Manuscript X-A@1 · X-A.tex · definition · dc7f5c0a

Definition: Switched scalar jet

Switched scalar action and coefficient equations › The switched solution with fixed free input · lines 1206–1221

X-A@1#def:rho, X-A@1#eq:rho1, X-A@1#eq:rho2

Fix a smooth real $ϕ_{0}$ with $(□ + m^{2}) ϕ_{0} = 0$ on $M_{T}$ . The switched solution $ρ_{g} [q; ϕ_{0}] = ρ_{0} + κ ρ_{1, g} + κ^{2} ρ_{2, g}$ is defined by $ρ_{0} = ϕ_{0}$ , $E_{g} [q] ρ_{g} = 0 (mod κ^{3})$ , and zero Cauchy data for $ρ_{1, g}, ρ_{2, g}$ . Explicitly,

(□ + m^{2}) ρ_{1, g} (□ + m^{2}) ρ_{2, g} = - V_{1, g} [h] ρ_{0}, = - V_{1, g} [h] ρ_{1, g} - V_{2, g} [h, k] ρ_{0}, ρ_{1, g} ρ_{2, g} = - E_{m} V_{1, g} [h] ϕ_{0}, = E_{m} V_{1, g} [h] E_{m} V_{1, g} [h] ϕ_{0} - E_{m} V_{2, g} [h, k] ϕ_{0} .

Displayed formulas in this passage

X-A@1#eq:rho1 · line 1212
$(□ + m^{2}) ρ_{1, g} (□ + m^{2}) ρ_{2, g} = - V_{1, g} [h] ρ_{0}, = - V_{1, g} [h] ρ_{1, g} - V_{2, g} [h, k] ρ_{0}, ρ_{1, g} ρ_{2, g} = - E_{m} V_{1, g} [h] ϕ_{0}, = E_{m} V_{1, g} [h] E_{m} V_{1, g} [h] ϕ_{0} - E_{m} V_{2, g} [h, k] ϕ_{0} .$
X-A@1#eq:rho2 · line 1212
$(□ + m^{2}) ρ_{1, g} (□ + m^{2}) ρ_{2, g} = - V_{1, g} [h] ρ_{0}, = - V_{1, g} [h] ρ_{1, g} - V_{2, g} [h, k] ρ_{0}, ρ_{1, g} ρ_{2, g} = - E_{m} V_{1, g} [h] ϕ_{0}, = E_{m} V_{1, g} [h] E_{m} V_{1, g} [h] ϕ_{0} - E_{m} V_{2, g} [h, k] ϕ_{0} .$

Definition: Switched scalar jet

Displayed formulas in this passage

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