence Theorem applies. Moreover, ${{\mathfrak{S}}}_{[{\pi}]}$ and ${{\mathfrak{S}}}_{[{\pi}]}^{\bm{\circ}}$ are fully symmetric, cf. [gohberg.krein:1965]. Now \threfQ113 combined with \threfQ119 and \threfQ152 yields

[TABLE]

where $\omega_{n}$ is buildt with $\kappa(t)\mathrel{\mathop{:}}=\sqrt{h_{1}(t)}$, $\varphi(t)\mathrel{\mathop{:}}=\sqrt{h_{2}(t)}$.

Matching notation shows that these numbers $\omega_{n}$ are just the same as the numbers written in \threfQ106. By a property of growth functions, it holds that

[TABLE]

and the proof is complete. ∎

5 Bounded invertibility, normalisation, and examples

5.1 Bounded invertibility

The present method also yields a condition for the model operator $A_{[{H}]}$ to be boundedly invertible.

5.1 Theorem**.**

\thlabel

Q172 Let H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pth_{3}\\[2.15277pt] h_{3}\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)} be a Hamiltonian on $[a,b)$ and assume that $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$. Then $0\notin\sigma(A_{[{H}]})$ if and only if

[TABLE]

Remember here \threfQ103.

This result implies [remling.scarbrough:1811.07067v1, Theorem 1.5].

Again we use the notation introduced in Section 3, in particular recall (3.1) and (3.2). The proof of \threfQ172 is based on the following AJPR-type theorem, which is a variant of [aleksandrov.janson.peller.rochberg:2002, Theorem 3.1] (proof details are given in Appendix A).

5.2 Theorem**.**

\thlabel

Q170 Let $-\infty\leq a<b\leq\infty$, let $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ be measurable functions with $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$, $c\in(a,b)$, and consider the integral operator $T$ on $L^{2}(a,b)$ with kernel (2.1). Then

$T$* is bounded ***

*** $\operatorname{Re}T$ is bounded*

$\sup_{n\in{\mathbb{N}}}\omega_{n}<\infty$

where $\omega_{n}$ are as in (3.2). ∎

Rewriting the sequential condition occuring from \threfQ170 to a continuous one is elementary; again details are deferred to Appendix B.

5.3 Lemma**.**

\thlabel

Q171 Letting notation be as above, we have

[TABLE]

∎

Proof of \threfQ172.

\thref

Q170 implies that the weak Matsaev Theorem holds in the operator ideal ${{\mathfrak{I}}}=\ell^{\infty}$ of all bounded operators. Hence the Independence Theorem applies, and together with \threfQ119 and \threfQ170 applied to $\operatorname{diag}H$ we find

[TABLE]

where $\omega_{n}$ is buildt with $\kappa(t)\mathrel{\mathop{:}}=\sqrt{h_{1}(t)}$, $\varphi(t)\mathrel{\mathop{:}}=\sqrt{h_{2}(t)}$. By \threfQ171

[TABLE]

and the proof of \threfQ172 is complete. ∎

5.2 The normalisation $\bm{\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty}$

In this section we provide the arguments announced in \threfQ103. Denote by $T_{\min}(H)$ and $T_{\max}(H)$ the minimal and maximal operators induced by the equation (1.1), cf. [hassi.snoo.winkler:2000, Section 3]. First observe that, in the cases of present interest, the space $L^{2}(H)$ always contains some constant.

5.4 Lemma**.**

\thlabel

Q174 Assume that $0\notin\sigma_{\operatorname{ess}}(A_{[{H}]})$. Then there exists $\phi\in{\mathbb{R}}$ such that

[TABLE]

This follows from [remling:2018, Theorem 3.8(b)] (use $t=0$); for the convenience of the reader we recall the argument.

Proof of \threfQ174.

Since $0\notin\sigma_{\operatorname{ess}}(A_{[{H}]})$, [math] is a point of regular type for $T_{\min}(H)$. Thus there exists a selfadjoint extension $\tilde{A}$ of $T_{\min}(H)$ such that $0\in\sigma_{{\rm p}}(\tilde{A})$ (see, e.g., [gorbachuk.gorbachuk:1997, Propositions 3.3 and 3.5]), and it follows that $\ker T_{\max}(H)\neq\{0\}$. This kernel, however, consists of all constant functions in $L^{2}(H)$. ∎

To achieve the normalisation $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$, equivalently, $\phi=0$ in (5.2), one uses rotation isomorphisms.

