Quadratic-cubic remainder after subtracting a trajectory

PDF p. 20: (4.6)

Assume the Axiom of Choice. Let $T>0$, let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and $\lambda \in \mathbb{R}\setminus \{0\}$ be as in the reference trajectory hypothesis (W), and let $w$ satisfy (W). Thus $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ and $i{\mathrm{\partial }}_{t}w+\mathrm{\Delta }w+\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}w=0$ in distributions. Let $\phi$ be the control cutoff and put $V_1=2\lambda \mathrm{\mid }w{\mathrm{\mid }}^2$ and $V_2=\lambda w^2$.

For a complex-valued state $v$, set $z=v-w$. The trajectory remainder is the pointwise expression

$F(w,z)=\lambda (2w\mathrm{\mid }z{\mathrm{\mid }}^2+\bar{w}{z}^2+\mathrm{\mid }z{\mathrm{\mid }}^{2}z)\mathrm{.}$

Here $\bar{w}$ is complex conjugation and $\mathrm{\mid }z\mathrm{\mid }$ is the complex modulus. If $v$ and $w$ have endpoint values and $v(0)=v_0$, $v(T)=v_T$, define $z_0=v_0-w(0)$ and $z_T=v_T-w(T)$.

For this distributional interpretation put $Q=(0,T)\times \mathrm{\Omega }$. Write $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^3(Q)$ when $u:Q\to \mathbb{C}$ is Lebesgue measurable and ${\int }_K\mathrm{\mid }u{\mathrm{\mid }}^{3}dtdx<\mathrm{\infty }$ for every compact $K\subset Q$; functions equal almost everywhere are identified. A measurable function $f$ is locally integrable on $Q$ when ${\int }_K\mathrm{\mid }f\mathrm{\mid }dtdx<\mathrm{\infty }$ for every such $K$. Each locally integrable term $f$ is interpreted as the regular distribution $⟨T_f,\eta ⟩={\int }_{Q}f\eta dtdx$ for $\eta \in C_c^{\mathrm{\infty }}(Q;\mathbb{C})$, with no conjugation.

For example, if $v\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^3((0,T)\times \mathrm{\Omega })$ and $\phi h$ is locally integrable, the cubic controlled equation for $v$ is equivalent, in space-time distributions, to

$i{\mathrm{\partial }}_{t}z+\mathrm{\Delta }z+V_{1}z+V_2\bar{z}+F(w,z)=\phi h\mathrm{.}$

When the chosen solution class also provides homogeneous Dirichlet traces and endpoint values, subtracting $w$ gives the same homogeneous boundary condition for $z$ and the endpoint conditions $z(0)=z_0$, $z(T)=z_T$. The distributional interior equation alone does not assert the existence of those traces or of a solution.

Reference trajectory hypothesis W

Linearized equation and its real-duality adjoint

Linear and nonlinear internally controlled exterior systems

Distributional equation convention for complex Schrödinger solutions

Real and imaginary parts, complex conjugation, and modulus

Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive

The Axiom of Choice

Locally integrable functions as regular distributions

The reference trajectory, its equation, and the inherited Choice assumption are specified by Reference trajectory hypothesis W and The Axiom of Choice. The cutoff and controlled cubic equation are those of Linear and nonlinear internally controlled exterior systems, while Linearized equation and its real-duality adjoint supplies the coefficients $V_1$ and $V_2$.

The conjugation and modulus conventions are those of Real and imaginary parts, complex conjugation, and modulus. Using the laws in Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive and ordinary complex multiplication, expansion gives $\mathrm{\mid }w+z{\mathrm{\mid }}^2(w+z)=\mathrm{\mid }w{\mathrm{\mid }}^{2}w+2\mathrm{\mid }w{\mathrm{\mid }}^{2}z+w^2\bar{z}+2w\mathrm{\mid }z{\mathrm{\mid }}^2+\bar{w}{z}^2+\mathrm{\mid }z{\mathrm{\mid }}^{2}z$. Hence the definition is precisely the part left after subtracting the trajectory cubic term and the terms real-linear in $z$.

The compact-local convention above defines $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^3(Q)$ directly using measurable representatives and the complex modulus. The meaning of local integrability and the regular-distribution construction are supplied by Locally integrable functions as regular distributions and its defining regular-functional prerequisite. The assumed Axiom of Choice covers that supplier's Countable Choice hypothesis; its construction and linearity themselves require no choice.

Under the displayed local integrability assumption, $w$ is locally bounded by Reference trajectory hypothesis W. For each compact $K\subset Q$, the bound $\mathrm{\mid }z{\mathrm{\mid }}^3\le 4(\mathrm{\mid }v{\mathrm{\mid }}^3+\mathrm{\mid }w{\mathrm{\mid }}^3)$ implies ${\int }_K\mathrm{\mid }z{\mathrm{\mid }}^3<\mathrm{\infty }$. The inequalities $\mathrm{\mid }z{\mathrm{\mid }}^j\le 1+\mathrm{\mid }z{\mathrm{\mid }}^3$ for $j=1,2$, the finite measure of $K$, and the local boundedness of $w$ show that every term of the expansion is locally integrable. By Locally integrable functions as regular distributions, these terms and the forcing define distributions through the complex-linear map $f\mapsto T_f$. Linearity of distributional derivatives and subtraction of the equation for $w$ yield the displayed remainder equation; reversing the same calculation recovers the equation for $v$. The interpretation of distributional equality and the limitation on endpoint and boundary traces are those of Distributional equation convention for complex Schrödinger solutions. In a class with linear trace and endpoint maps, subtracting the corresponding values gives the stated conditions for $z$.

Equation (4.6) on PDF page 20 of the original article supplies the attribution and notation. The algebraic calculation above uses the named mathematical suppliers; the article is not used as proof of its own claim.