Resolvents, moving Hilbert spaces and stable spectral bands

A comparison starts with maps, not similar spectra

Strong convergence asks whether each fixed vector has a convergent image. Operator-norm convergence asks for one error bound over all unit vectors, including directions that move with the cutoff. A sequence can pass every fixed-vector test while carrying an exceptional low mode in a direction that escapes those tests.

For unbounded Hamiltonians, the resolvent (H−z)⁻¹ is bounded at a nonreal spectral parameter. Strong- and norm-resolvent convergence apply those two topologies to this bounded operator. They avoid demanding operator-norm convergence of the unbounded generators themselves.

Lattice Hilbert spaces also change with spacing, volume and auxiliary resolution. Writing a difference of their resolvents is meaningless until comparison maps place them in one space. The source therefore separates common-space realization from the proof of norm-resolvent convergence. The maps must preserve the intended low-window data, rather than merely identify abstract Hilbert spaces of the same dimension.

For effective reduction, the comparison typically has the form (H_n−z)⁻¹ versus J_n(H_n^eff−z)⁻¹J_n*. This formula states exactly where effective vectors live in the full theory.

Source: Projected lattice gauge synthesis, Theorem targets 4.7–4.8, printed/PDF p.27.

Keep the complete Schur inverse

Write H = [[A,B],[B*,D]], assume D ≥ Γ and ‖B‖ ≤ b, and choose a real z < Γ. Set R_D = (D−z)⁻¹ and F(z) = A−z−BR_DB*. Whenever F is invertible, the exact inverse is

The low compression is exactly F⁻¹, but the complete resolvent also contains the two off-diagonal returns and a corrected high block. Estimating only BR_DB* does not by itself estimate the full difference from an embedded effective resolvent.

Let d = Γ−z and M = ‖F⁻¹‖. A convenient conservative bound for the difference from diag(F⁻¹,0) is 2bM/d + 1/d + b²M/d². It follows by bounding the four displayed blocks and summing their norms. This bound shows which quantities must remain controlled as Γ grows. In particular, a pole of F can overwhelm a small high-block inverse.

Source: Auxiliary-time lattice framework, Theorem 18.1, equations (132)–(136), printed pp.45–46 / PDF pp.52–53.

Vanishing high population can leave a finite energy shift

Consider H_Γ = [[2,√Γ],[√Γ,Γ]]. It is positive because its determinant is Γ and its trace is positive. Its low Schur function is

The low effective energy therefore tends to one, not the uncoupled low value two. Yet the high component of a normalized low eigenvector is proportional to √Γ/(Γ−λ), which tends to zero like Γ^−1/2. The auxiliary population disappears while its return contribution remains finite.

This example explains why decoupling is a comparison with an effective operator, rather than simply deleting all auxiliary matrix entries. If the cross scale b remains bounded, b²/Γ tends to zero and comparison with A may be justified. If b scales as √Γ, the finite self-energy belongs in the effective Hamiltonian. Larger scaling can invalidate the presumed low-energy regime.

For a gauge form, the cross block need not be bounded at all. The source’s normalized cross-form estimate replaces this scalar scaling discussion by a bound relative to the physical and high forms, including the vacuum and boundary sectors. Proving that estimate is the concrete adiabatic task.

Source: Auxiliary-time lattice framework, Corollary 18.2 and Remark 18.4, printed p.46 / PDF p.53; Requirement 18.5, printed p.47 / PDF p.54.

An isolated contour transports a whole band

Let a positively oriented contour C enclose the desired finite spectral cluster and remain outside every relevant spectrum. Its Riesz projection is Π = (2πi)⁻¹∮_C(ζ−L)⁻¹dζ. Uniform resolvent control on C gives the quantitative estimate

Once the right side is below one, the projections have the same rank. To see why, if x lies in Ran Π and Π_nx = 0, then ‖x‖ = ‖(Π−Π_n)x‖ < ‖x‖ unless x = 0. The restriction of Π_n is therefore injective on Ran Π. Interchanging the projections gives the opposite dimension inequality.

The contour condition is quantitative information. If the contour approaches an eigenvalue, its resolvent norm grows and a small operator error can produce a large projection error. Norm continuity of a band thus requires separation from neighboring spectrum, together with common-space control. Constant rank guessed from a few numerical samples does not replace that estimate.

Source: Projected lattice gauge synthesis, Theorem 4.11, printed/PDF pp.27–28.