5.5 Definition**.**

\thlabel

Q175 Let $\alpha\in{\mathbb{R}}$, and denote

[TABLE]

1. (i)

For a Hamiltonian $H$ defined on some interval $[a,b)$, we set

[TABLE] 2. (ii)

For a $2$-vector valued function defined on some interval $[a,b)$, we set

[TABLE]

∎

5.6 Remark*.*

\thlabel

Q176 The following facts hold (see, e.g., [kaltenbaeck.woracek:p5db, p.263]):

$\triangleright$

$\circlearrowleft_{{\alpha}}\mkern-5.0mu{{H}}$ is a Hamiltonian,

$\triangleright$

$\omega_{\alpha}$ induces an isometric isomorphism of $L^{2}(H)$ onto $L^{2}(\circlearrowleft_{{\alpha}}\mkern-5.0mu{{H}})$,

$\triangleright$

${\displaystyle T_{\min}(\circlearrowleft_{{\alpha}}\mkern-5.0mu{{H}})\circ\omega_{\alpha}=\omega_{\alpha}\circ T_{\min}(H)}$.

Consequently, the Hamiltonians $H$ and $\circlearrowleft_{{\alpha}}\mkern-5.0mu{{H}}$ will share all operator theoretic properties.

∎

In the context of diagonalisation, making a normalising rotation is inevitable. The reason being the following fact.

5.7 Lemma**.**

\thlabel

Q177 For a Hamiltonian $H$ there exists at most one angle $\alpha$ modulo $\frac{\pi}{2}$ such that $L^{2}(\operatorname{diag}\circlearrowleft_{{\alpha}}\mkern-5.0mu{{H}})$ contains a nonzero constant.

Proof.

Let $\alpha\in[0,\pi)$, write

[TABLE]

and assume that $\phi\in{\mathbb{R}}$ with

[TABLE]

Since

[TABLE]

and since $H$ (and with it $\tilde{H}$ and $\operatorname{diag}\tilde{H}$) is in limit point case, we conclude that $\phi\in\{0,\frac{\pi}{2}\}$. Thus, either $\binom{1}{0}$ or $\binom{0}{1}$ belongs to $L^{2}(\operatorname{diag}\tilde{H})$, and hence to $L^{2}(\tilde{H})$. From this we obtain that either $\xi_{\alpha}$ or $\xi_{\alpha+\frac{\pi}{2}}$ belongs to $L^{2}(H)$. There exists at most one angle $\phi$ modulo $\pi$ such that $\xi_{\phi}\in L^{2}(H)$, and therefore $\alpha$ is uniquely determined modulo $\frac{\pi}{2}$. ∎

5.3 Discussion of \threfQ109,Q132

For Hamiltonians of a particularly simple form the conditions given in \threfQ102,Q104,Q106 can be evaluated. We consider Hamiltonians which are defined on the interval $(0,1)$, where $h_{1}(t)=1$ a.e., and where $h_{2}(t)$ grows sufficiently regularly towards the singular endpoint $1$ in the following sense.

5.8 Definition**.**

\thlabel

Q178

1. (i)

A function $\psi\colon[1,\infty)\to(0,\infty)$ is called regularly varying with index $\rho_{\psi}\in{\mathbb{R}}$, if it is measurable and

[TABLE] 2. (ii)

We call a function $\varphi\colon[0,1)\to(0,\infty)$ regularly varying at $1$ with index $\rho\in{\mathbb{R}}$, if the function

[TABLE]

is regularly varying with index $\rho$.

∎

5.9 Lemma**.**

\thlabel

*Q179 Let $\varphi\colon[0,1)\to(0,\infty)$ be continuous and regularly varying at $1$ with index $\rho>0$, and set $\kappa(t)\mathrel{\mathop{:}}=1$, $t\in[0,1)$. Then the numbers $\omega_{n}$ constructed in (3.2) satisfy222We write $\alpha_{n}\asymp\beta_{n}$, if there exist $c_{1},c_{2}>0$ such that $c_{1}\alpha_{n}\leq\beta_{n}\leq c_{2}\alpha_{n}$, $n\in{\mathbb{N}}$. *

[TABLE]

Proof.

