# A solution of $-y'' + q(x)y= \lambda y$

Could you help me with the following problem (from Poschel and Trubowitz)?

I am looking for a solution of the differential equation $-y'' + q(x)y= \lambda y$, for $0 \leq x \leq 1$ with $\lambda \in \mathbb{C}$ and $q \in L_{\mathbb{C}}^2([0,1])$ subject to the initial conditions $y(0)=1$ and $y'(0)=0$.

Let $y(x,\lambda,q)$ is given as a power series in $q$, i.e.

$$y(x,\lambda,q)= C_0(x,\lambda) + \sum_{n \geq 1} C_n(x,\lambda,q),$$

where $C_n(x,\lambda,q)=C_n(x,\lambda,q_1,\cdots, q_n)|_{q_1=\cdots=q_n=q}$ and $C_n(x,\lambda,q_1,\cdots, q_n)$ is a bounded, multi-linear symmetric form on $L_{\mathbb{C}}^2\times \cdots\times L_{\mathbb{C}}^2$ for each $x$ and $\lambda$.

We already know that $C_n(x,\lambda,q)= \int_0^x \frac{\sin(\sqrt{\lambda}(x-t))}{\sqrt{\lambda}} q(t) C_{n-1}(t,\lambda,q) \,dt$, $n\geq 1$ is a solution with $C_0(x,\lambda)=\cos{\sqrt{\lambda}x}$.

How to show that

1) $y(x,\lambda,q)=y(x,0,q-\lambda)$.

2) $C_n(x,\lambda,q)= \int_{0\leq t_1 \leq \cdots \leq t_{n+1}= x} \prod_{i=1}^n (t_{i+1}-t_i)(q(t_i)-\lambda) \, dt_1 \cdots dt_n$.

Thank you!

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