First positive eigenvalue and extension-dependent ground states of the twisted Laplacian on a constant-curvature Möbius band

Abstract

We study the twisted Laplacian, the Laplace-Beltrami operator on sections of the orientation line bundle, on a constant-curvature Möbius band $M(W)$ of half-width $W$, built from a spherical band on $S^2(R)$ with the antipodal-edge identification. Its metric collapses one transverse fiber to a cone point, where the operator is not essentially self-adjoint in the constant transverse sector, so the bottom of the spectrum depends on the self-adjoint extension. For the Friedrichs extension the bottom is $0$, attained by a discontinuous piecewise-constant zero mode $\phi_0$, and the operator is unitarily equivalent to the Neumann Laplacian on a spherical lune. Every Boscain-Prandi bridging extension excludes $\phi_0$ and instead carries a single cone-localized bound state $\lambda_b(\delta_0) = -4e^{-2\gamma}/\delta_0^2\,(1 + O(\delta_0^2/R^2))$ below zero, set by a renormalization length $\delta_0$; by Friedrichs maximality, no self-adjoint extension has a strictly positive ground state. The first positive eigenvalue, by contrast, is stable across these realizations: for the Friedrichs extension and for bridging with $\delta_0 > 2R/e$ it equals $2/R^2$ for $0 < W \le \pi R/2$ and $\alpha_0(\alpha_0+1)/R^2$, $\alpha_0 = \pi R/(2W)$, for $\pi R/2 < W < \pi R$, doubly degenerate at the critical width. We prove this by separating the operator into transverse Neumann sectors and solving the reduced singular Sturm-Liouville problems in closed form via Legendre functions. For $W > \pi R/2$ the first positive eigenfunction changes sign, reflecting the nontriviality of the orientation bundle.

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1. Introduction

On a Möbius band the orientation line bundle has nontrivial holonomy: a section changes sign after one circuit of the core loop. We study how this twist affects the Laplacian when the metric also degenerates, the cone lying exactly where a transverse fiber collapses. On a smooth surface a nowhere-vanishing section would trivialize the bundle, so the twist already rules out a positive ground state (cf. [Shigekawa] for the magnetic spectral-positivity criterion); the cone determines how that obstruction appears in the spectrum, and the answer depends on the self-adjoint extension.

The setting is the twisted Laplacian (the Laplace-Beltrami operator on sections of the orientation bundle $\mathcal{L}$) on a constant-curvature Möbius band $M(W)$ constructed from a spherical band on $S^2(R)$ of half-width $W$. The metric $ds^2 = dy^2 + \cos^2(y/R)\,dw^2$ collapses a transverse fiber to a cone point $p_c$ at $y = \pi R/2$. At $p_c$ the operator is not essentially self-adjoint in the constant transverse sector, and a choice of self-adjoint extension is required. This choice is spectral, not cosmetic: it controls the bottom of the spectrum, and different natural realizations give qualitatively different ground states. The first positive eigenvalue, by contrast, is shared by the Friedrichs extension and the bridging family with $\delta_0 > 2R/e$, and it is this invariant that we compute. We leave open which realization is physically selected.

We first classify the bottom of the spectrum across realizations (Theorem 1.1), then isolate the invariant that survives the extension choice (Theorem 1.2).

Theorem 1.1 (Classification of the bottom of the spectrum). Let $A$ be any self-adjoint extension at the cone of the twisted Laplacian on $M(W)$, $0 < W < \pi R$. Then $\inf\sigma(A) \le 0$. Equality holds if and only if $A$ is a nonnegative extension; among these the Friedrichs extension is maximal, and its bottom is attained by the discontinuous piecewise-constant zero mode $\phi_0$ (eigenvalue $0$, the constant-sector kernel). The Boscain-Prandi bridging family is not nonnegative: at renormalization length $\delta_0$ it carries a single cone-localized bound state $\lambda_b(\delta_0) = -4e^{-2\gamma}/\delta_0^2\,(1 + O(\delta_0^2/R^2))$, with $\gamma$ the Euler-Mascheroni constant, and zero lies in the resolvent set. In particular no self-adjoint extension admits a strictly positive ground state.

Theorem 1.2 (First positive eigenvalue and width transition). The first positive eigenvalue of the twisted Laplacian on $M(W)$ is shared by the Friedrichs extension and by every bridging extension with $\delta_0 > 2R/e$, and equals

$\lambda_1(W) = \begin{cases} 2/R^2 & 0 < W \le \pi R/2, \\ \alpha_0(\alpha_0+1)/R^2, \quad \alpha_0 = \pi R/(2W) & \pi R/2 < W < \pi R. \end{cases}$

At $W = \pi R/2$ it is doubly degenerate. Both branches are realized by explicit eigenfunctions: $\sin(y/R)$, the $\ell = 1$ zonal Legendre polynomial, for the narrow regime, and $\lvert\cos(y/R)\rvert^{\alpha_0}\sin(\pi w/2W)$, with longitudinal factor $(1-x^2)^{\alpha_0/2}$ where $x = \sin(y/R)$, which reduces to the sectoral harmonic $P_{\alpha_0}^{\alpha_0}$ at integer degree, for the wide regime.

For the Friedrichs extension the operator is unitarily equivalent to the Neumann Laplacian on a spherical lune of opening $2W/R$ (Remark 4.2), so Theorem 1.2 is, in that case, the first nonzero Neumann eigenvalue of the lune, and its width transition is the crossing of the two lowest lune branches. What the lune does not capture is the persistence of this value across the bridging family for $\delta_0 > 2R/e$ and the extension-dependence of the bottom.

The proof separates the operator into transverse Neumann sectors and solves each reduced Sturm-Liouville problem in closed form via Legendre functions. Only the constant (zonal) sector carries the finite-energy extension ambiguity: there the cone is a limit-circle endpoint, while every other sector is limit-point except the wide-band first odd sector, which is limit-circle but where every finite-energy extension selects the regular branch. In the constant sector the cone parity splits the spectrum into a symmetric tower $\{2/R^2, 12/R^2, 30/R^2, \ldots\}$, common to the Friedrichs and bridging realizations, and an antisymmetric tower where they disagree. The Friedrichs antisymmetric tower is $\{0, 6/R^2, 20/R^2, \ldots\}$ with the zero mode at its base; the bridging condition deforms it through an explicit secular function and pushes the base below zero. So the narrow-regime first positive value $2/R^2$ comes from the symmetric tower and is robust, while the zero or negative ground state comes from the antisymmetric tower and is not.

