= Solution
Assume $\lambda>0$ for the attractive wells. Write $E=-\hbar^2\kappa^2/(2m)$ with $\kappa>0$. Between wells the solution is a linear combination of $e^{\kappa x}$ and $e^{-\kappa x}$. Across a well it is continuous and its derivative jumps by $-2\lambda\psi$. Free propagation by $a$ and the jump have matrices
$$
P_a=\begin{pmatrix}\cosh\kappa a&\kappa^{-1}\sinh\kappa a\\
\kappa\sinh\kappa a&\cosh\kappa a\end{pmatrix},\qquad
D=\begin{pmatrix}1&0\\-2\lambda&1\end{pmatrix}.
$$
Their product has <determinant> one and <trace> $2[\cosh\kappa a-(\lambda/\kappa)\sinh\kappa a]$. <Bloch theorem> $e^{\pm ika}$ therefore give the <negative-energy band of an attractive delta comb> equation
$$
\boxed{\cos ka=\cosh\kappa a-\frac\lambda\kappa\sinh\kappa a}.
$$
Negative energies are allowed precisely where the right side lies in $[-1,1]$. A cell <wavefunction> can be written explicitly as
$$
\psi(x)=\psi(0)\frac{\sinh\kappa(a-x)+e^{ika}\sinh\kappa x}{\sinh\kappa a},\quad 0<x<a,
$$
continued by <Bloch theorem>; the band equation enforces its derivative jump.
For $\lambda a\gg1$, the isolated well has $\kappa=\lambda$ and energy $E_0=-\hbar^2\lambda^2/(2m)$. Writing $\kappa=\lambda+\delta$, the band equation gives $\delta\sim2\lambda e^{-\lambda a}\cos ka$. Hence
$$
\boxed{E(k)\simeq E_0-2t\cos ka,\qquad t=\frac{\hbar^2\lambda^2}{m}e^{-\lambda a}}.
$$
This is the nearest-neighbour <tight-binding model>. Its hopping magnitude follows independently from isolated wavefunctions $\phi_n=\sqrt\lambda e^{-\lambda|x-na|}$: the neighbouring delta well contributes $\langle\phi_0,(H-E_0)\phi_1\rangle\simeq-(\hbar^2\lambda/m)\phi_0(0)\phi_1(0)=-t$. The subtraction removes the overlap contribution of $E_0$. More distant contributions are exponentially smaller. For repulsive or absent wells, $\lambda\le0$, the quadratic form is nonnegative and there are no negative-energy bands.
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