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Generic non-selfadjoint Zakharov–Shabat operators

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Abstract

In this paper we develop tools to study within a family of non-selfadjoint operators \(L(\varphi )\) depending on a parameter \(\varphi \) in a real Hilbert space, those with (partially) simple spectrum. As a case study we consider the Zakharov–Shabat operators \(L(\varphi )\) appearing in the Lax pair of the focusing NLS on the circle. In particular, the main result implies that the set of potentials \(\varphi \) of Sobolev class \(H^N\), \(N\ge 0\), so that all non real eigenvalues of \(L(\varphi )\) are simple, is path connected and dense.

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Notes

  1. A function \(F : V\rightarrow \mathbb C\), \(V\subseteq i L^2_r\), is called analytic if it is the restriction to \(V=V_c\cap i L^2_r\) of an analytic function \(\tilde{F} : V_c\rightarrow \mathbb C\) where \(V_c\) is an open set in \(L^2_c\).

  2. The periodic eigenvalues are counted with their algebraic multiplicities.

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Acknowledgments

We would like to thank the referee for his many thoughtful suggestions leading to significant improvements of the paper.

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Authors and Affiliations

Authors

Corresponding author

Correspondence to P. Topalov.

Additional information

T. Kappeler was supported in part by the Swiss National Science Foundation.

P. Lohrmann was supported in part by the Swiss National Science Foundation and the European Research Council under FP7 “New connections between dynamical systems and Hamiltonian PDE with small divisor phenomena”.

P. Topalov was supported in part by NSF Grant DMS-0901443.

Appendices

Appendix A: \(L^2\)-gradients of averaging functions

In this appendix, in a quite general set-up, we state and prove a theorem on multiple roots of characteristic functions applied in the proofs of Theorem 3.1 and Theorem 3.2.

Theorem 5.1

Let \(F, \chi : {\mathbb C} \times L^2_c \rightarrow {\mathbb C}\) be analytic maps and \(\psi \) an arbitrary but fixed element in \(L^2_c\). Assume that at \(z_\psi \in {\mathbb C}, \ \chi (\cdot , \psi )\) has a zero of order \(m \ge 1\). Then the following statements hold:

  1. (i)

    For any \(\varepsilon > 0\) sufficiently small there exists an open neighborhood \(V \subseteq L^2_c\) of \(\psi \) such that for any \(\varphi \in V\), \(\chi (\cdot ,\varphi )\) has exactly \(m\) roots \(z_1(\varphi ), \ldots , z_m(\varphi )\), listed with their multiplicities, in the open disk \(D^\varepsilon \equiv D ^\varepsilon (z_\psi ):= \{ \lambda \in {\mathbb C}\,|\,|\lambda - z_\psi |< \varepsilon \}\) and no roots on the boundary \(\partial D^\varepsilon \) of \(D^\varepsilon \).

  2. (ii)

    The functional \(F_\chi : V \rightarrow {\mathbb C}\), defined by

    $$\begin{aligned} F_\chi (\varphi ):= \sum ^m_{j=1} F(z_j(\varphi ), \varphi ), \end{aligned}$$

    is analytic and at \(\varphi = \psi \)

    $$\begin{aligned} \partial F_\chi =\Big ( m\,\partial F + \sum ^m_{j=0} a_j \partial ^{m-j}_\lambda \partial \chi \Big )\Big \arrowvert _{\lambda = z_\psi } \end{aligned}$$

    where \(a_j \in {\mathbb C}, \ 0 \le j \le m\), \(\partial \) denotes the \(L^2\)-gradient with respect to \(\varphi \), and \(\partial _\lambda \) denotes the derivative with respect to \(\lambda \). If \(F(\cdot , \psi )\) has a zero of order \(k \ge 1\) at \(z_\psi \), then \(a_0 = \cdots = a_{k-1} = 0\); if \(k = m\), then

    $$\begin{aligned} a_m = - \frac{1}{m!}\,\partial ^m_\lambda \Big (F(\lambda , \psi ) \frac{(\lambda - z_\psi )^{m+1} \partial _\lambda \chi (\lambda , \psi )}{\chi (\lambda , \psi )^2}\Big ) \Big \arrowvert _{\lambda = z_\psi } \not = 0 . \end{aligned}$$

Proof

  1. (i)

    By the analyticity of \(\chi (\cdot , \psi )\) there exists \(\varepsilon > 0\) so that \(\chi (\cdot , \psi )\) does not vanish on \(\overline{D^\varepsilon }{\setminus }\{z_\psi \}\). By the analyticity of \(\chi \) it then follows that there exists a neighborhood \(V\) of \(\psi \) in \(L^2_c\) so that for any \(\varphi \in V\), \(\chi (\cdot ,\varphi )\) does not vanish in a small tubular neighborhood of \(\partial D^\varepsilon \) in \(\mathbb C\). One then concludes by the argument principle that for any \(\varphi \in V\), \(\chi (\cdot ,\varphi )\) has precisely \(m\) zeros in \(D^\varepsilon \), when counted with their multiplicities.