We have $\|\mathds{1}_{(t,b)}\kappa\|^{2}=b-t$, and hence the sequence $(c_{n})_{n=0}^{\infty}$ is given as

[TABLE]

Since $\varphi$ is continuous, we find $t_{n}\in J_{n}$ with

[TABLE]

Set

[TABLE]

Since $\frac{x_{n-1}}{x_{n}}=\frac{1}{2}$, we have $y_{n}=k_{n}x_{n}$ with $k_{n}\in[\frac{1}{2},1]$. From the Uniform Convergence Theorem, see e.g. [bingham.goldie.teugels:1989, Theorem 1.5.2], we obtain that

[TABLE]

Passing back to $\varphi,c_{n},t_{n}$, this yields $\varphi(t_{n})\asymp\varphi(c_{n})$. ∎

Proof of \threfQ109.

We apply \threfQ179 with the function $\varphi(t)\mathrel{\mathop{:}}=\sqrt{h_{2}(t)}$. This is justified, since the corresponding function $\psi$ is

[TABLE]

and hence is regularly varying with index $\frac{\alpha}{2}$. Therefore the numbers $\omega_{n}$ which decide about the behaviour of the operator $S_{21}$ satisfy

[TABLE]

Using this relation, the stated spectral properties of $H_{\alpha;\alpha_{1},\alpha_{2}}$ follow immediately from the sequential characterisations given in (4.3), (3.3), (4.2), and (5.1). Let us go through the cases.

$\triangleright$

First of all the Krein-de Branges formula implies

[TABLE]

$\triangleright$

If $(\alpha>2)$ or $(\alpha=2,\alpha_{1}<0)$ or $(\alpha=2,\alpha_{1}=0,\alpha_{2}<0)$, then $\lim_{n\to\infty}\omega_{n}=\infty$, and hence [math] belongs to the essential spectrum.

$\triangleright$

If $(\alpha=2,\alpha_{1}=\alpha_{2}=0)$, then $\omega_{n}\asymp 1$, and hence the spectrum is not discrete, but bounded invertibility takes place.

$\triangleright$

If $(\alpha<2)$ or $(\alpha=2,\alpha_{1}>0)$ or $(\alpha=2,\alpha_{1}=0,\alpha_{2}>0)$, then $\lim_{n\to\infty}\omega_{n}=0$, and hence the spectrum is discrete.

$\triangleright$

If $(\alpha=2,\alpha_{1}>0)$, then the convergence exponent of $(\omega_{n})_{n=1}^{\infty}$ equals $\frac{2}{\alpha_{1}}$, while in the case $(\alpha=2,\alpha_{1}=0,\alpha_{2}>0)$, the convergence exponent of $(\omega_{n})_{n=1}^{\infty}$ is infinite. From this and (5.3) it follows that (for $\alpha=2$)

[TABLE]

$\triangleright$

If $(\alpha=2,\alpha_{1}\in(0,2),\alpha_{2}\in{\mathbb{R}})$ and ${\mathscr{g}}(r)\mathrel{\mathop{:}}=r^{\frac{2}{\alpha_{1}}}(\log r)^{\gamma}$, then

[TABLE]

This shows that for $\gamma=-\frac{\alpha_{2}}{\alpha_{1}}$ we have $n\cdot{\mathscr{g}}(\frac{1}{\omega_{n}})^{-1}\asymp 1$. Since the sequence $(\omega_{n})_{n=1}^{\infty}$ is comparable to a monotone sequence, it follows that

[TABLE]

∎

Proof of \threfQ132.

With a simple trick properties of $\mathring{H}_{\alpha_{1},\alpha_{2}}$ can be obtained from \threfQ109. To explain this, we start in the reverse direction. Consider the Hamiltonian $H_{2;\alpha_{1},\alpha_{2}}$, and set

[TABLE]

where

[TABLE]

Moreover, let $q$ be the Weyl-coefficient of $H_{2;\alpha_{1},\alpha_{2}}$ and $\mathring{q}$ the one of $\mathring{H}_{2;\alpha_{1},\alpha_{2}}$. Then, by [kaltenbaeck.winkler.woracek:bimmel, Theorem 4.2], we have

[TABLE]

Thus the spectra of $A_{[{\mathring{H}_{2;\alpha_{1},\alpha_{2}}}]}$ and $A_{[{H_{2;\alpha_{1},\alpha_{2}}}]}$ are together discrete or not. If these spectra are discrete, then the convergence exponent of $\sigma(A_{[{H_{2;\alpha_{1},\alpha_{2}}}]})$ is twice the convergence exponent of $\sigma(A_{[{\mathring{H}_{2;\alpha_{1},\alpha_{2}}}]})$. This yields

