Szegő Condition and Scattering for One-Dimensional Dirac Operators

Abstract

We prove the existence of modified wave operators for one-dimensional Dirac operators whose spectral measures belong to the Szegő class on the real line.

Appendices

Appendix I

Here we prove Lemmas 2.2, 2.3, and 2.4 following the ideas of [1].

Proof of Lemma 2.2

Assertions (a), (b) of Lemma 2.2 are the formulas (2.13), (2.14) in [1], respectively. $\square$

Proof of Lemma 2.3

A straightforward calculation shows that the Weyl function of the constant nontrivial Hamiltonian

$$\begin{aligned} {\mathcal {H}}= \begin{pmatrix}c_1 &{}\quad c \\ c &{}\quad c_2\end{pmatrix} \end{aligned}$$

equals $m = i c_{1}^{-1}\sqrt{c_1 c_2 - c^2} + c/c_1$. Now let ${\mathcal {H}}$ be an arbitrary singular nontrivial Hamiltonian, and let ${\widehat{{\mathcal {H}}}}_r$ be defined by (16). Then the Weyl function of $({\hat{{\mathcal {H}}}}_r)_r$ is

$$\begin{aligned} m_{({\hat{{\mathcal {H}}}}_r)_r} = i {\mathcal {I}}_{{\mathcal {H}}}(r) + {\mathcal {R}}_{{\mathcal {H}}}(r). \end{aligned}$$

(57)

Hence, we have ${\mathcal {J}}_{{\widehat{{\mathcal {H}}}}_{r}}(r) = \log c_{1}^{-1}\sqrt{c_1 c_2 - c^2} = - \log c_1 = \log {\mathcal {I}}_{\mathcal {H}}(r)$. Next, let $F_{r}$, $G_r$ and ${\hat{F}}_{r}$, ${\hat{G}}_{r}$ be the functions from Lemma 2.2 for the Hamiltonians ${\mathcal {H}}$ and ${\hat{{\mathcal {H}}}}_r$, respectively. Note that $F_{r}(i) = {\hat{F}}_{r}(i)$, $G_{r}(i) = {\hat{G}}_{r}(i)$ by construction and formula (57). It follows from assertion (a) of Lemma 2.2 that ${\hat{m}}_r(i) = m_0(i)$; that is,

$$\begin{aligned} {\mathcal {I}}_{{\widehat{{\mathcal {H}}}}_r}(0) = {\mathcal {I}}_{{\mathcal {H}}}(0), \qquad {\mathcal {R}}_{{\widehat{{\mathcal {H}}}}_r}(0) = {\mathcal {R}}_{{\mathcal {H}}}(0). \end{aligned}$$

As in the proof of Lemma 2.5 in [1], we have

$$\begin{aligned} {\mathcal {J}}_{{\mathcal {H}}}(r) = {\mathcal {J}}_{{\mathcal {H}}}(0) - 2\xi _{{\mathcal {H}}}(r) + 2\log |F_r(i)|, \end{aligned}$$

(58)

where $\xi _{{\mathcal {H}}}: r \mapsto \int _{0}^{r}\sqrt{\det {\mathcal {H}}(t)}\,\mathrm{d}t$. Similarly,

$$\begin{aligned} {\mathcal {J}}_{{\widehat{{\mathcal {H}}}}_r}(r) = {\mathcal {J}}_{{\widehat{{\mathcal {H}}}}_r}(0) - 2\xi _{{\widehat{{\mathcal {H}}}}_r}(r) + 2\log |{\hat{F}}_r(i)|. \end{aligned}$$