  2. (ii)

    Again by the argument principle, for any \(\varphi \in V\) one has

    $$\begin{aligned} F_\chi (\varphi ) = \frac{1}{2\pi i} \int \limits _{\partial D^\varepsilon } F(\lambda , \varphi )\,\frac{\dot{\chi }(\lambda , \varphi )}{\chi (\lambda , \varphi )}\,d\lambda \end{aligned}$$
    (5.1)

    where \(\dot{\chi }= \partial _{\lambda }\chi \). Note that the integrand in (5.1) is analytic on \(\partial D^\varepsilon \times V\), whence \(F_\chi \) is analytic on \(V\). Recall from Sect. 2 that for \(g = (g_1, g_2)\), \(h= (h_1, h_2) \in L^2_c\), we defined

    $$\begin{aligned} \langle g, h \rangle _r = \int \limits ^1_0 (g_1 h_1 + g_2 h_2) dx . \end{aligned}$$

    Then, by the definition of the \(L^2\)-gradient, one has at \(\varphi = \psi \),

    $$\begin{aligned} \langle \partial F_\chi , h \rangle _r&= \frac{d}{ds} \Big \arrowvert _{s=0} F_\chi (\psi + sh) \\&= \frac{1}{2\pi i} \int _{\partial D^\varepsilon }\frac{d}{ds}\Big |_{s=0} \left( F(\lambda , \psi + sh) \frac{\dot{\chi }(\lambda , \psi + sh)}{\chi (\lambda , \psi + sh)} \right) d\lambda . \end{aligned}$$

By the product rule one gets at \(\varphi = \psi \)

$$\begin{aligned} \langle \partial F_\chi , h\rangle _r = \frac{1}{2\pi i}\int _{\partial D^\varepsilon } \left[ \langle \partial F, h \rangle _r\, \frac{\dot{\chi }}{\chi } + F\cdot \left( \frac{1}{\chi }\,\langle \partial \dot{\chi }, h \rangle _r - \frac{1}{\chi ^2}\,\langle \partial \chi , h \rangle _r \dot{\chi }\right) \right] \,d\lambda . \end{aligned}$$

Hence \(\partial F_\chi \) is given by

$$\begin{aligned}&\frac{1}{2\pi i} \int _{\partial D^\varepsilon }\!\!\! \Big ( \frac{\dot{\chi }}{\chi } \partial F + \frac{1}{(\lambda - z_\psi )^m} \cdot \frac{( \lambda - z_\psi ) ^m F }{\chi } (\partial \chi )^\cdot \nonumber \\&\quad - \frac{1}{(\lambda - z_\psi )^{m+1}} \cdot \frac{(\lambda - z_\psi )^{m+1} F}{\chi ^2} {\dot{\chi }}{\partial \chi } \Big )\, d\lambda . \end{aligned}$$
(5.2)

Here we used that \(\partial \dot{\chi }=(\partial \chi )^\cdot \) and that \(\partial F, \partial \chi : {\mathbb C} \rightarrow L^2_c\) are analytic and hence in particular, the maps \(\partial F, \partial \chi , (\partial \chi )^\cdot : {\mathbb C} \rightarrow L^2_c\) are continuous. Furthermore, as by assumption, \(\chi (\cdot , \psi )\) has a zero of order \(m\) at \(\lambda = z_\psi \), the argument principle implies that, at \(\varphi = \psi \),

$$\begin{aligned} \frac{1}{2\pi i} \int _{\partial D^\varepsilon } \frac{\dot{\chi }}{\chi } \partial F d \lambda = m\,\partial F \Big \arrowvert _{\lambda = z_\psi }\,. \end{aligned}$$

Moreover, \(\frac{(\lambda - z_\psi )^mF}{\chi }(\partial \chi )^\cdot \) and \(\frac{(\lambda - z_\psi )^{m+1} F\dot{\chi }}{\chi ^2}\partial \chi \) are both analytic functions on \(\overline{D^\varepsilon }\) with values in \(L^2_c\). Hence by Cauchy’s integral formula,

$$\begin{aligned}&\frac{1}{2\pi i} \int _{\partial D^\varepsilon } \frac{1}{(\lambda - z_\psi )^m} \frac{(\lambda - z_\psi )^mF}{\chi }(\partial \chi )^\cdot \,d\lambda \\&\quad = \frac{1}{(m-1)!} \partial ^{m-1}_\lambda \Big \arrowvert _{\lambda = z_\psi } \left( \frac{(\lambda - z_\psi )^m F}{\chi } (\partial \chi )^\cdot \right) \end{aligned}$$

and

$$\begin{aligned}&\frac{1}{2\pi i} \int _{\partial D^\varepsilon } \frac{1}{(\lambda - z_\psi )^{m+1}} \frac{(\lambda - z_\psi )^{m+1}F\dot{\chi }}{\chi ^2} \partial \chi d\lambda \\&\quad = \frac{1}{m!} \partial ^{m}_{\lambda } \Big \arrowvert _{\lambda = z_\psi } \left( \frac{(\lambda - z_\psi )^{m+1} F\dot{\chi }}{\chi ^2} \partial \chi \right) . \end{aligned}$$

