Bound-State Moments and Embedded Singular Spectrum

Definitions and mathematical statement

The source asks whether finiteness of bound-state moments for two-dimensional Schrödinger operators implies absence of embedded singular spectrum. A conditional standard setting is a self-adjoint realization \[ H=-\Delta+V\quad\text{on }L^2({\mathbb{R}}^2) \] with real potential $V$ and essential spectrum $[0,\infty)$. The excerpt supplies neither conditions ensuring this realization and essential spectrum nor a specific potential class.

If $E_j<0$ are the discrete negative eigenvalues, counted with multiplicity, a typical bound-state moment is \[ M_\gamma(H)=\sum_j|E_j|^\gamma\qquad(\gamma>0), \] whereas the zeroth counting condition is $N_-(H)<\infty$. Finiteness of a positive moment allows infinitely many eigenvalues accumulating at zero and is not equivalent to finitely many bound states. The source does not specify the exponent, possible weights or logarithmic modifications, nor whether assumptions concern only $-\Delta+V$ or both signs $-\Delta\pm V$.

In the displayed essential-spectrum setting, exclusion of embedded singular spectrum on $(0,\infty)$ means that the spectral measures of $H$ restricted to that interval are absolutely continuous: both positive-energy eigenvalues and singular-continuous spectral measure are absent. In spectral-projection notation the requested conclusion is \[ P_{\mathrm{pp}}\,1_{(0,\infty)}(H)=0, \qquad P_{\mathrm{sc}}\,1_{(0,\infty)}(H)=0. \] Whether the threshold zero is included is a separate issue not resolved by the excerpt. Thus the intended implication must be read as a question with an unspecified moment hypothesis and potential class, not as a theorem for all real potentials with some finite eigenvalue sum.

The supplied literature assessment checked 21 August 2026 describes the report's question using finitely many bound states, reports results in one dimension and on a half-line, and says the two-dimensional case was left open. This differs from the extracted phrase finite bound-state moments; the two hypotheses must not be silently identified. The full cited theorem and report context are needed to resolve that tension and any two-sign assumptions. No independent current literature conclusion is asserted.

Short English statement

Does controlling negative bound states force purely absolutely continuous positive-energy spectrum in two dimensions? The precise moment and potential assumptions are missing, and the assessment's finite-number hypothesis differs from the statement's finite-moment wording. The supplied partial results concern one dimension and the half-line.

Sources

• Local UnsolvedMath registry, permanent ID 30000152, exact source code catalogue entry; Bound-State Moments and Embedded Singular Spectrum. Proposed year 2004, status open, exact supplied wording.
• Supplied report citation: Spectral Analysis of Partial Differential Equations (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/53, pp. 2839--2912, https://doi.org/10.4171/owr/2004/53. Title year differs from DOI/proposed year.
• Supplied literature assessment checked 2026-08-21: "original OWR report", https://doi.org/10.4171/owr/2004/53; "Damanik--Killip--Simon", https://arxiv.org/abs/math-ph/0409074. This dated OWR-derived triage requires independent verification of time-sensitive status and cited claims.