The background comes from a few directions. Cheeger [Cheeger1979, Cheeger1983] founded the spectral geometry of conic singularities, with the Friedrichs extension and its link-geometry deficiency indices. Boscain and Prandi [BP] classify the self-adjoint extensions of the model conic metric $ds^2 = dx^2 + \lvert x\rvert^{-2\alpha_{\mathrm{BP}}}\,d\theta^2$ in the cone regime $\alpha_{\mathrm{BP}} \in (-3, -1]$, which contains our $\alpha_{\mathrm{BP}} = -1$, distinguishing disjoint (Friedrichs) and bridging dynamics. The local asymptotics are regular-singular in the sense of Brüning and Seeley [BS], and the limit-point/limit-circle alternative is governed by the inverse-square theory of Kalf, Schmincke, Walter, and Wüst [KSWW, RS]. The bridging condition is a Kirchhoff-type vertex condition [BK]; its closest predecessor is the length-parametrized cone-apex extension of Kay and Studer [KayStuder], whose logarithmic bound-state branch is the analogue of ours, though without the Möbius topology or the width transition. The anti-periodic half-integer longitudinal quantization is the line-bundle holonomy spectrum of Ballmann, Brüning, and Carron [BBC].

The closest Möbius-strip predecessor is Kalvoda, Krejčiřík, and Zahradová [KKZ], who study the Laplacian on a Möbius strip embedded in $\mathbb{R}^3$ with Dirichlet boundary conditions. Their thin-strip asymptotics yield effective one-dimensional operators. Our setting differs in three respects: constant positive curvature rather than an embedded strip, Neumann boundary conditions (which supply the constant transverse mode that activates the extension ambiguity), and the cone singularity absent from the KKZ geometry. The result is an exact finite-width spectral calculation rather than a thin-strip asymptotic.

Section 2 sets up the geometric model and Section 3 the operator domain, the self-adjoint extensions, and Friedrichs maximality. We then analyze the constant sector by parity (§4) and classify the bottom of the spectrum (§5). Section 6 resolves the nonconstant sectors and the width-dependent first positive eigenvalue, and Section 7 discusses the sign-changing eigenfunction and the status of the renormalization length $\delta_0$.

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2. The geometric model $M(W)$

Consider the round 2-sphere $S^2(R)$ of radius $R$. Let $C$ be a great circle, and let $y$ denote arclength along a meridian measured from $C$, so that $y = 0$ and $y = \pi R$ correspond to antipodal arcs of $C$ reached on opposite sides of the pole at $y = \pi R/2$. Let $w$ denote arclength along $C$, with $w \in [-W, W]$ for a half-width parameter $0 < W < \pi R$. The induced metric on the band $\{(y, w) : 0 \le y \le \pi R,\; \lvert w\rvert \le W\}$ is

ds^2 = dy^2 + \cos^2(y/R)\,dw^2. \tag{2.1}

The Gauss curvature is $K = 1/R^2$ on the smooth locus and the scalar curvature is $R_{\mathrm{sm}} = 2/R^2$.

The map $\tau(y, w) = (\pi R - y, -w)$ is an orientation-preserving isometric involution ($D\tau = \operatorname{diag}(-1, -1)$, $\det D\tau = +1$) with a single fixed point at $(\pi R/2, 0)$. Identifying $(0, w) \sim (\pi R, -w)$ via $\tau$ produces the non-orientable surface

M(W) = [0, \pi R] \times [-W, W] \;/\; (0, w) \sim (\pi R, -w) \tag{2.2}

with boundary $\partial M \cong S^1$, since the two arcs $w = \pm W$ join into a single closed curve under the identification. The non-orientability arises from the edge identification itself: transporting a frame $(\partial_y, \partial_w)$ across the seam sends $\partial_w \mapsto -\partial_w$, reversing the frame orientation.

At $y = \pi R/2$, the metric coefficient $\cos(y/R)$ vanishes, collapsing the transverse fiber $\{y = \pi R/2\} \times [-W, W]$ to a single point $p_c$ in the metric space. Both boundary arcs pass through $p_c$ (the boundary meridians meet at the pole), so $p_c \in \partial M$ and the boundary curve has a corner at the cone vertex.

The local model is the one-point union of two half-cones meeting at the apex $p_c$: a punctured neighborhood of $p_c$ is disconnected into the $y < \pi R/2$ and $y > \pi R/2$ halves, since the geodesic distance to $p_c$ equals $\lvert y - \pi R/2\rvert$ and crossing between halves requires the collapsed fiber, every point of which is $p_c$. The link is two disjoint intervals $[-W, W]$, each of angular length $2W/R$; each half-cone has opening angle $2W/R$; and the closed boundary curve passes through $p_c$ twice, a figure-eight through the apex, with a corner on each side. This two-sided structure is what the four-dimensional cone boundary space and the deficiency count of §3 already encode.

Under Neumann boundary conditions at $w = \pm W$, the transverse problem unfolds onto a circle of circumference $4W/R$ (§3.4), recovering the circle-link framework of Boscain and Prandi [BP] with parameter $\alpha_{\mathrm{BP}} = -1$. Although $p_c$ lies on $\partial M$ as a surface point, it sits at $y = \pi R/2$, interior to the longitudinal interval $(0, \pi R)$. The reduced one-dimensional eigenvalue problems of §§4 and 6 inherit this interior singular-point structure, and the extension theory of §3 applies in full.

The cone angle $2W/R$ equals $\pi$ at the critical width $W = \pi R/2$. For $W < \pi R/2$ (narrow band), the cone angle is less than $\pi$; for $W > \pi R/2$ (wide band), it exceeds $\pi$. As $W \to \pi R$, each half of the band approaches a hemisphere.

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3. Twisted sections and the operator domain

3.1 The orientation line bundle

Since $M(W)$ is non-orientable, scalar functions on it do not capture the full spectral problem. The natural domain for the Laplacian is the space of sections of the orientation line bundle $\mathcal{L}$. A smooth function on the covering band $\widetilde{M} = [0, \pi R] \times [-W, W]$ descends to a section of $\mathcal{L} \to M$ if and only if it satisfies the equivariance condition

\psi(\pi R, -w) = -\psi(0, w) \qquad \text{for all } w \in [-W, W]. \tag{3.1}

This sign reversal is the defining property of sections of $\mathcal{L}$: traversing the core loop once reverses the orientation, so a section of the orientation bundle picks up a factor of $-1$.