Thus \(\partial F_\chi \) at \(\varphi = \psi \) is given by

$$\begin{aligned} \partial F_\chi&= m\,\partial F \Big \arrowvert _{\lambda = z_\psi } + \frac{1}{(m-1)!}\, \partial ^{m-1}_\lambda \Big \arrowvert _{\lambda = z _\psi } \left( \frac{(\lambda - z_\psi )^m F}{\chi }\,(\partial \chi )^\cdot \right) \\&- \frac{1}{m!}\,\partial ^m_\lambda \Big \arrowvert _{\lambda = z _\psi } \left( \frac{(\lambda - z_\psi )^{m+1} F \dot{\chi }}{\chi ^2}\,\partial \chi \right) . \end{aligned}$$

The claimed formula for \(\partial F_\chi \) at \(\varphi = \psi \) then follows from the Leibniz rule. If \(F(\cdot , \psi )\) has a zero of order \(k \ge 1\) at \(\lambda = z_\psi \), then at \(\varphi = \psi \)

$$\begin{aligned} \partial F_\chi = \left( m\,\partial F + \sum ^m_{j=k} a_j \partial ^{m-j}_\lambda \partial \chi \right) \Big |_{\lambda =z_\psi }, \end{aligned}$$

i.e., \(a_j = 0\) for \(0 \le j \le k - 1\). If \(k = m\), then

$$\begin{aligned} \partial F_\chi = \Big (m\,\partial F + a_m \partial \chi \Big )\Big \arrowvert _{\lambda = z_\psi } \end{aligned}$$

where in this case

$$\begin{aligned} a_m = - \frac{1}{m!}\,\partial ^m_\lambda \Big (F(\lambda , \psi ) \frac{(\lambda - z _\psi )^{m+1} \dot{\chi }(\lambda , \psi )}{\chi (\lambda , \psi )^2}\Big ) \Big |_{\lambda = z_\psi } \not = 0 . \end{aligned}$$

\(\square \)

Finally we record a few simple facts from linear algebra, needed in Sect. 3. Consider \(f = (f_1, f_2)\) in \(L^2_c\) and denote by \(\ell _f\) the \({\mathbb R}\)-linear functional on the \({\mathbb R}\)-vector space \(iL^2_r\) induced by \(f\),

$$\begin{aligned} \ell _f : iL^2_r\rightarrow {\mathbb C}, h \mapsto \langle f , h \rangle _r , \end{aligned}$$

where

$$\begin{aligned} \langle f, h \rangle _r = \int ^1_0 (f_1 h_1 + f_2 h_2) dx . \end{aligned}$$

Write \(\ell _f (h)\) as \(\ell _{f,R}(h) + i \ell _{f,I}(h)\) where \(\ell _{f,R}\) and \(\ell _{f,I}\) are the elements in the dual \({\mathcal L}(iL^2_r, {\mathbb R})\) of \(iL^2_r\) given by

$$\begin{aligned} \ell _{f,R}(h) = \mathrm{Re} (\langle f,h \rangle _r) \quad \text{ and }\quad \ell _{f,I}(h) = \mathrm{Im} \, (\langle f, h\rangle _r). \end{aligned}$$
(5.3)

They can be expressed in terms of \(f\) and \(\hat{f} = - (\overline{f}_2, \overline{f}_1)\) as follows

$$\begin{aligned} \ell _{f,R}(h) = \Big \langle \frac{f + \hat{f}}{2}, h\Big \rangle _r \text{ and } \ell _{f,I}(h) = \Big \langle \frac{f - \hat{f}}{2i}, h\Big \rangle _r . \end{aligned}$$
(5.4)

As the subspace \(iL^2_r\subseteq L^2_c\) is the subset of all elements \(\varphi \in L^2_c\) satisfying \(\varphi = \hat{\varphi }\) it follows that \(\frac{f + \hat{f}}{2}\) and \(\frac{f - \hat{f}}{2i}\) are in \(iL^2_r\). Using that \(f \mapsto \hat{f}\) is an involution and that for any \(c\) in \({\mathbb C}\), \(\widehat{(c f)} = \overline{c} \hat{f}\), the following lemma can be proved in a straightforward way.

Lemma 5.1

  1. (i)

    \(\ell _{f,R}\), \(\ell _{f,I}\) are \({\mathbb R}\)-linearly independent iff \(\frac{f + \hat{f}}{2}, \frac{f - \hat{f}}{2i}\) are \({\mathbb R}\)-linearly independent.

  2. (ii)

    \(\ell _{f,R}\), \(\ell _{f,I}\) are \({\mathbb R}\)-linearly independent iff \(\frac{f + \hat{f}}{2}\), \(\frac{f - \hat{f}}{2i}\) are \({\mathbb C}\)-linearly independent.

  3. (iii)

    For any \(\lambda \in {\mathbb C}{\setminus }\{ 0\}\), \(\ell _{f,R}\), \(\ell _{f,I}\) are \({\mathbb R}\)-linearly independent iff \(\ell _{\lambda f,R}\), \(\ell _{\lambda f,I}\) are \(\mathbb R\)-linearly independent.