[TABLE]

Integrating by parts gives

[TABLE]

The function $(h_{2}(t)(1-t))^{2}$ is again of the form (1.5) with $\alpha=2$, but with the parameters $2\alpha_{1}$ and $2\alpha_{2}$ instead of $\alpha_{1}$ and $\alpha_{2}$. Thus the spectrum of $A_{[{\mathring{H}_{\alpha_{1},\alpha_{2}}}]}$ has the same asymptotic behaviour as the spectrum of $A_{[{\mathring{H}_{2;\frac{\alpha_{1}}{2},\frac{\alpha_{2}}{2}}}]}$. ∎

Appendix A Proof of AJPR-type theorems

In this appendix we give a detailed proof of \threfQ120,Q152,Q170.

Recall the relevant notation. Given a finite or infinite interval $(a,b)$, and measurable functions $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ with $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$, $c\in(a,b)$, we consider the possibly unbounded integral operator $T$ on $L^{2}(a,b)$ acting as

[TABLE]

on its natural maximal domain. Let $c_{0}\mathrel{\mathop{:}}=a<c_{1}<c_{2}<\ldots<b$ is a sequence with

[TABLE]

and we denote

[TABLE]

Moreover, let ${{\mathfrak{I}}}$ be an operator ideal (remember that we do not distinguish notationally between an operator ideal and its sequence space). The task is to prove the implications in the triangle

[TABLE]

in the following situations.

1. (i)

For \threfQ170: ${{\mathfrak{I}}}=\ell^{\infty}$. 2. (ii)

For \threfQ120: ${{\mathfrak{I}}}=c_{0}$. 3. (iii)

For \threfQ152: ${{\mathfrak{I}}}\subsetneq{{\mathfrak{S}}}_{\infty}$, assuming that ${{\mathfrak{I}}}$ is fully symmetric for the downwards implication, and assuming that ${{\mathfrak{I}}}$ is symmetrically normed and Matsaev’s Theorem holds in ${{\mathfrak{I}}}$ for the upwards implication.

Proof of “ $\operatorname{Re}T\in{{\mathfrak{I}}}\Rightarrow(\omega_{n})_{n=1}^{\infty}\in{{\mathfrak{I}}}$”.

Let ${{\mathfrak{I}}}$ be $\ell^{\infty}$, $c_{0}$, or a fully symmetric operator ideal, and assume that $\operatorname{Re}T\in{{\mathfrak{I}}}$. Moreover, denote by $P_{n}$ the orthogonal projection $P_{n}f\mathrel{\mathop{:}}=\mathds{1}_{J_{n}}f$ of $L^{2}(a,b)$ onto its subspace $L^{2}(J_{n})$.

Since ${{\mathfrak{I}}}$ is of one of the stated forms, we obtain that $\sum_{n=1}^{\infty}P_{n}(\operatorname{Re}T)P_{n+1}\in{{\mathfrak{I}}}$. For ${{\mathfrak{I}}}=\ell^{\infty}$ or ${{\mathfrak{I}}}=c_{0}$, this is obvious. For ${{\mathfrak{I}}}$ being fully symmetric, it is a consequence of [gohberg.krein:1965, Theorem II.5.1]333This is actually a variant of [gohberg.krein:1965, Theorem II.5.1] which is easy to obtain in the present situation since all spaces $L^{2}(J_{n})$ have the same Hilbert space dimension. Choose unitary operators $U_{n}\colon L^{2}(J_{n})\to L^{2}(J_{n+1})$, let $S\colon L^{2}(a,b)\to L^{2}(a,b)$ be the block shift

S\mathrel{\mathop{:}}=\left(\begin{array}[]{ccccc}0&&&&\\[3.0pt] U_{1}&0&&&\\[3.0pt] &U_{2}&0&&\\[1.0pt] &&\ddots&\ddots&\end{array}\right)\colon\begin{pmatrix}L^{2}(J_{1})\\ \oplus\\ L^{2}(J_{2})\\ \oplus\\ \vdots\end{pmatrix}\to\begin{pmatrix}L^{2}(J_{1})\\ \oplus\\ L^{2}(J_{2})\\ \oplus\\ \vdots\end{pmatrix},

and apply [gohberg.krein:1965, Theorem II.5.1] to the operator $(\operatorname{Re}T)S$. .