The twisted Laplacian is the Laplace-Beltrami operator acting on sections of $\mathcal{L}$. It is the standard Laplace-Beltrami operator on the smooth locus; the adjective "twisted" refers to the domain (anti-equivariant sections) rather than to a modification of the operator itself. The eigenvalue problem $-\Delta\psi = \lambda\psi$ is posed for sections of $\mathcal{L}$ throughout this paper.

3.2 The formal operator

On the smooth locus $M^\circ = M \setminus \{p_c\}$ (where $p_c$ denotes the cone point at $y = \pi R/2$), the Laplace-Beltrami operator associated to the metric $ds^2 = dy^2 + f(y)^2\,dw^2$ with $f(y) = \cos(y/R)$ is

\Delta\psi = \psi_{yy} + \frac{f'}{f}\psi_y + \frac{1}{f^2}\psi_{ww} = \psi_{yy} - \frac{1}{R}\tan\!\Bigl(\frac{y}{R}\Bigr)\psi_y + \frac{1}{\cos^2(y/R)}\psi_{ww}. \tag{3.2}

The area element is $dA = \lvert f(y)\rvert\,dy\,dw$, and $L^2(M, dA)$ denotes the Hilbert space of square-integrable sections of $\mathcal{L}$ with respect to this measure. Although the formal operator is written using $f$ on each smooth coordinate patch, the global measure and the reduced Sturm-Liouville forms use $\lvert f\rvert$, which is nonnegative across the two sides of the collapsed fiber.

The boundary $\partial M$ consists of two geodesic arcs at $w = \pm W$. We impose Neumann boundary conditions: $\partial_w\psi\big|_{w = \pm W} = 0$.

3.3 The involution and the cone point

The involution $\tau(y, w) = (\pi R - y, -w)$ is an isometry of the covering band $\widetilde{M}$ with a unique fixed point at $(\pi R/2, 0)$, the cone point $p_c$. The Möbius identification restricts $\tau$ to the boundary edges $\{0\} \times [-W, W]$ and $\{\pi R\} \times [-W, W]$.

At $p_c$, $f(\pi R/2) = \cos(\pi/2) = 0$. Setting $\delta = y - \pi R/2$:

f(y) = \cos(y/R) = -\sin(\delta/R) \quad \Longrightarrow \quad \lvert f\rvert \sim \frac{\lvert\delta\rvert}{R} \quad \text{as } \delta \to 0. \tag{3.3}

The transverse fiber $[-W, W]$ has metric length $2W\lvert f(y)\rvert$, which collapses to zero. On the covering $S^2$, the coordinate $y = \pi R/2$ is a smooth pole; on the edge-identified Möbius band, the same collapse produces a conical singularity with cone angle $2W/R$.

The surface is smooth away from $p_c$, with constant Gaussian curvature $K = 1/R^2$ and smooth-locus scalar curvature $R_{\mathrm{sm}} = 2K = 2/R^2$. Distributional curvature at $p_c$ is not invoked in this paper.

3.4 Transverse decomposition and cone asymptotics

Separation of variables $\psi(y,w) = u(y)\,\Phi(w)$ with $-\Phi'' = \mu\,\Phi$ subject to Neumann conditions $\Phi'(\pm W) = 0$ yields two families of transverse eigenmodes on $[-W, W]$.

Even family (symmetric under $w \mapsto -w$):

\Phi_k^{\mathrm{e}}(w) = \cos(k\pi w/W), \qquad \mu_k^{\mathrm{e}} = (k\pi/W)^2, \qquad k = 0, 1, 2, \ldots \tag{3.4}

The case $k = 0$ is the constant mode $\Phi_0^{\mathrm{e}} \equiv 1$ with $\mu_0 = 0$.

Odd family (antisymmetric under $w \mapsto -w$):

\Phi_j^{\mathrm{o}}(w) = \sin\bigl((2j+1)\pi w/(2W)\bigr), \qquad \mu_j^{\mathrm{o}} = \bigl((2j+1)\pi/(2W)\bigr)^2, \qquad j = 0, 1, 2, \ldots \tag{3.5}

The Möbius identification $\psi(\pi R, -w) = -\psi(0, w)$ interacts with the transverse parity as follows. For even $\Phi(w) = \Phi(-w)$, the identification forces $u(\pi R) = -u(0)$: the longitudinal function is anti-periodic. For odd $\Phi(-w) = -\Phi(w)$, it forces $u(\pi R) = u(0)$: the longitudinal function is periodic.

A Neumann eigenmode on $[-W, W]$ extends by even reflection to a periodic function on a circle of angular circumference $4W/R$ in the variable $\theta = w/R$. Under this unfolding, the even family $\Phi_k^{\mathrm{e}}$ corresponds to Fourier modes $e^{i(2k)\theta}$ (even harmonics) and the odd family $\Phi_j^{\mathrm{o}}$ to $e^{i(2j+1)\theta}$ (odd harmonics) on that circle. The $k = 0$ constant mode is therefore the analogue of the $k = 0$ Fourier sector in [BP], and the non-essential-self-adjointness of the constant-sector reduced operator at the cone (Theorem 1.6(ii) of [BP]) is the same phenomenon in both settings.

Lemma 3.1 (Neumann unfolding). After the change of angular variable $\theta=w/R$, even reflection at the Neumann endpoints identifies each transverse Neumann eigensector on $[-W,W]$ with a Fourier eigensector on the circle link of angular circumference $4W/R$ (unitarily, up to the harmless normalization factor coming from reflection). For each transverse sector, the reduced one-dimensional operator in $y$ has the same singular endpoint at $\delta = 0$, the same weight $\lvert\delta\rvert/R$, and the same indicial equation. Since the maximal domain, Lagrange bracket, and deficiency indices depend only on the local behavior at the singular endpoint [Weidmann], the Boscain-Prandi extension classification [BP] transfers from the circle-link model cone to the Neumann half-cone: the disjoint and bridging extensions are characterized by the same boundary data in both settings.

In each sector, the reduced eigenequation for the longitudinal function $u(y)$ takes the Sturm-Liouville form

-\bigl(\lvert f\rvert\,u'\bigr)' + \frac{\mu}{\lvert f\rvert}\,u = \lambda\,\lvert f\rvert\,u, \qquad f(y) = \cos(y/R), \tag{3.6}

on $L^2\bigl((0, \pi R),\,\lvert f\rvert\,dy\bigr)$, where $\mu$ is the transverse eigenvalue of the sector. Near the cone ($\delta \to 0$, $\lvert f\rvert \sim \lvert\delta\rvert/R$), this becomes

-\Bigl(\frac{\lvert\delta\rvert}{R}\,u'\Bigr)' + \frac{\mu R}{\lvert\delta\rvert}\,u = \lambda\,\frac{\lvert\delta\rvert}{R}\,u. \tag{3.7}

Substituting $u = \lvert\delta\rvert^s$ into the leading-order balance gives the indicial equation $s^2 = \mu R^2$.

