Enter, I-B-VP-6-v2, 2026, pp. 1-39
I-B-VP-6 Longitudinally non-product self-adjoint extensions on the prescribed conical spacetime
Ngo Cheung
Version 1
Abstract
We develop the longitudinally non-product extension row of the I-B-VP ledger without changing its prescribed conical spacetime. If no transverse angular channel is limit circle, the full spatial minimal operator is essentially self-adjoint. Otherwise, all extensions strongly commuting with longitudinal translations are parametrized by measurable fields of finite-dimensional self-adjoint boundary relations. Product extensions are exactly the almost everywhere constant fields. We give an explicitly normalized cone Weyl function and prove the variable-boundary direct-integral resolvent formula, including the operator domain.
The essential spectrum is the union of the nonnegative continuum and the essential ranges of the momentum-dependent bound-channel energies. Every spectral point is essential, but point spectrum need not be absent: a tuned logarithmic relation produces an infinite-multiplicity eigenvalue at any prescribed negative energy, separated from the nonnegative continuum by a gap. Conversely, longitudinal momentum can compensate negative transverse eigenvalues, yielding nonnegative, non-Friedrichs untwisted extensions. These examples show precisely why product-class positivity and no-point-spectrum conclusions cannot be transferred.
We also prove an exact rigidity statement: within the declared time-independent spatial boundary class, covariance of all positive-transverse-energy boundary traces under longitudinal boosts holds if and only if the relation field is essentially constant. Beyond translation invariance, a rank-one change of the normalized deficiency-space unitary gives a self-adjoint, longitudinally nonlocal extension with a rank-one resolvent difference from Friedrichs and unchanged essential spectrum.
The potentially useful contribution is this combination of spectral counterexamples, boost rigidity, and a domain-complete cone implementation. Symmetry reduction and boundary-extension theory themselves are established subjects. No physical preference among extensions, spectral ground state, covariance wavefront theorem, one-point observable, backreaction, or continuum acquisition by an Enter report is constructed.
Bibliographic record
- Title
- I-B-VP-6 Longitudinally non-product self-adjoint extensions on the prescribed conical spacetime
- Author
- Cheung, Ngo
- Publication date
- 2026
- Institution
- Enter
- Report number
- I-B-VP-6-v2
- Publisher
- Cheung Ngo Medical Limited
- First page
- 1
- Last page
- 39
- Language
- en