Clearly, $P_{n}TP_{n+1}=(\textvisiblespace\,,\mathds{1}_{J_{n+1}}\kappa)\mathds{1}_{J_{n}}\varphi$. The adjoint of $T$ is the integral operator with kernel

[TABLE]

and hence $P_{n}T^{*}P_{n+1}=0$. Together,

[TABLE]

and hence a_{n}\big{(}\sum_{n=1}^{\infty}P_{n}(\operatorname{Re}T)P_{n+1}\big{)}=2^{-\frac{1}{2}}\omega_{n}^{*}. We see that $(\omega_{n})_{n=1}^{\infty}\in{{\mathfrak{I}}}$. ∎

The upwards implication in the triangle (A.2) is a bit more involved.

Proof of “ $(\omega_{n})_{n=1}^{\infty}\in{{\mathfrak{I}}}\Rightarrow T\in{{\mathfrak{I}}}$”.

Let ${{\mathfrak{I}}}$ be $\ell^{\infty}$, $c_{0}$, or a symmetrically normed ideal in which Matsaev’s Theorem holds, and assume that $(\omega_{n})_{n=1}^{\infty}\in{{\mathfrak{I}}}$. Note that in every case $(\omega_{n})_{n=1}^{\infty}$ is bounded. StepsCount191

The crucial point is to handle the diagonal cell sum $\sum_{n=1}^{\infty}P_{n}TP_{n}$. Our aim is to show that this series converges to an operator in ${{\mathfrak{I}}}$.

The summand $P_{n}TP_{n}$ is the integral operator in $L^{2}(a,b)$ with kernel

[TABLE]

Since $\mathds{1}_{J_{n}}\kappa,\mathds{1}_{J_{n}}\varphi\in L^{2}(a,b)$, it is compact and

[TABLE]

The sequence $(\omega_{n})_{n=1}^{\infty}$ is bounded, and hence the series $\sum_{n=1}^{\infty}P_{n}TP_{n}$ converges strongly, and its sum is a bounded operator with

[TABLE]

This settles the case that ${{\mathfrak{I}}}=\ell^{\infty}$. If $\lim_{n\to\infty}\omega_{n}=0$, the series converges w.r.t. the operator norm and hence its sum is a compact operator. This settles the case that ${{\mathfrak{I}}}=c_{0}$.

Consider the remaining case. Then, in particular, $\lim_{n\to\infty}\omega_{n}=0$. Let $Q_{0}$ be the compact operator given by the Schmidt-series

[TABLE]

Then $a_{n}(Q_{0})=\omega_{n}^{*}$, and hence $Q_{0}\in{{\mathfrak{I}}}$. Since Matsaev’s Theorem holds in ${{\mathfrak{I}}}$, the triangular truncation transformator ${\mathcal{C}}$, cf. [gohberg.krein:1967], is defined on all of ${{\mathfrak{I}}}$ and maps ${{\mathfrak{I}}}$ boundedly into itself. Thus ${\mathcal{C}}Q_{0}\in{{\mathfrak{I}}}$, and

[TABLE]

However, {\mathcal{C}}\big{(}(\textvisiblespace\,,\mathds{1}_{J_{n}}\kappa)\mathds{1}_{J_{n}}\varphi\big{)}=P_{n}TP_{n}. Thus we have $\sum_{n=1}^{\infty}P_{n}TP_{n}\in{{\mathfrak{I}}}$.

The rest of the proof merely uses completeness. For $l\in{\mathbb{N}}$ let $Q_{l}$ be the compact operator given by the Schmidt-series

[TABLE]

Then $Q_{l}\in{{\mathfrak{I}}}$ and $\|Q_{l}\|_{{{\mathfrak{I}}}}=2^{-l/2}\|(\omega_{n})_{n=1}^{\infty}\|_{{{\mathfrak{I}}}}$. Hence, the series $\sum_{l=1}^{\infty}Q_{l}$ converges w.r.t. $\|\textvisiblespace\|_{{{\mathfrak{I}}}}$ and its sum belongs to ${{\mathfrak{I}}}$.