The constant sector ($\mu = 0$, even family $k = 0$). The indicial equation gives $s^2 = 0$ (double root). The two local solutions are

$u_{\mathrm{reg}} = a_0 + a_2\delta^2 + O(\delta^4), \qquad u_{\mathrm{log}} = \log\lvert\delta\rvert + b_2\delta^2 + O(\delta^4).$

Both are square-integrable against the weight $\lvert\delta\rvert/R\,d\delta$: $\int \lvert\delta\rvert^0 \cdot \lvert\delta\rvert\,d\delta < \infty$ and $\int (\log\lvert\delta\rvert)^2 \cdot \lvert\delta\rvert\,d\delta < \infty$. The cone is a limit-circle endpoint, and a self-adjoint extension must be chosen.

Nonconstant sectors with $\alpha \ge 1$. Write $\alpha = \sqrt{\mu}\,R$ for the indicial exponent. When $\alpha \ge 1$, the singular branch $\lvert\delta\rvert^{-\alpha}$ fails the $L^2$ condition against the weight:

$\int u^2\,\lvert f\rvert\,d\delta \sim \int \lvert\delta\rvert^{-2\alpha} \cdot \lvert\delta\rvert\,d\delta = \int \lvert\delta\rvert^{1-2\alpha}\,d\delta = \infty \qquad \text{for } \alpha \ge 1.$

The cone is limit-point. Self-adjointness is unique; no extension choice arises.

Nonconstant sectors with $0 < \alpha < 1$. This regime occurs only for wide bands: $W > \pi R/2$ for the $j = 0$ odd mode. Both branches $\lvert\delta\rvert^{\pm\alpha}$ are now $L^2$ against the weight (since $1 - 2\alpha > -1$). The cone is limit-circle. The branches are distinguished by the Dirichlet energy. For the singular branch $u = \lvert\delta\rvert^{-\alpha}$,

$\int (u')^2\,\lvert f\rvert\,d\delta \sim \alpha^2\!\int \lvert\delta\rvert^{-2\alpha-1}\,d\delta = \infty \qquad \text{for all } \alpha > 0,$

so the singular branch has infinite form energy. The Friedrichs extension selects the regular branch $\lvert\delta\rvert^{+\alpha}$, and every self-adjoint extension with finite form energy coincides with it there.

The limit-point/limit-circle classification at a singular endpoint depends only on the local behavior of the coefficients [Weidmann, KSWW]. Near $p_c$, the weight $\lvert f\rvert$ vanishes linearly in $\delta$, matching the flat-cone case $\alpha_{\mathrm{BP}} = -1$ in the notation of [BP]. On the model cone of [BP] (angular period $2\pi$), only $k = 0$ has $\alpha < 1$. On the Möbius band, the unfolded period $4W/R$ produces fractional exponents, so for wide bands ($W > \pi R/2$) the first odd sector also enters the limit-circle regime; but there the singular branch has infinite form energy, so the extension ambiguity with a nonzero apex value remains confined to the constant sector. The intrinsic deficiency indices of the full minimal operator are therefore $(2,2)$ for narrow bands and $(4,4)$ for wide bands $W > \pi R/2$, where the first odd sector, with exponent $\alpha_0 = \pi R/(2W)$, is also limit-circle (at $W = \pi R/2$ exactly, $\alpha_0 = 1$ and the singular branch leaves $L^2$, so the count stays $(2,2)$). The singular odd-sector branch $\lvert\delta\rvert^{-\alpha_0}$ has infinite form energy for every $\alpha_0 \in (0,1)$, however, so it lies in the form domain of no semibounded extension; the two extra deficiency degrees of freedom parametrize only non-semibounded realizations. Among finite-energy (semibounded) self-adjoint extensions the singular branch is excluded in every nonconstant sector, so each finite-energy realization is determined by its constant-sector cone condition.

Lemma 3.2 (Vanishing of nonconstant modes at the cone). Let $\psi(y,w) = u(y)\,\Phi(w)$ be a section of $\mathcal{L}$ that is continuous on $M(W)$, where $\Phi$ is any nonconstant Neumann transverse eigenmode. Then $u(\pi R/2) = 0$.

Proof. At the cone point $p_c$, the metric coefficient $f(y) = \cos(y/R)$ vanishes, collapsing the entire transverse fiber to a single geometric point. A continuous section must take a single well-defined value at $p_c$. Since $\Phi$ is nonconstant, a product $u(\pi R/2) \cdot \Phi(w)$ with $u(\pi R/2) \ne 0$ would assign multiple values to $p_c$, contradicting continuity. $\square$

For the constant mode $\Phi_0^{\mathrm{e}} \equiv 1$, the product $u(\pi R/2) \cdot 1$ assigns a single value to $p_c$ regardless of $w$. The constant sector is the only sector where a nonzero apex value is compatible with continuity. This explains geometrically why the extension ambiguity is confined to the constant sector. Rigorously it follows from the limit-point/limit-circle classification and the infinite-form-energy exclusion of the singular branch above; continuity at $p_c$ is not the operative criterion, since it fails for the actual ground states (the discontinuous $\phi_0$ and the logarithmically divergent bridging mode).

3.5 The cone boundary form and the self-adjoint extensions

By §3.4, an extension choice arises only in the constant sector ($\mu = 0$), where both local solutions are $L^2$. Near the cone, the general constant-sector solution has the asymptotic form

u(\delta) = u_N\,\ln\frac{\lvert\delta\rvert}{2R} + u_D + o(1) \qquad (\delta \to 0), \tag{3.8}

with a log coefficient $u_N$ (the flux datum) and a regularized value $u_D$ (referenced to the length $2R$), independently on each side of the cone. We write $u_D^\pm$ and $u_N^\pm$ for the limits from $\delta \to 0^\pm$. The cone boundary form produced by Green's identity is

[u, v]_{p_c} = \frac{1}{R}\sum_{\pm}\bigl(u_D^\pm\,v_N^\pm - u_N^\pm\,v_D^\pm\bigr), \tag{3.9}

where the logarithmic divergences cancel pairwise: a direct computation from (3.7) gives $\lvert f\rvert(u v' - u' v) \to (\operatorname{sgn}\delta/R)(u_D v_N - u_N v_D)$ as $\delta \to 0$, and the two-sided Green identity sums these limits with the outward-normal orientation, yielding (3.9). A self-adjoint extension corresponds to a two-dimensional Lagrangian plane (a subspace on which (3.9) vanishes identically) in the four-dimensional boundary space $(u_D^+, u_N^+, u_D^-, u_N^-)$.