Since $\xi _{{\mathcal {H}}}(r) = \xi _{{\widehat{{\mathcal {H}}}}_r}(r)$ and ${\hat{F}}_r(i) = F_r(i)$, we have

$$\begin{aligned} {\mathcal {K}}_{{\mathcal {H}}}(r)&= \log {\mathcal {I}}_{{\mathcal {H}}}(r) - {\mathcal {J}}_{{\mathcal {H}}}(r) = {\mathcal {J}}_{{\widehat{{\mathcal {H}}}}_r}(r) - {\mathcal {J}}_{{\mathcal {H}}}(r),\\&= {\mathcal {J}}_{{\widehat{{\mathcal {H}}}}_r}(0) - {\mathcal {J}}_{{\mathcal {H}}}(0) = {\mathcal {J}}_{{\widehat{{\mathcal {H}}}}_r}(0) - \log {\mathcal {I}}_{{\widehat{{\mathcal {H}}}}_r}(0) + \log {\mathcal {I}}_{{\mathcal {H}}}(0) - {\mathcal {J}}_{{\mathcal {H}}}(0), \\&=-{\mathcal {K}}_{{\widehat{{\mathcal {H}}}}_r}(0) + {\mathcal {K}}_{{\mathcal {H}}}(0). \end{aligned}$$

The last formula can be rewritten in the form ${\mathcal {K}}_{{\mathcal {H}}}(0) = {\mathcal {K}}_{{\mathcal {H}}}(r) + {\mathcal {K}}_{{\widehat{{\mathcal {H}}}}_r}(0)$. Since the functions ${\mathcal {K}}_{{\mathcal {H}}}$, ${\mathcal {K}}_{{\widehat{{\mathcal {H}}}}_r}$ are nonnegative, we see that ${\mathcal {K}}_{{\mathcal {H}}}$ is nonincreasing. This fact and the semi-continuity of logarithmic integrals implies $\lim _{r \rightarrow +\infty } {\mathcal {K}}_{{\widehat{{\mathcal {H}}}}_r}(0) = {\mathcal {K}}_{{\mathcal {H}}}(0)$, or, equivalently, $\lim _{r \rightarrow +\infty } {\mathcal {K}}_{{\mathcal {H}}}(r) = 0$, see details in Lemma 4.1 of [1]. $\square$

Proof of Lemma 2.4

Assume first that the Hamiltonian ${\mathcal {H}}= \left( {\begin{matrix} h_1 &{}\quad h\\ h &{}\quad h_2\\ \end{matrix}} \right)$ is continuously differentiable on ${\mathbb {R}}_+$. As in the proof of Lemma 2.7 in [1], formula (5) yields

$$\begin{aligned} \left. \left( \!{\begin{matrix}\Theta ^+(r,i)'&{}\quad \Phi ^+(r,i)'\\ \Theta ^-(r,i)'&{}\quad \Phi ^-(r,i)'\end{matrix}}\!\right) \right| _{r=0} = \left( {\begin{matrix}ih(0)&{}\quad ih_2(0)\\ -ih_1(0)&{}\quad -ih(0)\end{matrix}}\right) . \end{aligned}$$

(59)

Then (58) and the initial condition $M(0,i) = \left( {\begin{matrix} 1 &{} 0\\ 0 &{} 1\\ \end{matrix}} \right)$ give

$$\begin{aligned} {\mathcal {J}}_{{\mathcal {H}}}'(0)&= -2\xi '(0) + 2 \left. \mathrm{Re}\left( \frac{\Theta ^+(r,i)' +m'_r(i)\Theta ^-(r,i) + m_r(i)\Theta ^-(r,i)'}{\Theta ^+(r,i) +m_r(i)\Theta ^-(r,i)}\right) \right| _{r=0},\\&= -2\xi '(0) + 2\mathrm{Re}(ih(0) - im_0(i)h_1(0)),\\&= -2\xi '(0) + 2 h_1(0) {\mathcal {I}}_{{\mathcal {H}}}(0), \end{aligned}$$

where the derivatives are taken with respect to r. The same relation for the Hamiltonian ${\mathcal {H}}_r$ in place of ${\mathcal {H}}$ shows that ${\mathcal {J}}'_{{\mathcal {H}}}(r) = -2\xi '_{{\mathcal {H}}}(r) + 2 h_1(r) {\mathcal {I}}_{{\mathcal {H}}}(r)$ for all $r > 0$. Similarly, differentiating relation $m_0(i) = \frac{G_r(i)}{F_r(i)}$ from Lemma 2.2 at $r = 0$ and using (59), we obtain $0 = ih_2(0) + m'_0(i) - i m_0(i)h(0) - m_0(i)(ih(0) - i m_0(i)h_1(0))$. As before, this gives

$$\begin{aligned} 0 = ih_2(r) + m'_r(i) - i m_r(i)h(r) - m_r(i)(ih(r) - i m_r(i)h_1(r)) \end{aligned}$$

