The differentiated Benjamin--Ono spectral problem

Definitions and mathematical statement

The record crosses a presenter boundary. Its opening asks for the spectrum of an operator "(A)," its relation to the known spectrum of $cv+Kv-\varphi v=\lambda v$, and a possible two-dimensional formulation. The aid identifies this opening as J. Albert's differentiated Benjamin--Ono spectral question, using the preceding source item to recover the missing operator. The subsequent K. Zumbrun viscous-shock setup is separate context.

For the aid's explicitly chosen realization, let $c>0$, let $|D|$ be the Fourier multiplier $|\xi|$ on $L^2({\mathbb{R}})$, and put \[ \varphi_c(x)=\frac{4c}{1+c^2x^2},\qquad L_c=|D|+c-\varphi_c,\quad\mathcal D(L_c)=H^1({\mathbb{R}}),\qquad A_c=\partial_xL_c,\quad\mathcal D(A_c)=H^2({\mathbb{R}}). \] The space is unweighted, and the domain is the stated maximal realization. The supplied partial result is \[ \sigma_{\mathrm{ess,Fredholm}}(A_c)=\sigma(A_c)=i{\mathbb{R}}. \] Thus there are no eigenvalues off the imaginary axis; this spectral-set assertion does not classify every possible embedded nonzero imaginary eigenvalue. At zero, the aid asserts \[ \ker A_c=\operatorname{span}\{\varphi_c'\},\qquad \bigcup_{j\geq1}\ker A_c^j =\operatorname{span}\{\varphi_c',\partial_c\varphi_c\},\qquad A_c(\partial_c\varphi_c)=-\varphi_c'. \] Zero is embedded in essential spectrum, so the two-dimensional generalized root space is not described as a Riesz algebraic multiplicity.

The aid's cancellation-aware correspondence says that an off-axis eigenfunction $v$ automatically has an $L^2$ antiderivative $w$. With the Fourier convention $\widehat{\partial_xv}=i\xi\widehat v$, it is represented by \[ \widehat w(\xi)=\frac{\widehat{\varphi_cv}(\xi)}{i\xi(|\xi|+c)-\lambda},\qquad \partial_xw=v. \] This avoids an unjustified division by $\xi$ at zero. The aid combines it with the known one-negative-direction and constrained-positivity properties of the selfadjoint $L_c$ to exclude off-axis eigenvalues. The obstruction to extending the translation/speed chain is reported with magnitude $4\pi$; in the present normalization, \[ \left|\left\langle\varphi_c',\partial_x^{-1}\partial_c\varphi_c\right\rangle_{L^2}\right|=4\pi. \] The antiderivative here is the $L^2$ one. These inputs require verification, and the source supplies no full reference for the known Hessian spectrum.

For harmonic extension $V(x,y)=e^{-y|D|}v(x)$ to $y>0$, $\Delta V=0$ and $-\partial_yV|_0=|D|v$. The selfadjoint problem is the Robin condition \[ -\partial_yV+(c-\varphi_c)V=\lambda V\quad(y=0), \] whereas the differentiated problem is \[ \partial_x\left[-\partial_yV+(c-\varphi_c)V\right]=\lambda V\quad(y=0). \] These boundary problems are related but are not interchangeable.

The adjacent contribution, K. Zumbrun's "Strong Shocks in one or multiD of viscous conservation laws," considers \[ u_t+\sum_{j=1}^d\partial_{x_j}f^j(u) =\sum_{j,k=1}^d\partial_{x_j}\left(B^{jk}(u)u_{x_k}\right),\qquad u\in{\mathbb{R}}^n, \] with smooth fluxes $f^j:{\mathbb{R}}^n\to{\mathbb{R}}^n$ and smooth matrices $B^{jk}:{\mathbb{R}}^n\to{\mathbb{R}}^{n\times n}$. It cites small-amplitude one-dimensional shock stability work and notes that shock strength supplies a fast--slow parameter for weak shocks, while an analogous structure for strong shocks is not apparent. No parabolicity, end states, profile, or strong-shock stability criterion is given. This separate setup and its references are retained without turning it into the Benjamin--Ono operator.

Short English statement

The opening question concerns a differentiated Benjamin--Ono operator, while the trailing material concerns viscous conservation laws. In the aid's unweighted realization, the differentiated operator has spectral set $i{\mathbb{R}}$ and an embedded translation/speed chain of length two. Harmonic extension distinguishes the Robin problem from its tangentially differentiated version; neither solves the separate strong-shock program.

Sources

• Local UnsolvedMath registry, permanent ID $20002448$, catalogue entry; aim-pdes-notes.json, notes[59], source item 12.
• American Institute of Mathematics, Stability criteria for multi-dimensional waves and patterns, https://aimath.org/WWN/multidimwaves/multidimwaves.pdf. Supplied verification identifies a presenter splice and reports recovery of J. Albert's entry from the original PDF and preceding AIM-PDES-0059; no full recovered entry follows.
• Adjacent source attribution: K. Zumbrun. H. Freisthler and P. Szmolyan, Spectral stability of small shock waves, I, Arch. Rat. Mech. Analysis 164 (2002), 287--309; spelling reproduced from the supplied record.
• R. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, J. Discrete and Continuous Dynamical Systems 10 (2004), no. 4, 885--924, as cited in the source.
• Research attempt 1: partial result, problem partially solved at run, rigor label "proved," novelty confidence low. Registry classification partial progress, status partially solved, statement status exact. No authoritative separate citation for the Benjamin--Ono spectral claims is supplied.