The Friedrichs (disjoint) extension. The Friedrichs extension [Friedrichs] is characterized by Theorem 1.8 of [BP] (applied at $\alpha_{\mathrm{BP}} = -1$): its domain requires

u_N^+ = 0 \qquad\text{and}\qquad u_N^- = 0, \tag{3.10}

with no condition relating $u_D^+$ to $u_D^-$. This is two conditions on four parameters, and the plane $\{u_N^+ = u_N^- = 0\}$ is Lagrangian: the bracket (3.9) vanishes when both fluxes are zero. The domain imposes no matching condition at $p_c$, so the two cone traces $u_D^\pm$ are unconstrained and the constant-sector form $q_0$ decomposes as $q_0^+ \oplus q_0^-$ on the two halves $M^\pm = \{\pm(y - \pi R/2) > 0\}$ (coupled only through the seam identification, not through $p_c$). This is the "disjoint dynamics" case $c^+ = c^- = 0$ of [BP].

The bridging extension at renormalization length $\delta_0$. Boscain and Prandi (Theorem 1.8 of [BP]) classify the conic extensions by a matrix $U \in \mathrm{SL}_2(\mathbb{R})$, and their Remark 1.9 names the $U = \mathrm{Id}$ case the bridging extension. Equipping it with a renormalization length $\delta_0 > 0$ is the standard two-dimensional point-interaction renormalization [AGHH, KayStuder], realized inside the conic framework: re-express the data relative to $\delta_0$,

u(\delta) = u_N\,\ln\frac{\lvert\delta\rvert}{\delta_0} + \tilde u_D + o(1), \qquad \tilde u_D = u_D + u_N\,\ln\frac{\delta_0}{2R}, \tag{3.11}

and imposes

\tilde u_D^+ = \tilde u_D^- \qquad \text{(continuity of the regularized value),} \tag{3.12}

u_N^+ + u_N^- = 0 \qquad \text{(Kirchhoff flux matching).} \tag{3.13}

This is two conditions on four parameters. The plane is Lagrangian: since $u_D^\pm = \tilde u_D^\pm - u_N^\pm\ln(\delta_0/2R)$, the $\ln(\delta_0/2R)$ terms cancel pairwise in (3.9), and for $u, v$ both satisfying (3.12)–(3.13),

$[u, v]_{p_c} = \frac{1}{R}\Bigl[\tilde u_D\bigl(v_N^+ + v_N^-\bigr) - \tilde v_D\bigl(u_N^+ + u_N^-\bigr)\Bigr] = 0.$

The value condition (3.12) is continuity across the cone; the flux condition (3.13) is the sign-flipped (Kirchhoff) matching dictated by the opposite outward normals on the two sides. Different choices of $\delta_0$ yield distinct self-adjoint extensions with distinct spectra; the renormalization length is a genuine parameter, set neither by the metric nor by the bundle alone.

For any constant $c \ne 0$, the discontinuous piecewise-constant section

\phi_0(y, w) = \begin{cases} +c & \text{if } y < \pi R/2, \\ -c & \text{if } y > \pi R/2, \end{cases} \tag{3.14}

has $u_N^\pm = 0$ and $\tilde u_D^+ = -c \ne +c = \tilde u_D^-$, so it lies in the Friedrichs domain (it satisfies (3.10)) but is excluded from every bridging domain (it fails (3.12)). Since $-\Delta\phi_0 = 0$ on the smooth locus and $\phi_0$ satisfies (3.1), it is a genuine eigenfunction with eigenvalue $0$ for the Friedrichs extension. The constant sector is the only sector in which it lives, by Lemma 3.2.

The covering geometry constrains, but does not by itself fix, the extension. A section of $\mathcal{L}$ that lifts to a function smooth at the pole of the covering $S^2$ is regular there ($u_N = 0$) and even in $\delta$; this forces the regular branch in the symmetric subsector (§4.2) but does not determine the cone condition in the antisymmetric subsector, where the natural candidates are the Friedrichs and bridging conditions above. The remaining choice is the antisymmetric cone condition, which is what separates Friedrichs from bridging in §5.

3.6 Friedrichs maximality

The constant-sector quadratic form

q[\psi] = \int_{M^\circ}\lvert\nabla\psi\rvert^2\,dA, \qquad M^\circ = M(W)\setminus\{p_c\}, \tag{3.15}

is nonnegative on $C_c^\infty(M^\circ)$ (twisted sections smooth and compactly supported away from $p_c$). The Friedrichs extension $\Delta_F$ is the self-adjoint operator associated with the closure of $q$; its form domain $\mathcal{D}(q_F) = \overline{C_c^\infty(M^\circ)}^{\,\lVert\cdot\rVert_q}$ is contained in the form domain of every other self-adjoint extension that is bounded below by the lower bound of $q$.

Lemma 3.3 (Friedrichs maximality). Let $A$ be any self-adjoint extension of the minimal twisted Laplacian on $M(W)$.

(i) The minimal operator is nonnegative, and $\Delta_F$ is its largest nonnegative extension in the Birman-Krein-Vishik order [AlonsoSimon]: $\mathcal{D}(q_F) \subseteq \mathcal{D}(q_A)$ for every nonnegative extension $A$, so by min-max $\lambda_n(A) \le \lambda_n(\Delta_F)$ for all $n$, and in particular $\inf\sigma(A) \le \inf\sigma(\Delta_F)$.

(ii) $\inf\sigma(\Delta_F) = 0$: the form $q \ge 0$ gives $\inf\sigma(\Delta_F) \ge 0$, and $\phi_0 \in \mathcal{D}(\Delta_F)$ with $\Delta_F\phi_0 = 0$ (Lemma 4.1) gives $\inf\sigma(\Delta_F) \le 0$.

(iii) Consequently $\inf\sigma(A) \le 0$ for every self-adjoint extension $A$: for nonnegative $A$ by (i)–(ii), and for non-nonnegative $A$ by definition. The argument uses only nonnegativity of the minimal operator, with no parity hypothesis.

