Read the equation
The displayed candidate is the first line of a multiline expression that adds a damped diagonal term and a sum of nearby couplings. Such a row sum is not generally an eigenvalue of a nonuniform matrix. The body also uses index m in its estimate but m+1 in the actual diagonal construction, so even a row-sum interpretation requires care.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| m | Cross-helix index |
| gamma_0 | Diagonal amplitude |
| r | Helix radius |
| xi | Positive decay scale |
| eta | Neighbor coupling |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
M_[m][mp] = params_.eta * phase;
}
}
}
}
// Eigenvalues (Section 11.3):
// λ_cross(m) = γ₀·e^{-4m·r·sin(π/8)/ξ}·cos(8πm/10.6) +
// η·Σ cos(2π(m-m')/10.6)
double eigenvalue(int m) const {
double decay = std::exp(-4.0 * m * HELIX_RADIUS *
std::sin(M_PI / 8.0) / params_.xi);
double cos_8 = std::cos(8.0 * M_PI * m / BASE_PAIRS_PER_TURN);
double main_term = params_.gamma_0 * decay * cos_8;
// Sum over neighbors |m' - m| ≤ 3
double neighbor_sum = 0.0;
for (int mp = std::max(0, m - 3); mp <= std::min(CROSS_HELIX_BONDS - 1, m + 3); ++mp) {Exact original notation
λ_cross(m) = γ₀·e^{-4m·r·sin(π/8)/ξ}·cos(8πm/10.6) +