# Determining a constant with conditions on a limit and an integral

$$\pmb{Problem} :$$

Given the differential equation :

$$(E) : -u''(x) + tanh(x). u(x) = -\lambda.u(x)$$

Let $$f_n(x) = C_n.cos(\sqrt{\lambda-1}. x). \varphi(\frac{x-n^2}{n})$$ where $$\varphi \in C^{\infty}(\mathbb{R})$$ and $$\varphi(x) = \left\{ \begin{array}{c} 1 & if \ \ x \in [-1, 1]\\ 0 & if \ \ \lvert x\rvert \geq 2 \\ g(x) &else \end{array} \right.$$ where $$g \in C^{\infty}$$ (That's all we know about the function g)

Let $$\alpha_n = \int_{-\infty}^\infty f_n^2(x) .dx$$

Determine $$C_n$$ such that $$\displaystyle\lim_{n\to\infty} \alpha_n \neq 0$$ $$and \int_{-\infty}^\infty L^2(f_n)(x).dx \to 0$$ when $$n \to +\infty$$

where $$L(f_n)(x) = -f''_n(x) + tanh(x).f_n(x) - \lambda.f_n(x)$$

$$\pmb{Discussion :}$$

For instance, I started by calculating $$f_n ^2(x)$$ to evalute $$\alpha_n$$ in terms of n. I also tried to solve the differential equation $$(E)$$ but I couldn't do that due to the hyperbolic tangent (non constant coefficients). In addition, I calculated the first derivative of $$f_n$$ for the hope of finding a simpler expression of $$f_n$$ after integrating but all of this didn't get me anything. So I literally can't find where to begin to address this problem.