Silent Orbits and Cancellations in the Wave Trace

The uploaded paper’s Theorem 1.1 states that for any ellipse Ee and any N, there exist arbitrarily C∞-small deformations Ω for which the even wave trace wΩ(t) is locally C^N near some length L in the length spectrum, achieved by canceling Balian–Bloch–Zelditch (BBZ) invariants at L via multiple same-length orbits with carefully arranged Maslov phases . The paper outlines the strategy: localize the trace near L using the BBZ parametrix, create four families of orbits yielding phases ±1, ±i by mixing elliptic/hyperbolic types and parity of periods, and orchestrate cancellations across families; the construction hinges on “length spectral resonances” in ellipses (two distinct rational caustics with the same orbit length) for a dense set of eccentricities, the design of controllable deformations that both prescribe curvature jets and eliminate stray orbits so L is isolated, and a Vandermonde/inverse-function/fixed-point argument to solve the nonlinear matching system . The candidate solution mirrors these steps almost verbatim: it uses the BBZ expansion to motivate cancellation, constructs (near ellipses) controllable small deformations yielding finitely many same-length orbits with prescribed elliptic/hyperbolic type and Maslov phases, isolates L in the length spectrum by removing stray orbits, and solves a finite system in curvature jets with Vandermonde structure, closing via inverse function/fixed point to cancel the first N+1 coefficients and hence obtain C^N smoothness near L. Each stage has a direct counterpart in the paper: (i) the localized resolvent/wave-trace asymptotics and the implication from coefficient cancellation to local smoothness , (ii) the four-family phase engineering via conjugate Maslov indices , (iii) dense length-spectral resonances in e via analyticity of Mather’s β-function and elliptic integrals , (iv) controllable families and the nonlocal/local deformation split that destroys stray orbits and prescribes jets , and (v) the Vandermonde Jacobian and fixed-point/inverse-function method to solve the nonlinear system . Minor presentational differences (e.g., the model counts cancellations from j=0 to N, while the paper states them from j=1 to m and notes the corresponding smoothness shift) do not affect substance. Consequently, the paper’s proof and the model’s solution are aligned and substantively the same.