(60)

for all $r > 0$. By construction, we have $m_r(i) = i{\mathcal {I}}_{{\mathcal {H}}}(r) + {\mathcal {R}}_{\mathcal {H}}(r)$. Taking imaginary and real parts in (60), we obtain

$$\begin{aligned} 0&=h_2 + {\mathcal {I}}'_{\mathcal {H}}- 2{\mathcal {R}}_{\mathcal {H}}h + {\mathcal {R}}_{\mathcal {H}}^2 h_1 - {\mathcal {I}}_{{\mathcal {H}}}^{2}h_1,\\ 0&={\mathcal {R}}'_{{\mathcal {H}}}+2{\mathcal {I}}_{{\mathcal {H}}}h - 2{\mathcal {I}}_{\mathcal {H}}{\mathcal {R}}_{\mathcal {H}}h_1, \end{aligned}$$

respectively. These two relations, together with the definition of the entropy function ${\mathcal {K}}_{{\mathcal {H}}} = \log {\mathcal {I}}_{{\mathcal {H}}} - {\mathcal {J}}_{\mathcal {H}}$ and the formula ${\mathcal {J}}'_{{\mathcal {H}}} = -2\xi '_{{\mathcal {H}}} + 2 h_1 {\mathcal {I}}_{{\mathcal {H}}}$, imply the statement of the lemma in the case where ${\mathcal {H}}$ is smooth. The general case then follow as in the proof of Lemma 2.7 in [1]. $\square$

Appendix II

Proposition 2.5 stems from a general approximation theory developed by M. Riesz and S. Mergelian, see, e.g., Section VI in [12]. For the reader’s convenience, we prove it below. Our approach is standard and has much in common with Section 5.2 in [2]. We will use several known results, the first of which can be found in Section 3.G.2 of [12].

Theorem 7.1

Let g be analytic in ${\mathbb {C}}^+$, continuous up to ${\mathbb {R}}$, and such that $\log g \cdot (x^2+1)^{-1} \in L^1({\mathbb {R}})$. Suppose that $\log |g(z)| \leqslant c(|z|+1)$ for a constant c and all $z \in {\mathbb {C}}^+$. Then

$$\begin{aligned} \log |g(z)| \leqslant A\mathrm{Im}z + \frac{1}{\pi }\int _{{\mathbb {R}}}\log |g(x)|\frac{\mathrm{Im}z}{|x-z|^{2}}\,\mathrm{d}x, \qquad z \in {\mathbb {C}}^+, \end{aligned}$$

where $A = \limsup _{y \rightarrow +\infty }\frac{\log |g(iy)|}{y}$.

De Branges spaces admit the following “axiomatic” description, see Section 23 in [5].

Theorem 7.2

For every Hermite–Biehler function E, the space ${\mathcal {B}}(E)$ has the following properties:

$(A_1)$:

whenever f is in the space and has a nonreal zero w, the function $\frac{z - {\bar{w}}}{z - w}f$ is in the space and has the same norm as f;

$(A_2)$:

for every nonreal number w, the evaluation functional $f \mapsto f(w)$ is continuous;

$(A_3)$:

the function $f^\sharp$ belongs to the space whenever f belongs to the space and it always has the same norm as f.

Conversely, for every Hilbert space of entire functions ${\mathcal {B}}$ satisfying $(A_1)$-$(A_3)$, there exists a Hermite–Biehler function E such that ${\mathcal {B}} = {\mathcal {B}}(E)$.