  4. (iv)

    \(\ell _{f,R}, \ell _{f,I}\) are \({\mathbb R}\)-linearly dependent iff there exists \(\lambda \in {\mathbb C} \backslash \{ 0 \} \) so that

    $$\begin{aligned} \frac{\lambda f + \widehat{(\lambda f)}}{2} = 0 . \end{aligned}$$

Appendix B: Examples

In this appendix we consider potentials in \(iL^2_r\) of the form \((a \in \mathbb C, k \in \mathbb Z\))

$$\begin{aligned} \varphi _{a,k}(x)=(a e^{2\pi i k x}, -\bar{a} e^{-2 \pi i k x}). \end{aligned}$$
(6.1)

Most of the results presented in this section can be found in [13]. We include them for the convenience of the reader. First we show that we can easily relate various spectra of \(L(\varphi _{a,k})\) with the corresponding ones for \(k=0\). More generally, for an arbitrary potential \(\varphi \in L^2_c\), various spectra of \(L(\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x})\) are related to the corresponding spectra of \(L(\varphi _1, \varphi _2)\) by the following lemma which can be verified in a straightforward way.

Lemma 6.1

Assume that \(f=(f_1,f_2)\) is a solution of \(L(\varphi )f= \lambda f\) where \(\varphi \in L^2_c\) is arbitrary. Then \((f_1 e^{i \pi k x}, f_2e^{- i \pi k x} )\) is a solution of

$$\begin{aligned} L(\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x})g=(\lambda - k \pi ) g. \end{aligned}$$

Corollary 6.1

For any \(\varphi \in L^2_c\) and \(k \in \mathbb Z\), the fundamental solution

$$\begin{aligned} M^{(k)}(x, \lambda ) \equiv M(x, \lambda , (\varphi _1 e^{2i\pi k x}, \varphi _2e^{-2i\pi k x})) \end{aligned}$$

of \(L(\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x})\) is related to the fundamental solution \(M(x, \lambda )\) of \(L(\varphi _1, \varphi _2)\) by

$$\begin{aligned} M^{(k)}(x,\lambda ) = \text {diag }(e^{i \pi k x}, e^{-i \pi k x}) \cdot M(x, \lambda + k \pi ). \end{aligned}$$

Corollary 6.1 yields the following application.

Proposition 6.2

For any \(\varphi =(\varphi _1, \varphi _2) \in L^2_c\) and any \(k \in \mathbb Z\),

$$\begin{aligned} {\mathop {\mathrm{spec}}\nolimits _{p}} (L(\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))= {\mathop {\mathrm{spec}}\nolimits _{p}}(L(\varphi _1, \varphi _2)) - k\pi \end{aligned}$$

and

$$\begin{aligned} {\mathop {\mathrm{spec}}\nolimits _{D}} (L(\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))= {\mathop {\mathrm{spec}}\nolimits _{D}} (L(\varphi _1, \varphi _2)) - k\pi \end{aligned}$$

(with multiplicities).

Proof

Recall that the characteristic functions \(\chi _{p}\) and \(\chi _{D}\) are given by

$$\begin{aligned} \chi _{p}(\lambda )&=(\grave{m}_1(\lambda )+\grave{m}_4(\lambda ))^2-4 \\ 2 i \chi _{D}(\lambda )&= \grave{m}_4(\lambda )+\grave{m}_3(\lambda )-\grave{m}_2(\lambda )-\grave{m}_1(\lambda ). \end{aligned}$$

By Corollary 6.1,

$$\begin{aligned} \chi _{p}(\lambda , (\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))=\chi _{p}(\lambda + k\pi , \varphi ) \end{aligned}$$

and

$$\begin{aligned} \chi _{D}(\lambda , (\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))=(-1)^{k}\chi _{D}(\lambda + k\pi , \varphi ). \end{aligned}$$

As \({\mathop {\mathrm{spec}}\nolimits _{p}} (L(\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))\) and \({\mathop {\mathrm{spec}}\nolimits _{D}} (L(\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))\) are the zero sets (with multiplicities) of \(\chi _{p}(\lambda , (\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))\) respectively \(\chi _{D}(\lambda , (\varphi _1 e^{2\pi i k x}, \varphi _2e^{-2\pi i k x}))\), the claimed identities follow. \(\square \)

In view of Proposition 6.2, instead of the potentials \(\varphi _{a,k}\) defined by (6.1), it suffices to consider the case \(k=0\),

$$\begin{aligned} \varphi _a \equiv \varphi _{a,0}=(a, - \bar{a}), \quad a \in \mathbb C. \end{aligned}$$

In a straightforward way one verifies the following

Lemma 6.2

For any \(a \in \mathbb C\),

$$\begin{aligned} M(x, \lambda , \varphi _a)= \left( \begin{array}{cc} \cos ( \kappa x)- i \lambda \, \frac{ \sin (\kappa x) }{ \kappa } &{} ia\,\frac{\sin (\kappa x)}{\kappa } \\ i \bar{a}\, \frac{\sin (\kappa x)}{\kappa } &{} \cos ( \kappa x)+ i \lambda \,\frac{\sin (\kappa x)}{\kappa } \end{array} \right) \end{aligned}$$
(6.2)

where

$$\begin{aligned} \kappa \equiv \kappa (\lambda , a)= \sqrt{\lambda ^2+|a|^2} \end{aligned}$$
(6.3)

Remark 6.3

Note that \(\kappa \) depends only on the modulus \(|a|\) of \(a\) and that the right hand side of (6.2) does not depend on the choice of the sign of the root \(\sqrt{\lambda ^2+|a|^2}\) as cosine is an even function whereas sine is odd. Furthermore, the right hand side of (6.2) is well defined at \(\kappa =0\) as \(\frac{\sin (\kappa x)}{ \kappa }=x+O(\kappa ^2)\) .

