Sharp Computational Bounds for Spectral Types of Schr\"odinger Operators

1. Introduction

• Spectral type is a basic task in classifying quantum phases and propagation, and the paper asks whether it can be determined from finite descriptions of Hamiltonians.The input model uses a finite source code computing all derivatives of a smooth potential and supplying certified derivative bounds.Source: §1 · p. 1

1.2.1. Lower Bounds on the Difficulty of Computing Spectral Type.

• Pure point and absolutely continuous spectral sets cannot generally be recovered with one limit, while singular continuous spectrum cannot generally be recovered with two.The obstructions persist even with certified information about all derivatives of the potential.Source: §1 · p. 1, §1.2.1 · p. 3

1.2.2. Upper Bounds Showing the Lower Bounds Are Sharp.

• Matching upper bounds make the hierarchy sharp: two limits suffice for pure point and absolutely continuous spectrum, and three for singular continuous spectrum.The upper algorithms compute resolvent matrix elements in compactly supported wavelet bases for self-adjoint differential operators with locally bounded variation coefficients and quantitative local variation control.Source: §1 · p. 2, §1.2.2 · p. 4

1.1. Multiple Limits and Finite Descriptions.

• Prior two-limit procedures for discrete Schrödinger spectral sets are optimal, and the paper extends the singular-continuous lower bound to a smooth one-dimensional continuum class.The earlier singular-continuous lower bound used a countable direct sum; this paper uses inverse spectral embeddings, within a finite-source-code input model.Source: §1.1 · p. 2

1.2.3. Consequences for Formal Certification.

• For each spectral type, the paper constructs true spectral inclusion and exclusion statements for finitely described smooth operators that are unprovable in an arithmetically sound, effectively axiomatized theory.For ZFC, the conclusion is conditional on its arithmetical soundness; the relative non-provability hierarchy mirrors the greater difficulty of singular continuous spectrum.Source: §1.2.3 · p. 4

1.3. Connections with Previous Work.

• The paper places spectral-type computation within the broader study of limiting complexity and distinguishes it from computing only the spectral set.Earlier work on spectral covers concerns inclusion and exclusion for the spectral set, whereas this paper identifies sharper obstacles for its measure-theoretic type.Source: §1.3 · pp. 5–6

2.1. Spectral Type

• Every finite positive Borel measure has a unique sum of pure point, singular continuous, and absolutely continuous components [23, Theorem 4.3.2].The corresponding scalar spectral measures define closed reducing subspaces and spectral sets for each type.Source: §2.1 · p. 6

2.2. SCI Hierarchy

• The SCI framework measures computational complexity by successive limits, while the Attouch–Wets metric captures local convergence of closed spectral sets and permits points to escape to infinity.Convergence of distance functions uniformly on compact sets rules out spectral invisibility and persistent spectral pollution on bounded sets.Source: §2.2 · pp. 7–8

2.3. Markov Model

• In the Markov model, an operator is supplied as a finite source code, and a tower algorithm takes that code and its indices to produce finite subsets of Q + iQ.The algorithm must halt for every valid source code and return an index for a finite set.Source: §2.3 · p. 9

2.4. The Arithmetical Hierarchy

• The arithmetical hierarchy classifies sets by quantifier alternations, and reductions from complete arithmetical sets establish lower bounds for spectral problems.A tower below the claimed level would induce a tower at that lower level for the embedded set, contradicting its completeness.Source: §2.4 · p. 9, §2.5 · p. 10

2.5. The Mechanism for Lower Bounds

• A computable coding map with a gap between spectral-set inclusion in and exclusion from an open ball yields a height-one Markov tower converging in the Attouch–Wets metric.The gap removes the boundary case.Source: §2.5 · p. 10

2.5. The Mechanism for Lower Bounds.

• An eventually correct computable sequence Υ_n(e) yields S ∈ Δ0_2.The relation Υ_n(e) = 1 is decidable uniformly in n and e, and Υ_n(e) is eventually equal to 1_S(e).Source: §2.5 · pp. 10–11
• The stabilization test tracks the latest visit of rational approximants to two separated threshold sets, retaining its label between visits.Because the approximants converge and the sets have a positive gap, both sets cannot be visited infinitely often; the resulting inner sequence therefore stabilizes.Source: §2.5 · pp. 11–12
• A stabilized height-two construction gives lim_m Υ_m(e) = 1_Cof(e), contradicting Theorem 2.11 and ruling out the proposed height-two classification.The computable Υ_m,ℓ sequence stabilizes for each m, and the resulting outer sequence converges to the characteristic function of Cof.Source: §2.5 · pp. 11–12