A short computation using (A.1) shows that

[TABLE]

Since $T$ is closed, it follows that $T=\sum_{n=1}^{\infty}P_{n}TP_{n}+\sum_{l=1}^{\infty}Q_{l}$, and we conclude that $T\in{{\mathfrak{I}}}$.

∎

Appendix B Sequential vs. continuous conditions

In this section we give detailed proofs of \threfQ122,Q155,Q171. Recall the relevant notation: We are given a finite or infinite interval $(a,b)$, and measurable functions $\kappa,\varphi\colon(a,b)\to{\mathbb{C}}$ with $\kappa\in L^{2}(a,b)$ and $\mathds{1}_{(a,c)}\varphi\in L^{2}(a,b)$, $c\in(a,b)$. Further, $c_{0}\mathrel{\mathop{:}}=a<c_{1}<c_{2}<\ldots<b$ is a sequence with

[TABLE]

and $J_{n}\mathrel{\mathop{:}}=(c_{n-1},c_{n})$ and $\omega_{n}\mathrel{\mathop{:}}=\|\mathds{1}_{J_{n}}\kappa\|\cdot\|\mathds{1}_{J_{n}}\varphi\|$. Moreover, denote

[TABLE]

The proof of \threfQ122 and \threfQ171 is simple.

Proof of \threfQ122,Q171.

A sequence $(\alpha_{n})_{n=1}^{\infty}$ of nonnegative numbers is bounded (tends to [math]), if and only if the sequence \big{(}2^{-n}\sum_{k=1}^{n}2^{k}\alpha_{k}\big{)}_{n=1}^{\infty} is bounded (tends to [math], respectively). Applying this with

[TABLE]

yields the assertion. ∎

The proof of \threfQ155 is based on dualising and requires some technique for Orlicz ideals. We start with one preparatory lemma.

Let a growth function ${\mathscr{g}}$ with $\rho_{{\mathscr{g}}}>1$ be given. Passing to an equivalent growth function and modifying ${\mathscr{g}}$ on a finite interval does not change the truth value of the left side of the asserted equivalence. Hence, we may assume w.l.o.g. that the function $M(t)\mathrel{\mathop{:}}={\mathscr{g}}(t^{-1})^{-1}$ has all properties required in \threfQ154, cf. \threfQ156, and additionally that $M(t)\asymp t^{\rho_{\mathscr{g}}}$, $t\geq 1$. Note that, since $\rho_{\mathscr{g}}>1$,

[TABLE]

Using the language of [maligranda:1989, Chapter 8,p.47] this means that $M$ belongs to the class $N$.

B.1 Lemma**.**

\thlabel

Q192 Set $I\mathrel{\mathop{:}}={\mathbb{N}}$ and let $q\in(0,1)$.

1. (i)

Set $J\mathrel{\mathop{:}}=\{(n,k)\in I\times I\colon\mkern 3.0muk\leq n\}$ and let ${{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$ denote the Orlicz space of sequences indexed by $I$ or by $J$ depending on the context. For a sequence $(\alpha_{n})_{n\in I}$ define sequences $(\beta_{(n,k)})_{(n,k)\in J}$ and $(\beta_{(n,k)}^{\prime})_{(n,k)\in J}$ as

[TABLE]

If $(\alpha_{n})_{n\in I}\in{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$, then $(\beta_{(n,k)})_{(n,k)\in J},(\beta_{(n,k)}^{\prime})_{(n,k)\in J}\in{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$. There exists a constant $C_{1}>0$ such that

[TABLE] 2. (ii)

Consider a sequence $(\alpha_{n})_{n\in I}$ with $(q^{n}\alpha_{n})_{n\in I}\in{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$, and define a sequence $(\beta_{n})_{n\in I}$ as

[TABLE]

Then $(q^{n}\beta_{n})_{n\in I}\in{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$. There exists a constant $C_{2}>0$ such that

[TABLE]

Proof.

For the proof of item (i) let $(\alpha_{n})_{n\in I}\in{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$ with $\|(\alpha_{n})_{n\in I}\|_{{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}}\leq 1$ be given. Then, in particular, $|\alpha_{n}|\leq 1$, $n\in I$. Since $\rho_{\mathscr{g}}>1$, we have

[TABLE]

cf. [bingham.goldie.teugels:1989, Proposition 7.4.1,Theorem 1.5.6]. Note that $C\geq 1$. Thus we can estimate

[TABLE]

Since $|\beta_{n,k}|\leq|\alpha_{n}|\leq 1$ and $\frac{C^{2}}{1-q}\geq 1$, it follows that

[TABLE]

This shows that $\|(\beta_{(n,k)})_{(n,k)\in J}\|_{{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}}\leq\frac{C^{2}}{1-q}$.