Periodic spectrum of \(L(\varphi _a)\): By Lemma 6.2 one has \(\Delta (\lambda , \varphi _a)=2\cos \kappa (\lambda )\) and hence the characteristic function \(\chi _p(\lambda ,\varphi _a)\) of \(L(\varphi _a)\) is given by

$$\begin{aligned} \chi _{p}(\lambda , \varphi _a)=\Delta ^2(\lambda , \varphi _a)-4=-4\sin ^2(\kappa (\lambda )). \end{aligned}$$
(6.4)

The periodic eigenvalues of \(L(\varphi _a)\) are thus given by the \(\lambda \)’s satisfying \(\kappa (\lambda )=n \pi \) for some \(n \in \mathbb Z\), or

$$\begin{aligned} \lambda ^2+ |a|^2= n^2 \pi ^2. \end{aligned}$$
(6.5)

The monodromy matrix \(\grave{M}\) for such a \(\lambda \) is given by

$$\begin{aligned} \grave{M}=\left( \begin{array}{cc}(-1)^n &{}\quad 0 \\ 0 &{}\quad (-1)^n \end{array} \right) \end{aligned}$$
(6.6)

when \(n\ne 0\) and by

$$\begin{aligned} \grave{M} = \left( \begin{array}{cc} 1-i\lambda &{}\quad i a \\ i \bar{a} &{}\quad 1+i\lambda \end{array} \right) = \left( \begin{array}{cc} 1\pm |a| &{}\quad i a \\ i \bar{a} &{}\quad 1 \mp |a| \end{array} \right) \end{aligned}$$
(6.7)

when \(n=0\). It is convenient to list the periodic eigenvalues not in lexicographic ordering, but rather use the integer \(n \in \mathbb Z\) in (6.5) as an index. When listed in this way, we denote the periodic eigenvalues by \(\hat{\lambda }_n ^\pm \), \(n \in \mathbb Z\), which are defined as follows. For any \(n \in \mathbb Z\) with \(|n \pi | > |a|\) denote

$$\begin{aligned} \hat{\lambda }_n^+=\hat{\lambda }_n^-= \text {sgn}(n)\cdot \sqrt{n^2 \pi ^2 - |a|^2} \end{aligned}$$

where here and in the sequel \(\sqrt{x}\) denotes the branch of the square root on \(\mathbb {C}{\setminus }\mathbb {R}_{\le 0}\) determined by \(\sqrt{1}=1\). In view of (6.6), \(\hat{\lambda }_n^+\) defined above is a periodic eigenvalue of \(L(\varphi _a)\) of geometric multiplicity two. Using (6.3) and (6.4) one easily sees that \(\hat{\lambda }^+_n\) has algebraic multiplicity two. Further, for any \(n \in \mathbb Z\) with \(0<|n \pi | < |a|\) let

$$\begin{aligned} \hat{\lambda }_n^+=\hat{\lambda }_n^-= \text {sgn}(n)\cdot i\sqrt{|a|^2 -n^2\pi ^2}. \end{aligned}$$

Again, in view of (6.6), for \(n \in {\mathbb Z}\) with \(0 < |n\pi | < |a|\), \(\hat{\lambda }_n^+\) is a periodic eigenvalue of \(L(\varphi _a)\) of geometric multiplicity two and, by (6.3) and (6.4), its algebraic multiplicity is two. Next note that for \(n = 0\), one has \(\hat{\lambda }^\pm _0 = \pm i|a|\). In view of (6.7), for \(a \not = 0\) the geometric multiplicity of \(\hat{\lambda }_0^+\) as well as of \(\hat{\lambda }_0^-\) equals one. In view of (6.3) and (6.4) the algebraic multiplicity of \(\hat{\lambda }_0^+\) and the one of \(\hat{\lambda }_0^-\) is one. For \(a \not = 0\), the eigenfunctions corresponding to \(\hat{\lambda }_0^+\) and \(\hat{\lambda }_0^-\) are the constant vectors \(\left( a, i|a| \right) \) resp. \(\left( a, -i|a| \right) \). We then obtain the following result, used in the proof of Theorem 3.1.

Lemma 6.3

For any \(k \in {\mathbb Z}\), consider the potential \(\varphi _{a,-k} = (ae^{-2 i\pi k x}, -\overline{a}e^{2i \pi k x})\). Then \(\hat{\lambda }^\pm _0 = k \pi \pm i|a|\) are periodic eigenvalues of \(L(\varphi _{a,-k})\) of algebraic multiplicity one.

In the special case where \(|a|=n_a \pi \) for some \(n_a \in \mathbb Z_{>0}\) set \(\hat{\lambda }^{\pm }_{\pm n_a}=0\). The above computations yield

Corollary 6.4

  1. (i)

    For \(a \in \mathbb C\), \(\varphi _a\) is a standard potential iff \(|a|< \pi \).

