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3 papers · Page 1 of 1

Spectral Theory

Edition 1

Spectral transitions for concentrated zero-mean energy forms

Scott Schubr

Primary subject
Mathematics
Subject subfields
Spectral Theory · Functional Analysis
Paper type
Research Article
Manuscript stage
Preprint

We study the Riesz operators of concentrating signed potential forms in the derivative-energy space H₀¹(a,b). For finitely many fixed separated centers and nonzero even zero-mean profiles with inverse-square tails, exactly one positive and one negative eigenvalue per center escape to infinity. Their leading coefficients are the square roots of the eigenvalues of a full Dirichlet Green matrix weighted by the squared L² norms of the profile primitives. The remaining spectrum is uniformly bounded. Equal critical tails give a common logarithmic correction, and mean perturbations at the square-root scale yield a complete finite Hermitian crossover model with an operator-norm remainder under explicit moving isometric embeddings. We apply the results to localized log-curvature forms, including a positive-Fourier-inverse example with different even and odd escape coefficients. For real zeros, a trace-class Hardy localization identifies the absolutely continuous multiplicities and explains strong convergence with an unavoidable norm gap under genuine root fusion. The results concern bounded-interval energy operators; they assert no full-line theta sign theorem or polynomial-degree rate.

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    Number Theory and Special Functions

    Edition 1

    Beyond logarithmic powers: critical-tail localization of theta correction inertia

    Scott Schubr

    Primary subject
    Mathematics
    Subject subfields
    Number Theory and Special Functions · Spectral Theory · Complex Analysis
    Paper type
    Research Article
    Manuscript stage
    Preprint

    For the Riemann theta transform F(z)=xi((1+iz)/2)/4, let E(P)=(4π)⁻¹∫|(FP)′|², Q(P)=(8π)⁻¹∫(F′²−FF″)P² and D=E−Q. We obtain ordinary-polynomial subspaces, in either parity, with opposite signs of D and strictly positive original Q on their entire nonzero spans. Their dimensions are at least C exp(c√(log N))√(log N) for every sufficiently large degree N. A critical Meixner–Pollaczek receiver controls the derivative and absolute-potential norm globally. A simultaneous inverse-square window selection and a relative root-product budget improve the cost of localized targets. A finite-order compact-cutoff family with uniform low-order energies and a fully controlled growing differentiation order gives the stated scale. All local complex roots, multiplicities, exterior roots and mixed Gram entries are retained. The same growth is necessary for repair rank in a specified scalar-centered strict-contraction scheme. No negative original Q, RH conclusion, sharpness, an optimal constant or effective numerical onset is asserted.

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      Number Theory

      Edition 1

      Weighted polynomial localization and theta correction inertia

      Scott Schubr

      Primary subject
      Mathematics
      Subject subfields
      Number Theory · Spectral Theory · Complex Analysis
      Paper type
      Research Article
      Manuscript stage
      Preprint

      Let F(z)=xi((1+iz)/2)/4, G=F′²-FF″, and let D₀=E₀-Q₀ be the original endpoint correction, where E₀(P)=(4π)⁻¹∫|(FP)′|² and Q₀(P)=(8π)⁻¹∫GP². For every fixed 0<θ<1 and A>0, every sufficiently large degree N, and each polynomial parity, we construct a real subspace with a rational basis, dimension at least c_(A,θ)(log N)^A(log log N)², and Q₀≥(2-θ)E₀ on its entire span. The proof combines classical unconditional short-interval zero moments with bounded-population microscopic seed selection, a two-scale root-product budget, seed-averaged logarithmic root energy and higher fixed-order weighted approximation in a derivative and absolute-potential norm. All local complex zeros, multiplicities, exterior roots and mixed polynomial Gram entries are retained. Companion spaces have 0<Q₀/E₀≤θ at the same dimension scale, yielding the corresponding necessary repair-rank bounds for scalar-centered fixed strict contractions. A finite Gaussian zero-ideal approximation theorem and an exact generic current certificate delineate the scope of the method. The constructed correction-negative spaces have positive original Q₀; the all-order theta sign problem remains open.

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