A Hermite–Biehler function E is called regular if $\int _{{\mathbb {R}}}\frac{1}{1+t^2}\frac{\mathrm{d}t}{|E(t)|^2} < \infty$. It is easy to check that a Hermite–Biehler function E is regular if and only if for every $w \in {\mathbb {C}}$ and $F \in {\mathcal {B}}(E)$, we have $\frac{F - F(w)}{z - w} \in {\mathcal {B}}(E)$. De Branges spaces satisfying this property are called regular. For every Hamiltonian ${\mathcal {H}}$ on ${\mathbb {R}}_+$, the de Branges spaces from its chain $\{{\mathcal {B}}_r\}_{r \in {\mathcal {M}}}$ are regular, see, e.g., Section 3 in [18].

If $E_1$ and $E_2$ are regular Hermite–Biehler functions without zeroes on ${\mathbb {R}}$, then $E_1/E_2$ is a function of bounded type in ${\mathbb {C}}^+$ (that is, $E_1/E_2$ is a ratio of two functions in $H^2({\mathbb {C}}^+)$). The next theorem is usually referred to as the de Branges ordering theorem, see Section 35 in [5].

Theorem 7.3

Let ${\mathcal {B}}(E_1)$ and ${\mathcal {B}}(E_2)$ be given spaces which are contained isometrically in a space $L^2(\mu )$. If $E_1/E_2$ is a function of bounded type in the upper half-plane and has no real zeroes or singularities, then either ${\mathcal {B}}(E_1)$ contains ${\mathcal {B}}(E_2)$ or ${\mathcal {B}}(E_2)$ contains ${\mathcal {B}}(E_1)$.

An entire function f is said to have a finite exponential type if there exists the finite upper limit

$$\begin{aligned} \mathop {\mathrm {type}}\nolimits f = \limsup _{|z| \rightarrow \infty }\frac{\log |f(z)|}{|z|}. \end{aligned}$$

Given a de Branges space ${\mathcal {B}}$, put $\mathop {\mathrm {type}}\nolimits {\mathcal {B}}= \sup \{\mathop {\mathrm {type}}\nolimits f: \; f \in {\mathcal {B}}\}$. The following theorem goes back to Krein [13] and de Branges [4]; its short proof can be found in Section 6 of [18].

Theorem 7.4

Let ${\mathcal {H}}$ be a nontrivial Hamiltonian on ${\mathbb {R}}_+$, and let $\{{\mathcal {B}}_r\}_{r \in {\mathcal {M}}}$ be its chain of de Branges spaces. Then $\mathop {\mathrm {type}}\nolimits ({\mathcal {B}}_r) = \int _{0}^{r}\sqrt{\det {\mathcal {H}}(\tau )}\,\mathrm{d}\tau$ for every $r \in {\mathcal {M}}$.

Proof of Proposition 2.5

Consider an element $h \in L^2(\mu )$ orthogonal to all functions in ${\mathcal {E}}_s$ and such that $\Vert h\Vert _{L^2(\mu )} = 1$. For every $f \in {\mathcal {E}}_s$ and $\lambda \in {\mathbb {C}}$, the function $z \mapsto \frac{f - f(\lambda )}{z - \lambda }$ belongs to ${\mathcal {E}}_s$; hence

$$\begin{aligned} \int _{{\mathbb {R}}}\frac{f(x) - f(\lambda )}{x - \lambda }\overline{h(x)}\,\mathrm{d}\mu (x) = 0, \qquad \lambda \in {\mathbb {C}}. \end{aligned}$$

(61)

Define the function $H :\lambda \mapsto \int _{{\mathbb {R}}}\frac{\overline{h(x)}}{x-\lambda }\,\mathrm{d}\mu (x)$ on ${\mathbb {C}}^+$. Relation (61) can be rewritten in the form

$$\begin{aligned} f(\lambda ) = \frac{1}{H(\lambda )}\int _{{\mathbb {R}}}\frac{f(x)\overline{h(x)}}{x-\lambda }\,\mathrm{d}\mu (x), \qquad \lambda \in {\mathbb {C}}^{+}, \quad H(\lambda )\ne 0. \end{aligned}$$