  2. (ii)

    For \(a\in \mathbb C\), any multiple periodic eigenvalue \(\lambda \) of \(L(\varphi _a)\) satisfies \(m_p(\lambda )=2\) and \(m_g(\lambda )=2\) iff \(|a|>\pi \) and \(|a|\ne \pi \mathbb Z\).

  3. (iii)

    If \(a \in \mathbb C {\setminus } \left\{ 0 \right\} \) satisfies \(|a| \in \pi \mathbb Z\), then \(0\) is a periodic eigenvalue of \(L(\varphi _a)\) of algebraic multiplicity four.

Isospectral set \(\text {Iso}_0(\varphi _a)\): Denote by \(\text {Iso}_0(\varphi _a)\) the connected component containing \(\varphi _a\) of the set \(\text {Iso}(\varphi _a)\) of all potentials \(\varphi \in iL^2_r\) with \({\mathop {\mathrm{spec}}\nolimits _{p}}L(\varphi )={\mathop {\mathrm{spec}}\nolimits _{p}}L(\varphi _a)\). By the computations above one sees that

$$\begin{aligned} \{|a| e^{i \alpha } \, |\, \alpha \in \mathbb R \} \subseteq \text {Iso}_0(\varphi _a). \end{aligned}$$

For \(|a|\) sufficiently small, \(\varphi _a\) is in the domain of the Birkhoff map introduced in Theorem 1.1 in [10]. As the \(L_2\)-norm is a spectral invariant it then follows that, for \(|a|\) sufficiently small, all of \(\text {Iso}(\varphi _a)\) is contained in this domain. According to the computations of \({\mathop {\mathrm{spec}}\nolimits _{p}} L(\varphi _a)\) it then follows from Theorem 1.1 in [10] and its proof that \(\text {Iso}(\varphi _a)\) is homeomorphic to a circle. As a consequence

$$\begin{aligned} \text {Iso}(\varphi _a)=\text {Iso}_0(\varphi _a) =\{|a| e^{i \alpha } \, |\, \alpha \in \mathbb R \}. \end{aligned}$$

Most likely the latter identities remain true for any \(|a|< \pi \), but we have not verified this. For \(|a|>\pi \), Li and McLaughlin observed in [13] that \(\text {Iso}_0(\varphi _a)\) is larger than \(\{|a| e^{i \alpha } \, |\, \alpha \in \mathbb R \}\). Indeed, let \(\pi < |a| < 2 \pi \). Then \(\hat{\lambda }^+_{\pm 1}= \pm i \sqrt{|a|^2-\pi ^2}\) are periodic eigenvalues of geometric multiplicity two. In subsection 4.3 of [13], using Bäcklund transformation techniques, formulas of solutions of fNLS are presented which evolve on \(\text {Iso}_0(\varphi _a)\) and depend explicitly on \(x\). They are parametrized by the punctured complex plane \(\mathbb C^*:=\left\{ e^{\rho }e^{i\beta }\right\} \) with coordinates \((\rho , \beta ) \in \mathbb R \times \mathbb R / 2 \pi \mathbb Z\), whereas the angle variable \(\alpha \) in \(\{|a| e^{i \alpha } \, |\, \alpha \in \mathbb R \}\) is proportional to the time \(t\). As \(t\rightarrow \pm \infty \) these solutions approach the \(x\) independent solutions evolving on \(\{|a| e^{i \alpha } \, |\, \alpha \in \mathbb R \}\). Due to the trace formulas ([13], Sect. 2.4), on the orbits of these solutions, the periodic eigenvalues \(\hat{\lambda }^+_{\pm 1}\) have geometric multiplicity one.

Dirichlet spectrum of \(L(\varphi _a)\): By Lemma 6.2, the characteristic function of the Dirichlet spectrum of \(L(\varphi _a)\) is given by

$$\begin{aligned} \chi _{D}(\lambda , \varphi _a)=\frac{\sin \kappa }{\kappa }\left( \lambda + \frac{\bar{a} - a}{2} \right) . \end{aligned}$$
(6.8)

The Dirichlet eigenvalues of \(L(\varphi _a)\) are thus given by the \(\lambda \)’s satisfying

$$\begin{aligned} \kappa (\lambda )= n \pi \end{aligned}$$
(6.9)

for some \(n \in \mathbb Z{\setminus } \left\{ 0\right\} \) or

$$\begin{aligned} \lambda + \frac{\bar{a} - a}{2}=0. \end{aligned}$$
(6.10)

Note that by the definition of the Dirichlet boundary conditions, any Dirichlet eigenvalue is of geometric multiplicity one. It is convenient to list the Dirichlet eigenvalues not in lexicographic ordering, but rather use as in the case of periodic eigenvalues the integer \(n\) in (6.9) as an index. When listed in this way, we denote the Dirichlet eigenvalues by \(\hat{\mu }_n\), \(n \in \mathbb Z\), which are defined as follows. For all \(n \in \mathbb Z\) with \(|n\pi |>|a|\) denote

$$\begin{aligned} \hat{\mu }_n= \text {sgn}(n)\cdot \sqrt{n^2 \pi ^2-|a|^2}. \end{aligned}$$

