Pure point Dirichlet spectrum on quasi-conical domains

Abstract

For every dimension at least two, we construct a connected open quasi-conical domain whose Dirichlet Laplacian has a complete orthonormal eigenbasis, although its spectrum and essential spectrum are the entire nonnegative half-line. Its positive eigenvalues are dense, zero is not an eigenvalue, and both continuous spectral subspaces vanish. We use the expanding-cube and shrinking-window geometry of Krejčířík and Lotoreichik, choosing both cube sizes and positive apertures inductively. The additional construction transports a growing family of complete finite spectral clusters with summable norm errors. A repeated dense test family proves completeness and excludes singular continuous spectrum. This answers the existential spectral-type question in their discussion and the corresponding Oberwolfach contribution.

The idea in plain English

Think of an infinite chain of increasingly large rooms, connected through increasingly small openings. Waves vanish on the walls. The rooms become large enough to contain balls of any radius, yet the whole connected space has a complete collection of square-integrable standing-wave modes.

The surprising point is that these individual modes suffice to represent every square-integrable state, even though the spectrum fills the whole nonnegative half-line. The eigenvalues are dense: there are eigenvalues arbitrarily close to every positive energy. This does not mean that every energy is itself an eigenvalue.

Krejčířík and Lotoreichik introduced this domain geometry and proved that it can retain an embedded eigenvalue while excluding the absolutely continuous part. Their remaining spectral-type question concerns the singular continuous part. This paper addresses that question by constructing a complete eigenbasis, choosing the room sizes as well as the openings.

The additional ingredient is to preserve an expanding collection of whole finite-dimensional spectral clusters and repeatedly test it against a dense family of states. Keeping just one mode, or even many modes without proving completeness, would leave the central question unanswered.

The precise result

For every integer d ≥ 2 there is a connected open quasi-conical set Ω ⊂ Rᵈ whose Dirichlet Laplacian A has a complete orthonormal eigenbasis. The operator is defined by the closed energy form with domain H₀¹(Ω).

The spectrum and essential spectrum are [0, ∞), which is also the closure of the point spectrum. All eigenvalues are positive, zero is not an eigenvalue, and both the absolutely continuous and singular continuous spectral subspaces vanish.

The domain consists of expanding cubes joined through positive shrinking windows in their common faces. Both cube widths and window widths are chosen inductively. The result gives an existential construction within this geometric family.

Pure point here describes spectral type and the existence of a complete eigenbasis. It does not mean an isolated sequence of eigenvalues or compact resolvent.

Why the result holds

At a finite stage, closing one window separates two bounded pieces. Shrinking-window convergence gives norm control of resolvents and of whole isolated finite spectral clusters, including their multiplicities.

The next cube size avoids the finite set of protected spectral values. Its opening is then small enough that every previously protected projection changes by a summable amount.

New finite spectral families capture successively larger portions of a countable dense collection of cube-supported states. Repeating this requirement prevents states from escaping the protected subspaces.

At the infinite limit, each protected finite-dimensional subspace remains invariant under the resolvent and is spanned by eigenvectors. The dense test family proves that these subspaces together fill the Hilbert space, which removes both continuous spectral parts.

The arbitrarily large cubes supply Weyl sequences throughout the nonnegative half-line. The zero-energy equation would require an L² constant on the connected infinite-volume domain, so zero is not an eigenvalue.

Scope and limitations

The construction chooses future cube sizes to avoid finitely many protected eigenvalues. It does not prove the same conclusion for every prescribed sequence of cube sizes.

The published proof uses windows in zero-thickness interface screens and does not establish a smooth-boundary or positive-length-passage version of the complete-eigenbasis result. Extensions of earlier embedded-eigenvalue results do not automatically give those stronger variants.

The aperture choices are existential, without explicit numerical bounds that guarantee the infinite construction. The statement concerns the Dirichlet Laplacian, not arbitrary boundary conditions.

The positive eigenvalues form a dense set whose closure is the half-line. The point spectrum itself is not asserted to equal the whole half-line.

The geometry, shrinking-window perturbation theory and spectral-projection approach have substantial predecessors. The added claim is exhaustive cluster transport and completeness; the bounded literature comparison does not certify historical first priority.

Finite diagnostic calculations in the archive cannot certify Mosco convergence, the infinite limiting operator or completeness. Those conclusions rely on the analytical argument.

Status and AI use

AI tools were used extensively in developing, reproducing, checking and adversarially reviewing the argument and preparing the note. Internal AI-assisted review is not external human peer review or proof-assistant certification.

Read the paper

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Collaborate on this result

I welcome substantive collaboration on independent verification of the infinite spectral-projection argument, its relationship to domain-perturbation literature, and preparation for journal review. Expertise in spectral theory, quadratic forms or partial differential equations would be particularly useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Pure point Dirichlet spectrum on quasi-conical domains (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23410944

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Background and prior work

• Krejčířík and Lotoreichik, Quasi-conical domains with embedded eigenvalues: Introduces the expanding-cube and shrinking-window construction. Section 5 asks whether its singular continuous part can also be eliminated; Remarks 3.1 and 3.3 clarify spectral clusters and geometric variants.
• Lotoreichik, Quasi-conical domains with embedded eigenvalues, Oberwolfach Report 36/2023: Printed pages 2077–2078 state the corresponding question of obtaining dense point spectrum with both continuous spectral parts absent.
• Krejčířík, Spectral geometry: old questions and new answers: Open Problem 1 on PDF page 16 repeats the spectral-type question. The present construction retains freedom to choose cube sizes as well as openings.
• Hempel, Seco and Simon, The Essential Spectrum of Neumann Laplacians on Some Bounded Singular Domains: An earlier rooms-and-passages construction in a different boundary-condition setting. Proposition 1.4 explicitly handles spectral projections and multiplicities; tracking whole clusters is not a new general principle.