The sequence $(\beta_{(n,k)}^{\prime})_{(n,k)\in J}$ is handled in the same way. Namely

[TABLE]

from which we again obtain that $\|(\beta_{(n,k)}^{\prime})_{(n,k)\in J}\|_{{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}}\leq\frac{C^{2}}{1-q}$.

The proof of (ii) is based on dualising. For each $\gamma>0$ we have

[TABLE]

cf. [bingham.goldie.teugels:1989, Proposition 7.4.1]. Hence the Orlicz indices of $M$ at [math], cf. [maligranda:1989, Chapter 11,p.84], are both equal to $\rho_{\mathscr{g}}$. Let $M^{*}$ be the Orlicz function complementary to $M$, cf. [maligranda:1989, Chapter 8,p.48]. Then both Orlicz indices of $M^{*}$ are equal to $\frac{\rho_{\mathscr{g}}}{\rho_{\mathscr{g}}-1}$, cf. [maligranda:1989, Corollary 11.6]. From [maligranda:1989, Theorem 11.13] we obtain

[TABLE]

Now let $(\sigma_{n})_{n\in I}\in{{\mathfrak{S}}}_{\llbracket{M^{*}}\rrbracket}$. Then we can use (i), the Hölder inequality [maligranda:1989, Chapter 8,Corollary 3], and the relation [maligranda:1989, Theorem 1.1] between Amemiya- and Luxemburg norms, to estimate

[TABLE]

By [maligranda:1989, Theorem 8.6] it follows that $(q^{n}\beta_{n})_{n\in I}\in{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$ and

[TABLE]

∎

Proof of \threfQ155.

For $t\in J_{n}$ it holds that

[TABLE]

and we can estimate

[TABLE]

This shows that the implication “$\Leftarrow$” holds.

Conversely, we have for $t\in J_{n}$

[TABLE]

Assume that $(\omega_{n})_{n=1}^{\infty}\in{{\mathfrak{S}}}_{\llbracket{M}\rrbracket}$. Since \omega_{n}=\|\kappa\|\big{(}\frac{1}{\sqrt{2}}\big{)}^{n}\cdot\|\mathds{1}_{J_{n}}\varphi\|, we can apply \threfQ192(ii) with the sequence $\alpha_{n}\mathrel{\mathop{:}}=\|\mathds{1}_{J_{n}}\varphi\|$, $n\in{\mathbb{N}}$. This shows that

[TABLE]

and we obtain

[TABLE]

∎

Appendix C I.S.Kac’s compactness theorem

Let us recall [kac:1995, Theorem 1], which is stated as the main result in this paper. Unfortunately, Kac’s original proofs are not available, and we do not know any source where proofs are given.

C.1 Theorem** ([kac:1995]).**

\thlabel

Q141 Let H=\Bigl{(}\begin{smallmatrix}h_{1}\hskip 0.60275pt&\hskip 0.60275pth_{3}\\[2.15277pt] h_{3}\hskip 0.60275pt&\hskip 0.60275pth_{2}\end{smallmatrix}\Bigr{)} be a Hamiltonian on $[0,\infty)$ which is normalised such that $\operatorname{tr}H(t)=1$ a.e., and denote $m_{j}(t)\mathrel{\mathop{:}}=\int_{0}^{t}h_{j}(t)\mkern 4.0mudt$. For $K>0$ set

[TABLE]

Then (i)$\Rightarrow$(ii)$\Rightarrow$(iii), where

1. (i)

{\displaystyle\sup\Big{(}\bigcup_{K<\frac{1}{4}}A_{K}^{+}\cup\bigcup_{K<\frac{1}{4}}B_{K}^{+}\Big{)}=+\infty,\ \inf\Big{(}\bigcup_{K<\frac{1}{4}}A_{K}^{+}\cup\bigcup_{K<\frac{1}{4}}B_{K}^{+}\Big{)}=-\infty}. 2. (ii)

$\sigma(A_{[{H}]})$* is discrete with444This normalisation does not appear in [kac:1995]. However, without it the statement is false. * $\int_{0}^{\infty}h_{1}(s)\mkern 4.0muds<\infty$ or $\int_{0}^{\infty}h_{2}(s)\mkern 4.0muds<\infty$. 3. (iii)

$A_{1}^{+}\cup B_{1}^{+}={\mathbb{R}}\setminus\{0\}$.