From (6.8) it follows that \(\hat{\mu }_n\) has algebraic multiplicity one. For all \(n \in \mathbb Z\) with \(0<|n \pi | < |a|\) let

$$\begin{aligned} \hat{\mu }_n= \text {sgn}(n) \cdot i \sqrt{|a|^2-n^2 \pi ^2}. \end{aligned}$$

By the same arguments as in the case \(|n\pi |> |a|\), the algebraic multiplicity of \(\hat{\mu }_n\) is equal to one iff \(\hat{\mu }_n+ \frac{\bar{a} - a}{2} \not = 0\) and two otherwise. For \(n=0\) denote

$$\begin{aligned} \hat{\mu }_0= i \mathrm{Im} \,\, (a). \end{aligned}$$

Again, by the same arguments, \(\hat{\mu }_0\) has algebraic multiplicity equal to one if

$$\begin{aligned}&[\mathrm{Im} \,(a) \not =0 \text { and } \mathrm{Im} \,(a)\not = \pm \sqrt{|a|^2- n^2 \pi ^2}\,\, \forall \, 0 < |n \pi |< |a|] \text { or } [ \mathrm{Im} \,(a) \!=\!0 \text { and } |a| \not \in \pi \mathbb Z_{>0}] \end{aligned}$$

or two if

$$\begin{aligned} \mathrm{Im} \,(a) \in \left\{ \pm \sqrt{|a|^2- n^2 \pi ^2}\,|\,0< |n \pi |< |a|\right\} \end{aligned}$$

or three if

$$\begin{aligned} \mathrm{Im} \,(a) =0 \quad \text { and } \quad |a| \in \pi \mathbb Z_{>0}. \end{aligned}$$

In the special case where \(|a|= n_a\pi \) for some \(n_a \in \mathbb Z_{>0}\) one has \(\hat{\mu }_{n_a}=\hat{\mu }_{-n_a}=0\). The algebraic multiplicity of \(\hat{\mu }_{n_a}\) is two (\(\mathrm{Im} \,(a) \not = 0\)) or three (\(\mathrm{Im} \,(a) =0\)). These computations lead to the following

Corollary 6.5

Let \(a \in \mathbb C\). Then the following statements hold:

  1. (i)

    If \(|a|< \pi \), then the Dirichlet spectrum of \(L(\varphi _a)\) is simple.

  2. (ii)

    If \(|a| \not \in \pi \mathbb Z_{>0}\), then the only possible multiple Dirichlet eigenvalue is \(i \mathrm{Im} \,(a)\). It is at most of algebraic multiplicity two.

  3. (iii)

    If \(|a| \in \pi \mathbb Z_{>0}\), then \(0\) is a Dirichlet eigenvalue of algebraic multiplicity two or three.

  4. (iv)

    For any \(0 < n \pi < |a| \) or \(n \pi > |a|\), \(\hat{\mu }_n\) is a periodic eigenvalue of geometric and algebraic multiplicity two whereas for \(|a|= n \pi \in \pi \mathbb Z_{> 0}\), \(\hat{\mu }_n=0\) is a periodic eigenvalue of algebraic multiplicity \(4\).

Appendix C: Algebraic multiplicities

In this Appendix we prove that the algebraic multiplicity \(m_p(\lambda )\equiv m_p(\lambda ,\varphi )\) of a periodic eigenvalue \(\lambda \) of the ZS operator \(L(\varphi )\) with \(\varphi \in L^2_c\) equals its multiplicity \(m_r(\lambda )\equiv m_r(\lambda ,\varphi )\) as a root of the characteristic function \(\chi _p(\cdot ,\varphi )\). Throughout the appendix we use the notation introduced in Sect. 2. First we note that by functional calculus, the algebraic multiplicity \(m_p(\lambda )\) is equal to the dimension of the subspace of \({\mathop {\mathrm{dom}}\nolimits _{p}} L(\varphi )\), given by the image of the Riesz projector \(\Pi _\lambda (\varphi )\),

$$\begin{aligned} \Pi _\lambda (\varphi )= \frac{1}{2\pi i}\int _{\partial B(\lambda )} (z-L_p(\varphi ))^{-1}\,dz, \end{aligned}$$

where \(L_p(\varphi )\) denotes the operator \(L(\varphi )\) with domain \({\mathop {\mathrm{dom}}\nolimits _{p}} L(\varphi )\), \(B(\lambda )\) denotes the open disk centered at \(\lambda \) with sufficiently small radius so that \(\overline{B(\lambda )}\cap {\mathop {\mathrm{spec}}\nolimits _{p}} L(\varphi )=\{\lambda \}\), and the circle \(\partial B(\lambda )\) is counterclockwise oriented. By Proposition 2.1, \(\lambda \) is a root of \(\chi _p(\cdot ,\varphi )\).

Lemma 7.1

For any periodic eigenvalue \(\lambda \) of \(L(\varphi )\) with \(\varphi \in L^2_c\), \(m_r(\lambda )=m_p(\lambda )\).