Proof. (i) is the Birman-Krein-Vishik characterization of the Friedrichs extension as the largest nonnegative self-adjoint extension of a nonnegative symmetric operator [AlonsoSimon]: its form domain $\mathcal{D}(q_F) = \overline{C_c^\infty(M^\circ)}$ is contained in $\mathcal{D}(q_A)$ for every nonnegative extension $A$, and a smaller form domain yields larger Rayleigh quotients, so $\lambda_n(A) \le \lambda_n(\Delta_F)$ by min-max. (ii) The cutoff estimate of Lemma 4.1 below shows $\phi_0 \in \mathcal{D}(\Delta_F)$ with $\Delta_F\phi_0 = 0$, so $\inf\sigma(\Delta_F) \le 0$; and $q \ge 0$ gives $\inf\sigma(\Delta_F) \ge 0$. (iii) Immediate. $\square$

Lemma 3.3 proves the part of Theorem 1.1 that needs no separation of variables: every realization has bottom at most $0$, parity-mixing extensions included, since the argument uses only nonnegativity of the minimal operator.

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4. The constant sector by parity

The constant transverse mode ($k = 0$ even family, $\Phi_0^{\mathrm{e}} \equiv 1$) reduces the eigenvalue problem to a single ODE in $y$ with anti-periodic boundary conditions. This is the zonal sector, where the extension ambiguity lives. We analyze it by parity about the cone.

4.1 The reduced operator and parity reduction

On the constant-curvature band, the zonal eigenequation from §3.4 with $\mu = 0$ is

-\Delta u = -u'' + \frac{1}{R}\tan\!\Bigl(\frac{y}{R}\Bigr)u' = \lambda\,u \qquad \text{on } (0, \pi R), \tag{4.1}

with anti-periodic boundary conditions $u(\pi R) = -u(0)$ and $u'(\pi R) = -u'(0)$, and a cone condition at $y = \pi R/2$ from §3.5. Decompose the constant-sector functions into modes symmetric and antisymmetric under reflection about the cone $\delta \mapsto -\delta$. Anti-periodicity interacts with this parity as follows.

Symmetric modes satisfy $u(\pi R) = u(0)$ automatically, so the anti-periodic value condition forces $u(0) = 0$ (Dirichlet at the seam), while the derivative condition $u'(\pi R) = -u'(0)$ is then free. The symmetric sector is the half-interval problem on $(0, \pi R/2)$ with Dirichlet at the seam and a cone condition.

Antisymmetric modes satisfy $u(\pi R) = -u(0)$ automatically, so the value condition is free and the derivative condition forces $u'(0) = 0$ (Neumann at the seam). The antisymmetric sector is the half-interval problem on $(0, \pi R/2)$ with Neumann at the seam and a cone condition.

On the bridging side, a symmetric mode has equal log coefficients on the two sides, so the Kirchhoff condition (3.13) forces the log coefficient to vanish, and the symmetric mode lies in the Friedrichs domain. An antisymmetric mode has $\tilde u_D^+ = -\tilde u_D^-$, so the value condition (3.12) forces $\tilde u_D = 0$, and the log datum is active. The two subsectors decouple: the symmetric subsector gives the same spectrum for the Friedrichs and bridging extensions and is treated in §4.2; the antisymmetric subsector is where these extensions differ and is treated in §§4.3–4.4.

4.2 Legendre reduction and the symmetric tower

The substitution $x = \sin(y/R)$ maps $(0, \pi R/2)$ bijectively to $(0, 1)$ and converts the zonal operator to the Legendre equation. With $u(y) = v(\sin(y/R))$,

\frac{d}{dx}\!\Bigl[(1-x^2)\frac{dv}{dx}\Bigr] + \nu(\nu+1)\,v = 0, \qquad \lambda = \frac{\nu(\nu+1)}{R^2}, \tag{4.2}

where $\nu \ge 0$ is a continuous spectral parameter. The Legendre function of the first kind $P_\nu(x)$ is regular at $x = 1$ (the cone); the Legendre function of the second kind $Q_\nu(x)$ diverges logarithmically and supplies the log datum $u_N$. The substitution $x = \sin(y/R)$ is two-to-one ($y$ and $\pi R - y$ map to the same $x$) and produces exactly the modes symmetric about the cone.

In the symmetric subsector both the Friedrichs and bridging conditions force the regular branch (the log coefficient vanishes, by §4.1), so the admissible symmetric solutions are $P_\nu(\sin(y/R))$. The Dirichlet seam condition $u(0) = 0$ is $P_\nu(0) = 0$. Since

P_\nu(0) = \frac{\sqrt{\pi}}{\Gamma((1-\nu)/2)\,\Gamma(1+\nu/2)}, \tag{4.3}

this holds exactly when $(1-\nu)/2$ is a non-positive integer, i.e. $\nu = 1, 3, 5, \ldots$ The symmetric zonal tower is therefore

\sigma_{\mathrm{sym}} = \Bigl\{\frac{\ell(\ell+1)}{R^2} : \ell = 1, 3, 5, \ldots\Bigr\} = \Bigl\{\frac{2}{R^2}, \frac{12}{R^2}, \frac{30}{R^2}, \ldots\Bigr\}, \tag{4.4}

the same for the Friedrichs and bridging extensions. The lowest member is $\ell = 1$, with eigenfunction $P_1(\sin(y/R)) = \sin(y/R)$, the restriction of the zonal harmonic $Y_1^0 \propto \sin(y/R)$. It is strictly positive on $(0, \pi R)$, lies in the regular Frobenius branch at the cone, and gives

\frac{\ell(\ell+1)}{R^2}\Big|_{\ell=1} = \frac{2}{R^2} = R_{\mathrm{sm}}, \tag{4.5}

independent of the half-width $W$.

On the flat strip (Euclidean metric, anti-periodic condition, same period $\pi R$), the lowest eigenvalue is $1/R^2 = R_{\mathrm{sm}}/2$, with the same eigenfunction $\sin(y/R)$. The curvature term $(1/R)\tan(y/R)\,u'$ doubles this to $2/R^2 = R_{\mathrm{sm}}$, matching the familiar $\ell = 1$ eigenvalue on $S^2(R)$.

4.3 The Friedrichs antisymmetric tower and the kernel

Under the Friedrichs extension, the antisymmetric subsector (Neumann seam, regular cone) admits $\operatorname{sgn}(\delta)\,P_\nu(\sin(y/R))$ with the regular branch at the cone, and the Neumann seam condition is $P_\nu'(0) = 0$, i.e. $\nu = 0, 2, 4, \ldots$ The Friedrichs antisymmetric tower is

\sigma_{\mathrm{anti}}^{\mathrm{Fr}} = \Bigl\{\frac{\ell(\ell+1)}{R^2} : \ell = 0, 2, 4, \ldots\Bigr\} = \Bigl\{0, \frac{6}{R^2}, \frac{20}{R^2}, \ldots\Bigr\}. \tag{4.6}

The $\ell = 0$ member is the zero mode $\phi_0$ of (3.14).

