Computation of overlap matrix $P(n\vert m)$ and its profile

The quantity of interest is the matrix of overlaps $P(n\vert m) \equiv \vert\langle \psi_n(x) \vert \psi_m(0) \rangle_{{\mathcal{D}}'} \vert ^2$. The eigenstates $\psi_n(x)$ are dependent on the deformation parameter $x$. The overlap is taken in position-space ${\mathbf r}$ over the domain ${\mathcal{D}}'$, defined as the intersection of the undeformed domain ${\mathcal{D}}(0)$ and the deformed domain ${\mathcal{D}}(x)$. We seek a way to perform these $d$-dimensional domain integrals rapidly (in our case $d=2$). Appendix H shows that because the eigenvalues $k_n(x)$ and $k_m(0)$ are (in general) different, there exists a very simple formula (H.5) for computing this integral using only the boundary. This will be a huge saving in effort.

Because our eigenstates have Dirichlet boundary conditions, a simplification can be made. Define $\Gamma_1$ ($\Gamma_2$) as the part of $\Gamma(0)$ ($\Gamma(x)$) inside $\mathcal{D}(x)$ ( $\mathcal{D}(0)$). Then $\Gamma_1 + \Gamma_2 = \Gamma'$, where $\Gamma'$ is the boundary of the intersection region $\mathcal{D}'$. Then the overlap integral over $\mathcal{D}'$ can be written

$$\!\!\! \left\langle \psi_n(x) \left\vert\psi_m(0)\right.\ri... ...!\!\! d{\mathbf s} \,\psi_m^*(0) \partial_n \psi_n(x) \right].$$ (6.30)

In practice this formula is evaluated using the method of Appendix H, using the wavefunctions evaluated at the set of discretization points from $\Gamma(x)$ and $\Gamma(0)$ which lie in $\Gamma'$. The formula becomes inaccurate (depending on the accuracy of the eigenstates) when $\vert k_n(x) - k_m(0)\vert \rightarrow 0$. In practice this was only a problem at wavenumber differences well below the mean level spacing. The above overlap calculation technique is so efficient that it only took a few seconds to fill each $P(n\vert m)$ matrix of size about $450$ by $450$, at $k \approx 400$.

Given the overlap matrix $P(n\vert m)$ at each $x$, its average profile was found by the smoothing method of Section C.2. The smoothing width $\omega_s$ was chosen to be 0.02 in wavenumber units, which was a couple of times the mean level spacing $\Delta_k \approx 0.0088$.

The resulting sequences of profiles are shown in Fig. 6.14 and 6.15. Comparison with the first-order perturbation theory (FOPT) result,

$$P(n\vert m) \; = \; \frac{x^2}{(E_n - E_m)^2} \cdot \left\v... ...al x} \right)_{\!nm} \right\vert ^2 \hspace{1in} \mbox{FOPT},$$ (6.31)

is also made in these plots. $(\partial {\mathcal{H}} / \partial x)_{nm}$ is explicitly given by (C.2), and its band profile is described in Chapters 2 and 3. The FOPT result simply corresponds to a linearization of the wavefunction $\psi = z\cdot\partial_n \psi + O(z^2) \cdots$ using the notation of Appendix I. The profile shows complete breakdown of FOPT when $x$ reaches $\sim \lambda_{{\mbox{\tiny B}}}$, and in the case of a generic deformation the formation of a `core' region before this. For more analysis see our work [48].