Non-self adjoint spectral problems

The research direction is the one stated in the question: “The spectral problem with general data, both for NLS and also other related non-self-adjoint problems, however, is far from understood, and poses a great challenge whose resolution is still only in the initial stages.” A concrete model motivating this direction is the semiclassical Zakharov–Shabat eigenvalue problem

$$\epsilon\frac{d}{dx}\mathbf w=\begin{bmatrix}-i\lambda&\psi_0(x)\\-\psi_0(x)^*&i\lambda\end{bmatrix}\mathbf w,$$

where $\epsilon>0$ is small, $\lambda\in\mathbb C$ is the spectral parameter, and the discrete spectrum consists of those $\lambda$ for which there is a nonzero solution $\mathbf w\in L^2(\mathbb R)^2$. The associated question is how to determine the location and accumulation of these eigenvalues in the semiclassical limit for general data. The broad direction does not specify a class of general data or a single operator beyond such motivating examples. The WKB approximation of semiclassical eigenvalues of the Zakharov–Shabat problem The semi-classical limit of focusing NLS for a family of non-analytic initial data

Background

The Zakharov–Shabat problem is the spatial scattering problem in the Lax pair for focusing nonlinear Schrödinger (NLS); its discrete eigenvalues are part of the scattering data used to reconstruct the evolution. In the semiclassical regime, determining that data is itself a substantial obstacle: the eigenvalues become poles in the inverse-scattering Riemann–Hilbert problem, and their number is reported to grow on the order of $1/\epsilon$ for broad classes of initial data. For real data with a single maximum, the cited account reports that the eigenvalues lie on the imaginary axis, but exact determination is available only in special solvable cases. These features explain why asymptotics for the NLS solution and accurate information about the underlying eigenvalues are related but distinct objectives. Semiclassical limit of the scattering transform for the focusing Nonlinear Schrödinger Equation The semi-classical limit of focusing NLS for a family of non-analytic initial data

A central analytical tool in these studies is the inverse-scattering Riemann–Hilbert formulation and its nonlinear steepest-descent analysis. The relevant works developed results for selected data classes rather than a general spectral theory: earlier studies treated analytic soliton ensembles and particular analytic profiles, while later work examined a square barrier, whose discontinuities make the limiting problem especially delicate. The square-barrier analysis shows that rigorous semiclassical descriptions can extend beyond analytic initial data, but results for the evolving NLS field do not by themselves determine how accurately an approximate spectrum tracks the true eigenvalues. The semi-classical limit of focusing NLS for a family of non-analytic initial data

Known results

A rigorous WKB result established convergence of the relevant potentials pointwise for nonzero spatial locations, for certain classes of real-analytic potentials. The result justifies the formal WKB approximation at the level of potentials; it does not quantify the displacement of individual eigenvalues. The authors specifically identify that spectral comparison as necessary for connecting the approximation to semiclassical NLS, so this result is important but does not settle the eigenvalue-accumulation question. Asymptotics of Semiclassical Soliton Ensembles: Rigorous Justification of the WKB Approximation

For the square-barrier initial data $\psi_0(x)=q\chi_{[-L,L]}(x)$, with real amplitude $q$, a rigorous Riemann–Hilbert analysis gives pointwise semiclassical asymptotics for the focusing NLS solution globally in space and up to an order-one maximal time. This is a concrete result for non-analytic data, not a theorem on the spectrum for general data. In a separate WKB study, the proposed estimate $|\lambda^{\mathrm{wkb}}_{N,k}-\lambda_{N,k}|=O(N^{-2})$ as $N\to\infty$ is stated as a conjecture for the specified analytic Klaus–Shaw setting and quantized parameters; it should not be read as a proved eigenvalue error bound. The semi-classical limit of focusing NLS for a family of non-analytic initial data The WKB approximation of semiclassical eigenvalues of the Zakharov–Shabat problem

Another relevant advance gives explicit leading-order expressions for the semiclassical direct and inverse scattering transforms for decaying data of the form $q(x,0)=A(x)e^{iS(x)}$, using inversion of the leading-order modulation equations rather than WKB analysis of the Zakharov–Shabat system. This describes a limiting scattering map, but does not provide a general theorem locating each discrete eigenvalue or its error relative to an approximation. Together, these results make clear the distinction between asymptotics for potentials or scattering transforms and the finer problem of eigenvalue motion. Semiclassical limit of the scattering transform for the focusing Nonlinear Schrödinger Equation