Lemma 4.1 (Friedrichs kernel in the constant sector). In the constant transverse sector, the Friedrichs form decomposes as the direct sum of the two half-band forms on $M^+$ and $M^-$. Consequently $\phi_0$ of (3.14) belongs to $\mathcal{D}(\Delta_F)$ and satisfies $\Delta_F\phi_0 = 0$, and the zero eigenspace in the constant sector is one-dimensional, spanned by $\phi_0$.

Proof. Let $\chi_\varepsilon(\delta)$ be smooth, equal to $0$ for $\lvert\delta\rvert \le \varepsilon$, equal to $1$ for $\lvert\delta\rvert \ge \sqrt{\varepsilon}$, and satisfying $\lvert\chi_\varepsilon'(\delta)\rvert \le C/(\lvert\delta\rvert\,\lvert\log\varepsilon\rvert)$ for $\varepsilon < \lvert\delta\rvert < \sqrt{\varepsilon}$. Since $\lvert f(y)\rvert \sim \lvert\delta\rvert/R$ at the cone, the cutoff energy (the transverse integration contributing only the constant factor $2W$) obeys

$\int_{\varepsilon < \lvert\delta\rvert < \sqrt{\varepsilon}} \lvert\chi_\varepsilon'(\delta)\rvert^2\,\lvert f(y)\rvert\,d\delta \le \frac{C}{\lvert\log\varepsilon\rvert^2}\int_\varepsilon^{\sqrt\varepsilon}\frac{d\delta}{\delta} = O(\lvert\log\varepsilon\rvert^{-1}) \to 0.$

Thus the jump of $\phi_0$ across the collapsed point is invisible to the Friedrichs form closure: $\chi_\varepsilon\phi_0 \to \phi_0$ in the form norm, so $\phi_0 \in \mathcal{D}(q_F)$ and $q_F(\phi_0, \eta) = 0$ for every form-domain test section $\eta$, whence $\phi_0 \in \mathcal{D}(\Delta_F)$ and $\Delta_F\phi_0 = 0$. By contrast the logarithmic branch has $u'(\delta) \sim 1/\delta$, hence $\int \lvert u'\rvert^2\lvert f\rvert\,d\delta \sim \int d\delta/\lvert\delta\rvert = \infty$, and is excluded from the Friedrichs domain. Conversely, if $q_F[\psi] = 0$ in the constant sector then $\nabla\psi = 0$ a.e. on each connected half, so $\psi$ is constant on each of $M^\pm$; the anti-periodic seam condition forces the two constants to be opposite, giving a scalar multiple of $\phi_0$. $\square$

Remark 4.2 (Lune equivalence). The Friedrichs extension renders the topological twist spectrally invisible. Over the smooth locus $M^\circ$ the bundle $\mathcal{L}$ is trivial, and multiplication by $\phi_0$ (with $c = 1$) provides the trivializing section. Concretely, set $\tilde\psi = \psi$ on $\{y < \pi R/2\}$ and $\tilde\psi = -\psi$ on $\{y > \pi R/2\}$; in coordinates $t = y$ on the first half and $t = y - \pi R$ on the second (with $w' = w$ and $w' = -w$ respectively), the anti-equivariance (3.1) becomes plain continuity and $C^1$ matching of $\tilde\psi$ across $t = 0$, and the metric becomes $dt^2 + \cos^2(t/R)\,dw'^2$ on $t \in (-\pi R/2, \pi R/2)$, $w' \in [-W, W]$. Pointwise $\lvert\tilde\psi\rvert = \lvert\psi\rvert$ and $\lvert\nabla\tilde\psi\rvert = \lvert\nabla\psi\rvert$, so the $L^2$ norm and the energy form are preserved. The image is a spherical lune of opening angle $2W/R$ with the two pole corners corresponding to $p_c$; since a single point has zero capacity, the Friedrichs twisted Laplacian is unitarily equivalent to the Neumann Laplacian on the lune, and $\phi_0$ maps to the constant function. The trivialization and the zero-capacity estimate are sector-independent, so the equivalence is for the full Friedrichs operator, with the sector towers organized by longitude order. The Friedrichs spectrum is the lune Neumann spectrum, obtained by separation of variables: longitude Neumann modes of order $m_n = n\pi R/(2W)$ combined with colatitude Legendre functions regular at both poles give eigenvalues $(m_n + q)(m_n + q + 1)/R^2$ for $n, q \ge 0$. In particular, for the Friedrichs realization the first positive eigenvalue of Theorem 1.2 is the first nonzero Neumann eigenvalue of this lune: the $(n,q) = (0,1)$ and $(1,0)$ modes give $2/R^2$ and $\alpha_0(\alpha_0+1)/R^2$, crossing doubly at $W = \pi R/2$. The width transition is thus the crossing of the two lowest lune branches.

4.4 The deformed antisymmetric branch

The bridging extension deforms the tower (4.6). By §4.1, the antisymmetric subsector under bridging is the half-interval problem on $(0, \pi R/2)$ with Neumann at the seam and the regularized condition $\tilde u_D = 0$ at the cone. Let $u_\lambda$ denote the solution of (4.1) on $(0,\pi R/2)$ normalized by $u_\lambda'(0) = 0$, $u_\lambda(0) = 1$. Near the cone it has the expansion

u_\lambda(\delta) = \mathcal{N}(\lambda)\,\ln\frac{\lvert\delta\rvert}{2R} + \mathcal{D}(\lambda) + o(1), \tag{4.7}

with $\mathcal{N}(\lambda)$ the log coefficient and $\mathcal{D}(\lambda)$ the regularized value, both real-analytic. Define the secular function

G(\lambda) = -\frac{\mathcal{D}(\lambda)}{\mathcal{N}(\lambda)}. \tag{4.8}

Since the data referenced to $\delta_0$ satisfy $\tilde u_D = \mathcal{D}(\lambda) + \mathcal{N}(\lambda)\ln(\delta_0/2R)$ by (3.11), the bridging eigenvalue condition $\tilde u_D = 0$ is

G(\lambda) = \ln\frac{\delta_0}{2R}. \tag{4.9}

For $\nu > -1/2$ (equivalently $\lambda > -1/(4R^2)$), the substitution $x = \sin(y/R)$ and the Legendre closed forms give