Proof

First, note that a direct computation shows that the statement of the Lemma holds for the zero potential \(\varphi =0\). A simple perturbation argument involving Proposition 2.1, Lemma 2.3, the argument principle, and the properties of the Riesz projector (see the arguments below), then shows that the Lemma continues to hold in an open neighborhood of zero in \(L^2_c\).

Now, consider the general case. Take \(\varphi \in L^2_c\). As \([0,\varphi ]:=\{s\varphi \,|\, 0\le s \le 1 \}\) is compact in \(L^2_c\) there exists a connected open neighborhood \(W\) of \([0, \varphi ]\) in \(L^2_c\) so that the integer \(R \ge 1\) of Lemma 2.3 can be chosen independently of \(\psi \in W\). First consider the periodic eigenvalues in \(B_R\). For \(\psi \in W\) denote by \(\Pi _R(\psi )\) the Riesz projector

$$\begin{aligned} \Pi _R(\psi )= \frac{1}{2\pi i}\int _{\partial B_R}(z-L_p(\psi ))^{-1}\, dz. \end{aligned}$$

Note that by functional calculus

$$\begin{aligned} \mathrm{Image}\, \Pi _R(\psi )=\oplus _{\lambda \in B_R\cap {\mathop {\mathrm{spec}}\nolimits _{p}} L(\psi )}\mathcal {R}_\lambda (\psi ) \end{aligned}$$
(7.1)

where \(\mathcal {R}_\lambda (\varphi )\) is the root space corresponding to \(\lambda \). Moreover, standard arguments show that \(W\rightarrow {\mathcal L}(L^2_c,L^2_c)\), \(\psi \mapsto \Pi _R(\psi )\), is analytic. In particular, by the general properties of the projection operators the dimension of \(\mathrm{Image}\,\Pi _R(\psi )\) is independent on \(\psi \in W\) (see [9], Chapter III, §3). Consider the operator,

$$\begin{aligned} A(\psi )=\frac{1}{2\pi i}\int _{\partial B_R} z (z-L_p(\psi ))^{-1}\, dz\,. \end{aligned}$$

One easily sees that \(W\rightarrow {\mathcal L}(L^2_c,L^2_c)\), \(\psi \mapsto A(\psi )\), is analytic. By functional calculus

$$\begin{aligned} L_p(\psi )|_{\mathrm{Image}\,\Pi _R(\psi )}=A(\psi )|_{\mathrm{Image}\,\Pi _R(\psi )}, \end{aligned}$$

and hence

$$\begin{aligned} \det \Big (\big (\lambda -L_p(\psi )\big )|_{\mathrm{Image}\,\Pi _R(\psi )}\Big )= \det \Big (\big (\lambda -A(\psi )\big )|_{\mathrm{Image}\,\Pi _R(\psi )}\Big )\,. \end{aligned}$$

Hence the polynomial

$$\begin{aligned} Q(\lambda ,\psi ):=\det \Big (\big (\lambda -L_p(\psi )\big )|_{\mathrm{Image}\,\Pi _R(\psi )}\Big ) \end{aligned}$$

is well defined, analytic in \(\mathbb C \times W\), and has leading coefficient one. By (7.1), the roots of \(Q(\cdot ,\psi )\) are precisely the periodic eigenvalues of \(L(\psi )\) in \(B_R\) counted with their multiplicities. On the other hand, define

$$\begin{aligned} P(\lambda , \psi ):= \prod _{|j|\le R}(\lambda -\lambda _j^+)(\lambda -\lambda _j^-). \end{aligned}$$

Note that \(P(\lambda , \psi )\) is a polynomial in \(\lambda \) of degree \(4R+2\) with leading coefficient \(1\). By the argument principle and the last statement of Proposition 2.1, \(P(\lambda ,\psi )\) is analytic in \(\mathbb C \times W\). Hence, the coefficients of \(Q(\cdot ,\psi )\) and \(P(\cdot ,\psi )\) are analytic on \(W\). As \(Q(\cdot ,\psi )=P(\cdot ,\psi )\) in an open neighborhood of zero in \(L^2_c\) we get by analyticity that

$$\begin{aligned} Q(\cdot ,\psi )=P(\cdot ,\psi ) \end{aligned}$$

for any \(\psi \in W\). In particular, the Lemma holds also for any \(\lambda \in B_R\cap {\mathop {\mathrm{spec}}\nolimits _{p}} L(\psi )\). The same argument shows that the statement of the Lemma holds also for any \(\lambda \in D_n\cap {\mathop {\mathrm{spec}}\nolimits _{p}} L(\psi )\), \(|n|>R\). \(\square \)

Note that Lemma 1.1, stated in the introduction, is an immediate consequence of Lemma 2.3 and Lemma 7.1. Indeed, by Lemma 7.1, for any \(\varphi \in L^2_c\), the roots of \(\chi _p(\cdot ,\varphi )\) coincide with the eigenvalues of \(L_p(\varphi )\), together with the corresponding multiplicities. Lemma 1.1 thus follows from Lemma 2.3.

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Kappeler, T., Lohrmann, P. & Topalov, P. Generic non-selfadjoint Zakharov–Shabat operators. Math. Ann. 359, 427–470 (2014). https://doi.org/10.1007/s00208-013-1004-4

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