3. Lower Bound for Pure Point Spectrum

• For each program index e, e ∈ Tot yields an operator with no eigenvalues, while e ∉ Tot yields an eigenvalue in (−7/8, −5/8).The construction uses a generalized Gordon potential in the first case and a compactly supported potential containing a fixed well in the second.Source: §3 · p. 12
• When e ∈ Tot, the construction produces a generalized Gordon potential, which has empty pure point spectrum.The potential is approximated by periodic potentials agreeing with it on increasingly large intervals.Source: §3 · p. 15
• When e ∉ Tot, only finitely many construction stages are active, so the potential is compactly supported and the forced negative spectral point is an eigenvalue.The fixed well supplies the negative spectral point.Source: §3 · pp. 12, 15

4. Lower Bound for Absolutely Continuous Spectrum

• For e ∈ Tot, the high-barrier construction gives empty absolutely continuous spectrum; for e ∉ Tot, compact support leaves absolutely continuous spectrum [0, ∞).The paper uses this dichotomy for the absolutely continuous lower bound.Source: §4 · pp. 16–17
• The cited barrier criterion excludes absolutely continuous spectrum when barrier heights grow at both ends and satisfy h_nℓ_n^2 → ∞.The potential must also be bounded below and meet the stated barrier inequalities around points x_n.Source: §4 · p. 16
• The computable sequence a_e is eventually zero when e ∉ Tot and unbounded when e ∈ Tot, providing the coefficient values used to build the barriers.The recursion advances through inputs and increases its simulation bound when the current input has not halted.Source: §4 · p. 16

5. Lower Bound for Singular Continuous Spectrum

• The construction encodes the desired dichotomy in a measure, incorporates it into a half-line Schrödinger operator’s spectral measure, and smoothly extends the reconstructed potential by reflection.Source: §5 · p. 17

5.1. Roadmap of the Construction.

• Riesz-product measures encode the matrix columns, and their assembled measure has singular continuous mass on I exactly when some column has infinitely many ones.A computable absolutely continuous signed measure supported away from I is added to cancel all polynomial moments.Source: §5.1 · p. 18
• Moment cancellation makes the reconstructed potential flat at the origin, allowing smooth reflection to the whole line.The half-line spectral measure is realized through the Gelfand–Levitan equation.Source: §5.1 · p. 18
• The whole-line operator splits into Dirichlet and Neumann half-line channels; the Dirichlet channel carries the encoded dichotomy, while a Weyl m-function estimate excludes spurious singular continuous spectrum from the Neumann channel on the test interval.Source: §5.1 · p. 18

5.2. Spectral Measures and Infinite Binary Sequences.

• Finitely many ones give an absolutely continuous measure, whereas infinitely many give a singular continuous measure.The dichotomy is preserved after mapping the Riesz-product measure to I.Source: §5.2 · pp. 18–19
• A binary sequence selects which Riesz factors appear, and the finite products converge weakly to a measure used in the dichotomy.The construction maps the torus measure to I.Source: §5.2 · p. 18, §5.3 · p. 20
• The resulting measure ντ is supported on I = [5/4, 7/4].The construction establishes full support on I.Source: §5.2 · pp. 18, 20

5.3. Weighted Sums of Measures.

• The mth column contributes weight 2^-(m+1), so one infinite column suffices for singular continuous mass.When every column is finite, the whole sum remains absolutely continuous.Source: §5.3 · p. 20
• If any matrix column has infinitely many ones, νA has singular continuous mass on every non-empty open subinterval of I; if all columns are finite, νA is absolutely continuous.The singular continuous component has full support in I in the first case.Source: §5.3 · p. 20
• Integrals against νA are computable uniformly from a C2 function’s source code, derivative bounds, requested precision, and effective column data.The construction computes finite approximations and controls the omitted tail.Source: §5.3 · pp. 21–22

5.4. The Moment-Killing Measure.

• A computable real-valued Schwartz function h supported in [4, ∞) is constructed with the specified moments equal to −1.The construction also provides computable bounds on all Schwartz seminorms.Source: §5.4 · p. 22
• The scaled functions ht place the correction beyond the support of νA, and their νA-integral defines the absolutely continuous signed correction κA.The resulting density gA is computable and Schwartz, so κA has finite variation.Source: §5.4 · pp. 24–25
• Adding the signed correction κA to νA yields σA, the perturbation used to cancel moments of the free spectral measure.The construction combines the identities for the moments of the two measures.Source: §5.4 · p. 25
• A sufficiently small rational ε makes ρA = ρ0 + εσA positive and ensures 2ρA ≥ ρ0.The choice of ε is independent of A.Source: §5.4 · pp. 25–26

