Theorem: Complete product positivity ledger

For $0<\beta <1$:

1. The negative transverse eigenvalues are $-{\kappa }^2$ satisfying

   $$\det [H_W+V^{†}\Delta (\kappa )V]=0.$$

   Their number, with multiplicity, is the number of negative eigenvalues of $H_W$. There are no transverse zero eigenfunctions. Thus the full product operator is nonnegative exactly when $H_W\ge 0$.

2. In a full finite chart, rotation invariance is equivalent to diagonality of $H$. Rotation by $\gamma$ sends $H_{0,-1}$ to $e^{\mathrm{i}\gamma }{H}_{0,-1}$. On boundary charts the corresponding condition is invariance of the whole relation.

3. If the deficient orders are distinct, the dilation-invariant relations are exactly the $2^d$ coordinate choices of Friedrichs or ${\mathsf{E}}_0$. If $\beta =1/2$ and $\alpha >1/2$, there is in addition

   $$a\in \mathbb{C}v,b\perp v,[v]\in {\mathbb{CP}}^1.$$

   This is ${\mathsf{E}}_0$ on $v$ and Friedrichs on $v^{\perp }$.

For $\beta =0$, every ${\mathsf{L}}_L$ has one transverse eigenvalue $-{{\kappa }_L}^2$. Friedrichs is the only nonnegative massless extension in the untwisted product class.

Proof. A negative transverse eigenfunction is a linear combination of $K_{{\nu }_j}(\kappa r){\psi }_j$. Its boundary condition is precisely the displayed determinant condition, by the calculation in Theorem thm:krein.

The compressed matrix $V^{†}\Delta (\kappa )V$ increases strictly in Loewner order from zero to infinity. The ordered eigenvalues of its sum with $H_W$ therefore increase strictly and continuously from the eigenvalues of $H_W$ to positive infinity. Exactly its initially negative eigenvalues cross zero, counted with multiplicity. The resolvent difference has finite rank, so the transverse essential spectrum remains $[0,\infty )$. Zero-energy radial solutions $r^{\pm \nu }$, or $1,\log r$, cannot be square integrable at infinity.

Rotations multiply each boundary component by $e^{\mathrm{i}{q}_j\gamma }$. For two deficient modes their charge difference is one, giving the stated transformation.

Modulo a common scalar normalization, dilations act by $a_j\mapsto s^{{\nu }_j}{a}_j$, $b_j\mapsto s^{-{\nu }_j}{b}_j$. With distinct positive orders all weights are distinct, so an invariant Lagrangian is a coordinate sum containing exactly one of each paired $a_j,b_j$ axes. At equal orders an invariant Lagrangian is $W_a\oplus {W_a}^{\perp }$ in the $a,b$ weight spaces. The intermediate one-dimensional choices give ${\mathbb{CP}}^1$.

Finally $K_0(\kappa r)$ matches $\log (r/L)$ at $\kappa =2e^{-{\gamma }_E}/L$.