Neglecting an overall constant (Casimir energy).

This implies zero particle flow at $x=\pm L/2$ since the two fermion densities ${\rho }_{\pm }(x)=:{\psi }_{\pm }^{†}(x){\psi }_{\pm }(x):$ are equal at each boundary and the current is $\propto {\rho }_{+}-{\rho }_{-}$.

Equation (3) corresponds to the low-energy linearization of the spin-chain dispersion relation around its Fermi points $\pm k_{\mathrm{F}}$, followed by recentering the resulting fields so that they respectively describe right- and left-moving excitations around zero momentum.

Full PWT, as defined in Eq. (1), corresponds to Eq. (6) holding for all states $|{\mathrm{\Psi }}_1⟩$ in the Fock space with $|{\mathrm{\Psi }}_2⟩$ being their respective reflections.

Oppositely, the limiting case of an inhomogeneous CFT given by Möbius transformations so that $v(x)$ vanishes at $\pm L/2$ cannot exhibit PWT since $1/v_0$ diverges.

This applies to regular systems, to ensure that the Neumann BCs remain simple. If irregular, say, $K(x)$ and/or $v(x)$ vanishing at $\pm L/2$, this is irrelevant as the BCs are different and no longer couple right and left movers.

This generalizes [37] as $v(x)$ and $K(x)$ are unrelated, but in a way that is different from the approach in [61].

More precisely, twice continuously differentiable.

Here, these amount to $v(x)$ and $K(x)$ being strictly positive and $v(x)$, $K(x)$, ${\partial }_{x}p(x)$ [and $q(x)$ if nonzero] continuous on the whole interval $[-L/2,L/2]$.

Zero modes such as ${\phi }_0$ need separate treatments when $E_0=0$, but they have no consequence for our discussion of PWT, so we omit them.

Here, (b) follows from Property (iv) and (a) from Properties (i) and (ii) with ${\lambda }_0=0$.

A related prominent problem is the inverse scattering for integrable differential equations, solved by the Gelfand-Levitan-Marchenko equation. That case is on the infinite line, while we have a finite interval.

The BCs are that solutions must be sufficiently well behaved for Eq. (9) to be satisfied.

For even $v(x)>0$, they are $v_0{\partial }_y\mathfrak{u}(y)-v(x)\mathrm{\Lambda }(x)\mathfrak{u}(y)=0$ at $y=\pm L/2$ with $\mathrm{\Lambda }(x)\equiv {\partial }_x\ln \sqrt{K(x)}$ [61], which become Neumann BCs if $K^{\prime }(x){|}_{x=\pm L/2}=0$.

As $K(x)$ must be even, this also implies $K^{\prime }(x){|}_{x=\pm L/2}=0$, as assumed below Eq. (9).