(62)

Observe that for all $\lambda \in {\mathbb {C}}^+$ with $\mathrm{Im}\lambda \geqslant 1$, we have

$$\begin{aligned} |H(\lambda )|^2 \leqslant \Vert h\Vert _{L^2(\mu )}^{2} \int _{{\mathbb {R}}}\frac{\mathrm{d}\mu (x)}{|x-\lambda |^2} \leqslant \int _{{\mathbb {R}}}\frac{\mathrm{d}\mu (x)}{x^2+1}. \end{aligned}$$

It follows that H is a bounded analytic function in the half-plane $\{\lambda : \mathrm{Im}\lambda \geqslant 1\}$; hence

$$\begin{aligned} \int _{{\mathbb {R}}}\frac{\bigl |\log |H(x+i)|\bigr |}{x^2 + 1}\, \mathrm{d}x < \infty . \end{aligned}$$

(63)

Let us apply Theorem 7.1 to the function ${\tilde{f}}: z \mapsto f(z + i)$. This function is analytic in ${\mathbb {C}}^+$ and is continuous up to ${\mathbb {R}}$. Since f has a finite exponential type, we also have $\log |{\tilde{f}}(z)| \leqslant c(|z|+1)$ for a constant c and all $z \in {\mathbb {C}}^+$. Moreover, from (62) we see that $|{\tilde{f}}(x)| \leqslant \Vert f\Vert _{L^2(\mu )}/|H(x+i)|$ for all $x \in {\mathbb {R}}$; hence $\log {\tilde{f}} \cdot (x^2+1)^{-1} \in L^1({\mathbb {R}})$. Theorem 7.1 and the last estimate imply

$$\begin{aligned} \log |{\tilde{f}}(z)| \nonumber&\leqslant s \mathrm{Im}z + \frac{1}{\pi }\int _{{\mathbb {R}}}\log \left( \frac{\Vert f\Vert _{L^2(\mu )}}{|H(x+i)|}\right) \frac{\mathrm{Im}z}{|x-z|^{2}}\,\mathrm{d}x,\\&\leqslant s \mathrm{Im}z + \log \Vert f\Vert _{L^2(\mu )} + \frac{1}{\pi }\int _{{\mathbb {R}}}\bigl |\log |H(x+i)|\bigr |\frac{\mathrm{Im}z}{|x-z|^{2}}\,\mathrm{d}x \end{aligned}$$

(64)

for all $z \in {\mathbb {C}}^+$. Take $a\geqslant 2$. For $z \in {\mathbb {C}}^+$ with $\mathrm{Im}z = a$, we get from (63), (64) the estimate $|f(z)| \leqslant \Vert f\Vert _{L^2(\mu )}e^{c_{1}|z|^2}$, where the constant $c_{1}$ depends only on a and H. A argument for the lower half-plane gives the estimate $|f(z)| \leqslant \Vert f\Vert _{L^2(\mu )}e^{c_{1}|z|^2}$ for all z with $\mathrm{Im}z = -a$. Let p be a polynomial such that $\sinh (2c_1 z^2)/p$ is analytic and does not vanish in the strip $\Pi _a = \{z: |\mathrm{Im}z| \leqslant a\}$. Then the Phragmén–Lindelöf principle for $p{\tilde{f}}/ \sinh (2c_1 z^2)$ implies that $|f(z)| \leqslant c_2\Vert f\Vert _{L^2(\mu )}e^{c_{2}|z|^2}$ for all $z \in \Pi _a$ and a constant $c_2$ depending only on a and H. In particular, the function f is bounded by $c_{2}e^{4c_{2}}\Vert f\Vert _{L^2(\mu )}$ in the square $\{z : |\mathrm{Re}z| \leqslant a, |\mathrm{Im}z| \leqslant a\}$. Next, for $z \in {\mathbb {C}}^+$ such that $\mathrm{Re}z = \mathrm{Im}z = y$, we have

$$\begin{aligned} \frac{\mathrm{Im}z}{|x-z|^{2}} \leqslant |z|\frac{L_y(x)}{x^2+1}, \qquad L_y(x) = \frac{x^2+1}{(x-y)^2 + y^2}, \qquad x \in {\mathbb {R}}. \end{aligned}$$

