Let $H_0|r⟩=E_r|r⟩$. In the interaction picture,and the state obeysThe first term of the Dyson series, or equivalently first-order time-dependent perturbation theory, givesThus, if $|\psi (0)⟩={\sum }_r{a}_r|r⟩$,

The first Born approximation for a one-dimensional reflection coefficient follows by replacing the Jost solution inside its Volterra scattering equation by the free wave $e^{ikx}$:

To check the inverse problem, any shallow discrete state contributes only at $O({ϵ}^2)$ pointwise, so to first order the Marchenko input isDifferentiating and inserting the Born approximation giveswhere the second line is the Fourier inversion theorem. At first order the quadratic integral term in the Marchenko equation may be dropped, soThe reconstruction formula therefore returnsThus direct and inverse scattering agree to first order.

For one bound state at $-{\kappa }^2$ and no reflection, writeThe rank-one Gelfand-Levitan-Marchenko kernel has the form $K(x,y;t)=-A(x,t)e^{-\kappa y}$. Substitution into the integral equation givesand henceChoose $x_0$ by $C_0/(2\kappa )=e^{2\kappa x_0}$. ThenDifferentiating the reconstruction formula gives the one-soliton solutionIt is a negative soliton of amplitude $2{\kappa }^2$ travelling to the right with speed $4{\kappa }^2$, as required by this sign convention for KdV.

The Korteweg-De Vries equation is the compatibility condition $L_t=[P,L]$ for the Lax pairFor rapidly decaying initial data, first solve the direct KdV Schrodinger spectral problemComparing its Jost solutions at the two spatial infinities produces the KdV scattering data: the reflection coefficient $R(k)$ on the continuous spectrum, discrete eigenvalues $-{\kappa }_n^2$, and norming amplitudes $c_n$ for the bound states.

The Lax evolution is isospectral. More precisely, the Time evolution of KdV scattering data isThis is the simple linear evolution that makes the nonlinear initial-value problem tractable.

To invert the data, formSolve the Gelfand-Levitan-Marchenko equationand reconstructDirect scattering at $t=0$, linear evolution of the data, and this inverse step constitute the inverse scattering transform.

The Bendixson-Dulac criterion says the following. Let $D\subseteq {\mathbb{R}}^2$ be a simply connected domain, let $F=(f,g)$ be continuously differentiable, and suppose there is a continuously differentiable function $B$ such thathas one sign throughout $D$ and does not vanish identically on any open subset. Then $\dot{z}=F(z)$ has no periodic orbit contained in $D$.

Indeed, if a periodic orbit $\Gamma$ existed, simple connectivity would put its interior $U$ inside $D$. The vector field $F$, and hence $BF$, is tangent to $\Gamma$, so its outward flux is zero. The divergence theorem would givecontradicting the sign hypothesis.

The Poincaré-Bendixson theorem states that a nonempty compact omega-limit set of a continuously differentiable planar flow that contains no equilibrium point is a periodic orbit. More generally, when the omega-limit set contains only finitely many equilibria, it is either an equilibrium, a periodic orbit, or a union of equilibria and connecting trajectories.

For each $d$-element coordinate set $J\subseteq \{1,\dots ,p\}$, let ${\mathcal{G}}_J$ contain the lower orthants whose finite thresholds occur precisely in the coordinates in $J$. Ignoring the remaining coordinates identifies ${\mathcal{G}}_J$ with the lower-orthant class in ${\mathbb{R}}^d$, so part (a) gives $\mathrm{VC}({\mathcal{G}}_J)=d$. Moreover,is a union of $r=\left(\genfrac{}{}{0ex}{}{p}{d}\right)$ such classes. Part (c) and $\left(\genfrac{}{}{0ex}{}{p}{d}\right)\le p^d$ give

If $x_{1:n}$ is shattered by ${\bigcup }_{k=1}^r{\mathcal{F}}_k$, then all $2^n$ label vectors belong to the union of the label-vector sets generated by the individual classes. The shattering coefficient of a union and the Sauer-Shelah lemma therefore giveTaking base-two logarithms yieldsFor $r>1$, apply the stated elementary bound with $\alpha ={\log }_{2}r$ and $\beta =v\ge 3$:If $r=1$, the assertion follows directly from $n\le v$. Taking the largest shattered sample size proves

The Sauer-Shelah lemma states that if $\mathrm{VC}(\mathcal{F})=v<\infty$, thenIn particular, the Sauer-Shelah growth bound gives $s(\mathcal{F},n)\le (n+1{)}^v$.

A sample $x_{1:n}=(x_1,\dots ,x_n)$ is shattered by $\mathcal{F}$ when every binary labeling is realized:The shattering coefficient and VC dimension are

For the lower orthants in ${\mathbb{R}}^d$, the $d$ points $e_1,\dots ,e_d$ are shattered: to realize a subset $I$, choose $a_j=1$ for $j\in I$ and $a_j=1/2$ otherwise. Conversely, among any $d+1$ points choose, for each coordinate, one point attaining the maximum in that coordinate. At most $d$ points have been chosen, so there is an unchosen point $x$. Any lower orthant containing every chosen coordinate maximizer must contain every sample point, including $x$. It cannot realize the labeling that includes all chosen points and excludes $x$. This proves

Suppose the claim $H=(S_1{)}^2$, payable at time $1$, were replicable. Holdings chosen at time zero are constants, so the time-$1$ value of any self-financing portfolio has the affine formBut $S_1=S_0{Z}_1$ has a continuous positive density on $(0,\infty )$. No affine function of $S_1$ can equal the quadratic function $S_1^2$ almost surely. Therefore $H$ is not attainable and the market is not complete.

The same argument applies if $H$ is viewed as a time-$T$ claim: once $H$ is known at time $1$, its required discounted value at time $1$ is fixed, and the first-period portfolio value would still have to reproduce a nonaffine function of $S_1$.

There is no arbitrage. Indeed, part (iii) supplies an equivalent martingale measure, so the fundamental theorem of asset pricing implies absence of arbitrage. Directly, the discounted terminal value of a zero-cost admissible self-financing strategy is a ${\mathbb{P}}^{\ast }$-martingale with mean zero; it cannot be nonnegative almost surely and positive with positive probability.

Set $Y_n=\log Z_n$ andDefine a probability measure ${\mathbb{P}}^{\ast }$ by the positive densityThe density has expectation one and is strictly positive, so ${\mathbb{P}}^{\ast }$ is equivalent to $\mathbb{P}$. By exponential tilting, the variables $Y_k$ remain independent under ${\mathbb{P}}^{\ast }$ and have distribution $N({\mu }_{\ast },{\sigma }^2)$. Thus $Z_k$ has a log-normal distribution andFor the discounted stock ${\tilde{S}}_n=S_n/S_n^0$,Hence ${\mathbb{P}}^{\ast }$ is an equivalent martingale measure.

The model is a complete market when every contingent claim in the specified class, usually every ${\mathcal{F}}_T$-measurable payoff satisfying the required integrability condition, is the terminal value of some admissible self-financing portfolio.