Strong resolvents can lose an entire vacuum line

On ℓ² of the positive integers, let H e_j = j e_j. Define H_n to agree with H except that H_ne_n = 0. Every operator has compact resolvent. For n ≥ 2, H_n has the unique vacuum e_n and gap one above it.

At spectral parameter −1, the resolvent difference is exactly

It tends strongly to zero because the nth coordinate of every fixed ℓ² vector tends to zero. Its norm tends to one. The low Riesz projection onto the vacuum likewise tends strongly to zero while retaining norm one. The limit H has no zero eigenvector.

This example preserves compact resolvent and a uniform excitation gap, yet loses the intended vacuum because the normalized vacua converge only weakly to zero. It demonstrates why the source asks for strong vacuum convergence and norm transport of a dangerous cluster as distinct hypotheses. Strong resolvent convergence alone cannot identify a stable finite-rank low sector or its vacuum representation.

Source: Auxiliary-time lattice framework, Definition 29.1, printed p.59 / PDF p.66; projected comparison conditions are in Theorem 4.11, linked above.

Recovery energies carry the lower bound

Suppose q_n[u] ≥ Δ(‖u‖²−|〈Ω_n,u〉|²) with one Δ > 0. Let comparison maps place the theories in a common space, and assume Ω_n converges strongly to Ω. For each limiting form vector u, take a recovery sequence u_n converging strongly with q_n[u_n] → q[u]. The norms and vacuum overlaps converge, so the same inequality passes directly to q.

If u is orthogonal to Ω, one can instead subtract its finite vacuum overlap: v_n = u_n−〈Ω_n,u_n〉Ω_n. The zero-energy vacuum makes q_n[v_n] = q_n[u_n], while v_n converges to u. This is the explicit vacuum-compatible gap-transfer proof.

The liminf half of Mosco convergence alone has the wrong direction for this step. For example, q_n[u] = |u|² and a proposed q[u] = 0 satisfy q[u] ≤ liminf q_n[u_n] for strongly convergent sequences, but have no recovery energies at nonzero u. The recovery condition rules out that false limiting form.

Source: Auxiliary-time lattice framework, Theorem 29.2, printed p.59 / PDF p.66; projected-lattice Theorem 4.9 is on printed/PDF p.27.

A stable band is not yet a theory identification

Norm convergence of Π_n identifies the dimensions and geometry of an isolated band. It does not determine how a Wilson loop, curvature insertion or time translation acts within that band. Two theories can have equal eigenvalue lists and different matrix elements of their physical observables.

The source therefore requires local observable and time-translation convergence on a common dense core in addition to the form convergence. For bounded operators, a product comparison can be made explicit by telescoping: the difference of an m-fold product is a sum of m terms, each containing one operator difference and otherwise bounded factors. If each factor has norm at most C and error at most ε, the product error is at most mC^m−1ε.

Such a bound is useful only when the comparison maps, insertions and their norms refer to the same physical core. Extending matching from a finite cluster to an entire vacuum theory requires density, continuity and cyclicity. Defining a target functional to be the limit would not establish independent Yang–Mills identification.

Source: Auxiliary-time lattice framework, Definition 29.1, M4, printed p.59 / PDF p.66; Theorem 31.2, printed pp.61–62 / PDF pp.68–69.

The concrete comparison package to prove

A reviewable norm-resolvent result should provide the comparison maps, the exact operators and domains, a contour with a verified spectral distance, and a numerical or analytic bound on the resolvent error. Its band conclusion should state the resulting projection-norm error and the point beyond which the rank is stable.

A gap-transfer result additionally needs a uniform physical lower bound, normalized vacua converging strongly, recovery sequences with convergent energies, and compatibility with the limiting observable representation. These are different deliverables, even when one proof eventually supplies several of them.

The projected-lattice source identifies common-space band realization and isolated-contour transport as concrete near-term targets. The auxiliary framework adds the exact-vacuum form reduction and a nontrivial OS limit. The three-state puncture offers a finite checkpoint for the block identities, but its p-dependent high inverse already illustrates why uniform comparison constants need their own proof. A successful finite calculation makes the required data visible; it does not provide the infinite-volume or continuum comparison automatically.

Source: Projected lattice gauge synthesis, Remark 4.13, printed/PDF p.28; source programme scope is in §11, pp.52–53.

Continue reading

• The finite Schur checkpoint
• Spectral projections and bundles
• The limiting vacuum functional
• A concrete proof sequence