5.5. Identifying the Potential.

• The Gelfand–Levitan equation has a unique solution kernel, and its reconstruction produces qA whose Dirichlet spectral measure is ρA.The reconstruction formula is applied after verifying the conditions on ρA.Source: §5.5 · p. 27
• The reconstructed potential qA is smooth and all of its derivatives vanish at the origin.The flatness follows from the vanishing mixed derivatives of FA at (0, 0), which transfer to the diagonal kernel and then to qA.Source: §5.5 · pp. 29–30
• The construction provides a smooth source code for qA with certified uniform approximations and derivative bounds on compact intervals.It approximates FA, solves the Gelfand–Levitan equation, and differentiates the resulting kernel with explicit error control.Source: §5.6 · p. 31

• Spline approximations to F_A on [0,X]^2 have computable error bounds tending to zero with tolerance δ.Source: §5.6 · pp. 31–32
• Rescaling the Gelfand–Levitan equation to [0,1] reduces its spline approximation to finite-dimensional systems with uniform inverse bounds.Source: §5.6 · pp. 33–35
• The scheme computes uniform approximations to kernel-derived quantities on bounded intervals with certified error below any requested δ.Source: §5.6 · pp. 31, 34

• The reconstructed potential is smooth on R and has a quadratic lower bound, yielding an essentially self-adjoint whole-line operator.Source: §5.7 · pp. 36–37
• Evenness decomposes the whole-line operator into Neumann and Dirichlet half-line problems; the proof separately excludes Neumann singular-continuous spectrum in the test window.Source: §5.7 · pp. 38–39
• If some column of A has infinitely many ones, singular-continuous spectrum intersects D_1/16(3/2); if every column has finitely many ones, it does not.Source: §5.7 · pp. 38–39
• The Dirichlet spectral measure inherits the encoded dichotomy: infinite columns give singular-continuous mass on I, while finite columns make ν_A absolutely continuous there.Source: §5.7 · pp. 38–39

• A computable map sends each e to a binary matrix with an infinite-one column exactly when e belongs to Cof.Source: §5.8 · p. 39
• The column condition is implemented by simulating successive inputs: a total computation produces infinitely many ones, while a non-total one eventually leaves only zeros.Source: §5.8 · p. 40
• The construction uniformly computes smooth potential codes from binary matrices, and the resulting operator's singular-continuous spectrum in the test window follows the column dichotomy.Source: §5.8 · p. 40
• Assuming the stated Δ^M_3 computability would place Cof in Δ^0_3, contradicting its Σ^0_3-completeness.Source: §5.8 · p. 40

6. Upper Bounds

• The upper-bound method combines resolvent algorithms [24] with a compactly supported, sufficiently smooth wavelet basis whose finite span is a core.Source: §6 · p. 40

6.1. Computing the Resolvent.

• Boundary behavior of scalar Borel transforms recovers the pure point, absolutely continuous, and singular continuous supports with the stated SCI heights.Source: §6.1 · p. 41
• Certified least-squares algorithms compute resolvent vectors and matrix elements from operator matrix-element data.Source: §6.1 · p. 41

6.2. Constructing the Basis.

• Compactly supported wavelets of sufficient regularity have finite span that is a core for the operators in DBV_N,d.Source: §6.2 · pp. 42–43

6.3. Computation of Matrix Elements.

• Wavelet matrix elements required by the resolvent algorithm are computable uniformly from EvalBV_N,d, with a source-code Markov version as well.Source: §6.3 · pp. 44–45

7.1. Establishing Non-Provability from Non-Computability.

• When S or its complement is not computably enumerable, some corresponding true spectral-set inclusion or exclusion statement is not provable in an arithmetically sound, effectively axiomatized theory.The argument uses effective proof enumeration: if every relevant true statement were provable, the associated set would be computably enumerable.Source: §7.1 · p. 46

7.2. A Concrete Result.

• Affine normalization extends the fixed-region dichotomies to any non-empty rational open interval while preserving spectral type.The transformed potential remains smoothly source-coded under the stated computable affine parameters.Source: §7.2 · p. 46
• Assuming ZFC is arithmetically sound, some true spectral-type statements for finitely described smooth Schrödinger operators are not provable in ZFC.The stated condition is arithmetical soundness, which is stronger than consistency alone.Source: §7.2 · p. 47

7.3. A Hierarchy of Non-Provability.

• The lower bounds place pure point and absolutely continuous problems at the Σ^M_2 level and the singular continuous problem at the Σ^M_3 level, yielding a corresponding hierarchy of non-provability.Source: §7.3 · p. 47
• The theorem's Truthψ2 and Truthψ3 oracles decide instances of particular fixed predicates, not every true sentence at those arithmetical levels.The non-provability result is therefore predicate-relative.Source: §7.3 · p. 47