Szegő condition, scattering, and vibration of Krein strings

Abstract

We give a dynamical characterization of measures on the real line with finite logarithmic integral. The general case is considered in the setting of evolution groups generated by de Branges canonical systems. Obtained results are applied to the Dirac operators and Krein strings.

Appendix

In this Appendix, we collect a few auxiliary results and prove some statements made in the main text.

1.1 A.1 Proof of Proposition 2.8

The modification of the proofs in [8] yields the statement. Alternatively, one can argue as follows. First, we claim that (2.25) implies (2.26) for $\alpha _{n}=\lambda n$ with any $\lambda >0$. Indeed, if $m_{\lambda}$ and $\mu _{\lambda}$ denote Titchmarsh-Weyl function and spectral measure, respectively, of canonical system with Hamiltonian $\mathcal {H}(\lambda \tau )$, then $m_{\lambda}(z)=m_{1}(\lambda ^{-1} z)$ as follows from (1-5) in [8]. Now, it is enough to observe that $\mu _{\lambda}\in \mathrm{Sz}(\mathbb{R})\Longleftrightarrow \mu _{1} \in \mathrm{Sz}(\mathbb{R})$.

Second, we claim that, given intervals $I^{-}\subseteq I$, $|I|=1$, and $\varepsilon \in (0,1]$, the following implication holds

$$ \det \int _{I} \mathcal {H}(\tau ) d\tau - 1=\varepsilon \Rightarrow \det \int _{I^{-}} \mathcal {H}(\tau ) d\tau - |I^{-}|^{2}\lesssim \varepsilon , $$

(A.1)

for every non-negative Hamiltonian ℋ that satisfies $\det \mathcal {H}=1$ a.e. on $I$. Indeed, denote $A=\int _{I^{-}} \mathcal {H}d\tau$ and $B=\int _{I^{+}} \mathcal {H}d\tau$ where $I=I^{-}\cup I^{+}$. Then, we get (see, e.g., (A-1) in [8]):

$$ \det A\ge |I^{-}|^{2}, \quad \det B\ge |I^{+}|^{2} $$

and

$$ \det (A+B)= 1+\varepsilon . $$

Minkowski inequality for determinants yields

$$ \det (A+B)\ge (\sqrt{\det A}+\sqrt{\det B})^{2}. $$

Denoting $\sqrt{\det A}=x$ and $\sqrt{\det B}=y$, we get

$$ x+y\le (1+\varepsilon )^{\frac {1}{2}}, \quad |I^{-}|+|I^{+}|=1, \quad |I^{-}|\le x, \quad |I^{+}|\le y. $$

That implies (draw the corresponding domains on the plane), that $|I^{-}|\le x\le |I^{-}|+C\varepsilon$. Taking the square of the last bound yields the estimate on the right-hand side of (A.1).

Now, if $\det \mathcal {H}=1$ a.e., the statement in (2.26) holds by combining these two claims. Indeed, suppose $\mu \in \mathrm{Sz}(\mathbb{R}_{+})$ and we are given sequence $\{\alpha _{n}\}$. Then, there is $\lambda$ such that every interval $[\alpha _{n}, \alpha _{n+2}]$ is inside one of the intervals $[\lambda l, \lambda (l+2)]$ or $[\lambda (l-1),\lambda (l+1)]$ for some $l$. Since the sum in (2.26) converges for $\{\alpha _{l}\}=\{\lambda l\}$, we can apply (A.1) (with dilated and translated interval $I$) to get condition in the left-hand side of (2.26) satisfied for $\{\alpha _{n}\}$. Conversely, if the sum in (2.26) converges for some $\{\alpha _{n}\}$, then there is suitable $\lambda$ such that each interval $[\lambda n, \lambda (n+2)]$ is covered by either $[\alpha _{l}, \alpha _{l+2}]$ or $[\alpha _{l-1}, \alpha _{l+1}]$ for some $l$. Thus, applying (A.1) again, we get that the sum in (2.26) converges with $\{\alpha _{n}\}=\{\lambda n\}$ and $\mu \in \mathrm{Sz}(\mathbb{R}_{+})$. The case of general ℋ follows by making the change of variables in $\tau$ and using an approximation argument. □

1.2 A.2 Free evolution for canonical systems and Dirac operators

Recall that $\mathfrak {D}_{0}=\mathcal {D}_{\mathcal {H}_{0}}$. We now show that the free evolution for these operators is, in fact, equivalent to the shift on the real line and that relation is algebraic. To this end, we work in terms of Dirac operator and perform two elementary unitary transformations

$$ \widetilde {\mathfrak {D}}_{0}=-Z^{-1}\mathfrak {D}_{0} Z= \left ( \begin{smallmatrix} -i\partial _{\tau }& 0 \\ 0 & i\partial _{\tau}\end{smallmatrix} \right ), \quad Z=\tfrac{1}{\sqrt {2}} \left ( \begin{smallmatrix} i & -1 \\ 1 & -i\end{smallmatrix} \right ). $$