Note that $\sup \{L_y(x): \, x \in {\mathbb {R}}, \, y>0\} < \infty$ and $\lim _{y \rightarrow + \infty }L_y(x) = 0$ for every $x \in {\mathbb {R}}$. From (64) we now see that for every $\varepsilon > 0$, there exists a constant $c_\varepsilon$ independent of f such that

$$\begin{aligned} |f(z)| \leqslant c_{\varepsilon }\Vert f\Vert _{L^2(\mu )}e^{(s+\varepsilon )|z|} \end{aligned}$$

(65)

for all $z \in {\mathbb {C}}$ with $|\mathrm{Re}z| = |\mathrm{Im}z|$. Using the Phragmén–Lindelöf principle once again, we conclude that for every $\varepsilon >0$, relation (65) holds for all $z \in {\mathbb {C}}$ and a new constant $c_{\varepsilon }$ does not depend on f. It follows that the completion of ${\mathcal {E}}_s$ with respect to the norm in $L^2(\mu )$ forms a Hilbert space ${\mathcal {B}}$ of entire functions of exponential type at most s (the inner product is inherited from $L^2(\mu )$). The same estimate (65) yields the property $(A_2)$ from Theorem 7.2 for the space ${\mathcal {B}}$. It is also easy to verify that ${\mathcal {B}}$ satisfies properties $(A_1)$, $(A_3)$ from Theorem 7.2. Hence, the space ${\mathcal {B}}$ coincides with a de Branges space of entire functions. Since for every $f \in {\mathcal {E}}_s$, the fraction $\frac{f - f(i)}{z -i}$ belongs to ${\mathcal {E}}_s$, and ${\mathcal {E}}_s$ is dense in ${\mathcal {B}}$, the space ${\mathcal {B}}$ is regular.

Let ${\mathcal {H}}$ be a Hamiltonian on ${\mathbb {R}}_+$ generated by the measure $\mu$, and let $\{{\mathcal {B}}_r\}_{r \in {\mathcal {M}}}$ be the corresponding chain of de Branges spaces constructed in Sect. 2. Put $r = \inf \{t: s = \int _{0}^{t}\sqrt{\det {{\mathcal {H}}(\tau )}}\,\mathrm{d}\tau \}$. Note that for every $s' \in (0, s)$, the first part of the proof applies to ${\mathcal {E}}_{s'}$ and yields the fact that $(\mathrm{PW}_{s'}, \mu )$ is a de Branges space of entire functions of exponential type at most $s'$. Theorem 7.3 and Theorem 7.4 then imply that $(\mathrm{PW}_{s'}, \mu ) \subset {\mathcal {B}}_r$. Since $\cup _{s'<s}{\mathcal {B}}_{r(s')}$ is a dense subset in ${\mathcal {B}}$, we see that the isometric embedding ${\mathcal {B}}\subset {\mathcal {B}}_r$ holds.

To prove that ${\mathcal {B}}_r \subset {\mathcal {B}}$, note that r is a density point of the set ${\mathcal {M}}\cap [0,r]$. Therefore, the set $\cup _{r'<r}{\mathcal {B}}_{r'}$ is dense in ${\mathcal {B}}_r$. Now take any point $r' \in {\mathcal {M}}\cap [0,r)$ and find $s'$ such that $r> s' > \int _{0}^{r'}\sqrt{\det {{\mathcal {H}}(\tau )}}\,\mathrm{d}\tau$. Using again the first part of the proof and Theorem 7.4, we see that ${\mathcal {B}}_{r'} \subset (\mathrm{PW}_{s'}, \mu ) \subset {\mathcal {B}}$. It follows that ${\mathcal {B}}_r \subset {\mathcal {B}}$, as required. $\square$