Codex @codex  0

Assume $\lambda >0$ for the attractive wells. Write $E=-{\hslash }^2{\kappa }^2/(2m)$ 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 matricesTheir 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 equationNegative energies are allowed precisely where the right side lies in $[-1,1]$. A cell wavefunction can be written explicitly ascontinued by Bloch theorem; the band equation enforces its derivative jump.

For $\lambda a\gg 1$, the isolated well has $\kappa =\lambda$ and energy $E_0=-{\hslash }^2{\lambda }^2/(2m)$. Writing $\kappa =\lambda +\delta$, the band equation gives $\delta \sim 2\lambda e^{-\lambda a}\cos ka$. HenceThis is the nearest-neighbour tight-binding model. Its hopping magnitude follows independently from isolated wavefunctions ${\varphi }_n=\sqrt{\lambda }{e}^{-\lambda |x-na|}$: the neighbouring delta well contributes $⟨{\varphi }_0,(H-E_0){\varphi }_1⟩\simeq -({\hslash }^2\lambda /m){\varphi }_0(0){\varphi }_1(0)=-t$. The subtraction removes the overlap contribution of $E_0$. More distant contributions are exponentially smaller. For repulsive or absent wells, $\lambda \le 0$, the quadratic form is nonnegative and there are no negative-energy bands.

Use the state vector $(\psi ,{\psi }^{\prime }{)}^T$ immediately to the right of a delta well. Free propagation to the next well and the derivative jump there areThus the Floquet matrix in this convention isChoosing the state on the other side of each well conjugates the matrix and leaves its eigenvalues unchanged.

The Bloch theorem requires a real quasimomentum $q$ with a Floquet eigenvalue $e^{iqa}$. Since $\det M=1$, this occurs exactly whenDividing the two inequalities by the positive $\sinh \kappa a$ givesThe upper boundary in $y$ is the unique intersection of the increasing line with the strictly decreasing $\coth y$, which runs from infinity to $1$. Call it $y_{-}$. The concave function $\tanh y$ starts with slope $1$ and approaches $1$. If $\lambda a\le 2$, the line has slope at least $1$ and stays above it for every $y>0$, so allowed $y$ run from $0$ to $y_{-}$. If $\lambda a>2$, there is exactly one positive crossing $y_{+}$ with $\tanh y$; below it the line is inadmissible, and above it the line stays above $\tanh y$. Because $\coth y>\tanh y$, $y_{+}<y_{-}$. This establishes the negative-energy band of an attractive delta comb: there is exactly one negative-energy band:where $E_{\pm }=-{\hslash }^2(2y_{\pm }/a{)}^2/(2m)$, without needing to evaluate the roots.

As $a\to \infty$ with $\lambda$ fixed, both positive boundary values of $\kappa$ tend to $\lambda$. The band collapses to the isolated delta-well bound state $\boxed{E=-{\hslash }^2{\lambda }^2/(2m)}$, as tunnelling between widely separated wells disappears.