In the following theorem we make the connection to our present work.

C.2 Theorem**.**

\thlabel

Q140 Let notation be as in \threfQ141. Then the following items are equivalent.

1. (i)

There exists $K>0$ such that $\inf A_{K}^{+}=-\infty$ and $\sup A_{K}^{+}=+\infty$. 2. (ii)

$\sigma(A_{[{H}]})$* is discrete with $\int_{a}^{b}h_{1}(s)\mkern 4.0muds<\infty$.* 3. (iii)

${\displaystyle\bigcap_{K>0}A_{K}^{+}={\mathbb{R}}\setminus\{0\}}$.

The analogous statement holds when $A_{K}^{+}$ is replaced by $B_{K}^{+}$ and $h_{1}$ by $h_{2}$.

Concerning the normalisation in item (ii) remember again \threfQ105. Moreover, note that the implication “(iii)$\Rightarrow$(i)” is trivial.

Proof of “(i)$\Rightarrow$(ii)”.

Choose $c\in(0,\infty)$ such that $h_{2}$ does not vanish almost everywhere on $[0,c]$, and consider the function $F:[c,\infty)\times{\mathbb{R}}\to[0,\infty]$ defined as

[TABLE]

StepsCount191

We show that for every finite interval $[\mu_{1},\mu_{2}]\subseteq{\mathbb{R}}$ it holds that

[TABLE]

Let $\lambda\in(\mu_{1},\mu_{2})$, and write $\lambda=v\mu_{1}+(1-v)\mu_{2}$ with $v\in(0,1)$. The Hölder inequality gives

[TABLE]

and similarly

[TABLE]

Multiplying these inequalites, and using the elementary inequality

[TABLE]

which is seen by distinguishing cases whether $x,y$ are $\leq 1$ or $\geq 1$ and whether $v\leq\frac{1}{2}$ or $v\geq\frac{1}{2}$, we obtain (C.1).

Assume now that (i) holds. Let $0<\varepsilon<1$, and choose $\mu_{1}\in A_{K}^{+}\cap(-\infty,0)$ and $\mu_{2}\in A_{K}^{+}\cap(0,\infty)$ such that $K\mu_{j}^{-2}\leq\varepsilon^{2}$. Then (C.1) gives $\limsup\limits_{t\to\infty}F(t,0)\leq\varepsilon$. We conclude that

[TABLE]

in particular, $\int_{0}^{\infty}h_{1}(s)\mkern 4.0muds<\infty$. \threfQ102 implies that $\sigma(A_{[{H}]})$ is discrete.

∎

Proof of “(ii)$\Rightarrow$(iii)”.

StepsCount191

By \threfQ102 the condition (C.2) holds. The condition stated in (iii) is an extension of this relation to nonzero values of $\lambda$; obviously it holds if and only if

[TABLE]

Since we have the normalisation $\operatorname{tr}H(t)=1$, it holds that

[TABLE]

Note that $p$ is nonincreasing. Moreover, again from trace normalisation,

[TABLE]

In this step we show that

[TABLE]

To this end estimate

[TABLE]

Integrating by parts and using (C.4) gives

[TABLE]

and together (C.5) follows.

Under the assumption that $t_{0}>0$ is such that $p(t_{0})<\frac{1}{2|\lambda|}$, we show that for all $t\geq t_{0}$

[TABLE]

The product in (C.6) equals

[TABLE]

For the first integral we have

[TABLE]

For the second integral,

[TABLE]

Putting together, (C.6) follows.

Assume now that (C.2) holds. Then we find for every $\lambda\in{\mathbb{R}}\setminus\{0\}$ a suitable point $t_{0}$, and (C.6) implies (C.3).

∎

R. Romanov

Department of Mathematical Physics

Faculty of Physics, St Petersburg State University

7/9 Universitetskaya nab.

199034 St.Petersburg

RUSSIA

email: morovom@gmail.com

H. Woracek

Institute for Analysis and Scientific Computing

Vienna University of Technology

Wiedner Hauptstraße 8–10/101

1040 Wien

AUSTRIA

email: harald.woracek@tuwien.ac.at