Here, $\partial _{\tau}$ stands for the differentiation operator. Operator $\widetilde {\mathfrak {D}}_{0}$, taken with suitable boundary condition at 0: $f_{1}(0)=if_{2}(0)$, is self-adjoint on the same Hilbert space $\left ( \begin{smallmatrix} f_{1} \\ f_{2}\end{smallmatrix} \right )\in L^{2}(\mathbb{R}_{+},\mathbb{C}^{2})$. If one further maps

$$ \left ( \begin{smallmatrix} f_{1} \\ f_{2}\end{smallmatrix} \right )\mapsto g(x)= \textstyle\begin{cases} f_{1}(x), &x>0, \\ if_{2}(-x), &x< 0,\end{cases} $$

then, $\mathfrak {D}_{0}$ becomes unitary equivalent to $-i\partial _{x}$ on $L^{2}(\mathbb{R})$ with $e^{t\partial _{x}}g=g(x+t)$, which is the standard shift operator.

1.3 A.3 A formula for exponential type

Lemma A.1

If entire function $f$ has bounded type both in $\mathbb{C}_{+}$ and $\mathbb{C}_{-}$, then its exponential type can be computed by the formula

$$ \mathop{type}\nolimits f = \limsup _{y \to +\infty} \frac{\log \max (|f(iy)|, |f(-iy)|)}{y}. $$

(A.2)

Proof

Let us apply Theorem 2 in Lecture 16 of [60]. It says that for every entire function $f$ of bounded type in $\mathbb{C}_{+}$ and $\mathbb{C}_{-}$ we have

$$\begin{aligned} \log |f(z)| &= \sigma _{+} y + o(|z|), \qquad y \ge 0, \\ \log |f(z)| &= \sigma _{-} y + o(|z|), \qquad y \le 0, \end{aligned}$$

outside of a set of disks $\{z \in \mathbb{C}\colon |z - a_{j}| < r_{j}\}$ of finite view (the latter means that $\sum _{j} \frac{r_{j}}{|a_{j}|} < \infty$). Here $\sigma _{\pm} \in \mathbb{R}$ and $y = \mathop{\mathrm{Im}}z$. Take $\varepsilon > 0$ and denote $\sigma = \max (\sigma _{+}, -\sigma _{-})$. By the maximum principle for subharmonic functions, we have

$$ \log |f(z)| \le (\sigma + \varepsilon )|y| + o(|z|) $$

everywhere in ℂ as $z \to \infty$. Therefore, we have $\mathop{type}\nolimits f \le \sigma$. On the other hand, the set of disks of finite view cannot fill any half-axis, hence

$$ \sigma _{+} = \limsup _{y \to +\infty}\frac{\log |f(iy)|}{y}, \qquad - \sigma _{-} = \limsup _{y \to -\infty}\frac{\log |f(iy)|}{|y|}, $$

which proves the statement. □

1.4 A.4 Rotation matrices

The following result in the linear algebra has been used in the main text.

Lemma A.2

For every real $2\times 2$ matrix $A$ with non-negative determinant, there is a rotation matrix $\Sigma _{\varphi }$ of the form

$$ \Sigma _{\varphi }= \begin{pmatrix} \cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{pmatrix} , \qquad \varphi \in [0, 2\pi ), $$

such that $\Sigma _{\varphi }A \ge 0$.

Proof

This is immediate from the proof of the polar decomposition given in [31], p. 276. □

1.5 A.5 Robinson’s theorem

In the main text, we used the following variation of a result by Robinson [70], which is based on ideas dating back to Ruelle’s work [72].

Lemma A.3

Suppose $H$ is a Hilbert space, $\mathcal {D}$ is a densely defined self-adjoint operator, and $P_{\Delta}$ denotes the orthogonal projector for $\mathcal {D}$ relative to a set $\Delta \subseteq \mathbb{R}$. If $A$ is a bounded operator on $H$ and $AP_{[-\Lambda ,\Lambda ]}$ is compact for some $\Lambda \ge 0$, then

$$ \lim _{\mathbf {T}\to +\infty}\frac {1}{\mathbf {T}}\int _{0}^{\mathbf {T}} \|Ae^{it\mathcal {D}}P_{[-\Lambda ,\Lambda ]}\psi \|^{2}dt=\sum _{j}\|AP_{E_{j}}P_{[- \Lambda ,\Lambda ]}\psi \|^{2} $$

for every $\psi \in H$, where $P_{ E_{j}}$ denotes the orthogonal projection on the eigenspace that corresponds to eigenvalue $E_{j}$ and the sum is over all eigenvalues $\{E_{j}\}$ of $\mathcal {D}$.

Proof

The proof is an application of Theorem 2 in [70] to operator $\mathcal {D}P_{[-\Lambda ,\Lambda ]}$ with the perturbation taken as $AP_{[-\Lambda ,\Lambda ]}$ where both $\mathcal {D}P_{[-\Lambda ,\Lambda ]}$ and $AP_{[-\Lambda ,\Lambda ]}$ are considered as operators acting on $H$. □