G\bigl(\nu(\nu+1)/R^2\bigr) = -\gamma - \psi(\nu+1) - \frac{Q_\nu'(0)}{P_\nu'(0)}, \tag{4.10}

where $\gamma$ is the Euler-Mascheroni constant and $\psi = \Gamma'/\Gamma$ the digamma function (distinct from the section $\psi$). The derivative values at the origin,

$P_\nu'(0) = \frac{2\pi^{-1/2}\sin(\pi\nu/2)\,\Gamma(\tfrac{\nu}{2}+1)}{\Gamma(\tfrac{\nu}{2}+\tfrac{1}{2})}, \qquad Q_\nu'(0) = \frac{\pi^{1/2}\cos(\pi\nu/2)\,\Gamma(\tfrac{\nu}{2}+1)}{\Gamma(\tfrac{\nu}{2}+\tfrac{1}{2})}$

[DLMF, Eqs. 14.5.2 and 14.5.4], share the same Gamma quotient, which cancels in the ratio, leaving

G\bigl(\nu(\nu+1)/R^2\bigr) = -\gamma - \psi(\nu+1) - \frac{\pi}{2}\cot\frac{\pi\nu}{2}. \tag{4.11}

The cotangent has poles at the even integers $\nu = 0, 2, 4, \ldots$, which are exactly the Friedrichs antisymmetric tower (4.6); there $G$ has simple poles.

For $\lambda < -1/(4R^2)$, write $\nu = -\tfrac{1}{2} + i\kappa$ with $\kappa = (-\lambda R^2 - \tfrac{1}{4})^{1/2}$. The closed form continues real-analytically: $\cot(-\tfrac{\pi}{4} + \tfrac{i\pi\kappa}{2}) = -\operatorname{sech}(\pi\kappa) - i\tanh(\pi\kappa)$ and $\operatorname{Im}\psi(\tfrac{1}{2} + i\kappa) = \tfrac{\pi}{2}\tanh(\pi\kappa)$, and the imaginary parts cancel, giving

G(\lambda) = -\gamma - \operatorname{Re}\psi\bigl(\tfrac{1}{2} + i\kappa\bigr) + \frac{\pi}{2}\,\operatorname{sech}(\pi\kappa), \qquad \lambda \le -\frac{1}{4R^2}. \tag{4.12}

Lemma 4.3 (Monotonicity). $G$ is strictly increasing between consecutive poles; on each interval between consecutive poles it is a bijection onto $\mathbb{R}$, and on $(-\infty, 0)$ it has no poles and increases from $-\infty$ to $+\infty$.

Proof. A Green identity on $(0, \pi R/2)$ for the Neumann-normalized solutions at distinct parameters gives, after the log terms cancel in the cone bracket and the seam bracket vanishes by the shared Neumann data,

$(\lambda' - \lambda)\int_0^{\pi R/2}\lvert f\rvert\,u_\lambda u_{\lambda'}\,dy = \frac{\mathcal{N}(\lambda')\mathcal{D}(\lambda) - \mathcal{N}(\lambda)\mathcal{D}(\lambda')}{R},$

whence $G'(\lambda) = R\,\lVert u_\lambda\rVert^2_{\lvert f\rvert}/\mathcal{N}(\lambda)^2 > 0$ at points where $\mathcal{N}(\lambda) \ne 0$; at the even integers, where $\mathcal{N} = 0$, $G$ has simple poles, and monotonicity is asserted on each open interval between consecutive poles. The poles of $G$ are the even integers, which are nonnegative; so $G$ has no pole on $(-\infty, 0)$. The deep-negative asymptotic from $\psi(z) = \ln z - 1/(2z) + O(z^{-2})$ gives $G(\lambda) = -\gamma - \tfrac{1}{2}\ln(-\lambda R^2) + O((-\lambda R^2)^{-1}) \to -\infty$ as $\lambda \to -\infty$, while $G \to +\infty$ as $\lambda \to 0^-$ (the pole at $\nu = 0$). Strict monotonicity makes $G$ a bijection on $(-\infty, 0)$. $\square$

Proposition 4.4 (Bridging constant-sector spectrum). For every $\delta_0 > 0$, the bridging extension in the constant sector has exactly one negative eigenvalue

\lambda_b(\delta_0) = -\frac{4e^{-2\gamma}}{\delta_0^2}\bigl(1 + O(\delta_0^2/R^2)\bigr) \qquad (\delta_0 \to 0^+), \tag{4.13}

a cone-localized bound state independent of the half-width $W$ and strictly increasing in $\delta_0$ with $\lambda_b \to 0^-$ as $\delta_0 \to \infty$. Zero is not an eigenvalue: it lies in the resolvent set. The symmetric tower (4.4) is unchanged, and the deformed antisymmetric eigenvalues interlace the Friedrichs tower (4.6), one in each gap. Moreover, for

\delta_0 > \frac{2R}{e} \tag{4.14}

the lowest positive eigenvalue of the constant sector is $2/R^2$ and simple; at $\delta_0 = 2R/e$ it is still $2/R^2$ but doubly degenerate, the deformed antisymmetric root coinciding with the symmetric $\ell = 1$ value, and for $\delta_0 < 2R/e$ it lies strictly below $2/R^2$.

Proof. By Lemma 4.3, $G$ is a bijection of $(-\infty, 0)$ onto $\mathbb{R}$, so (4.9) has exactly one negative root for every $\delta_0$; solving the deep asymptotic against $\ln(\delta_0/2R)$ gives (4.13), and $G' > 0$ gives monotonicity in $\delta_0$ with $\lambda_b \to 0^-$ as $\delta_0 \to \infty$. At $\lambda = 0$ the Neumann-seam solution is $u_0 \equiv 1$, with $\mathcal{N}(0) = 0$ and $\mathcal{D}(0) = 1 \ne 0$, so $\tilde u_D = 1 \ne 0$ and the bridging condition excludes $\lambda = 0$. On each interval between consecutive poles (the Friedrichs tower) $G$ is a bijection onto $\mathbb{R}$, giving one deformed root per gap. Finally, at $\nu = 1$ ($\lambda = 2/R^2$), $\cot(\pi/2) = 0$, so $G(2/R^2) = -\gamma - \psi(2) = -\gamma - (1 - \gamma) = -1$; since $G$ is increasing on $(0, 6/R^2)$, the lowest positive deformed root exceeds $2/R^2$ for $\ln(\delta_0/2R) > -1$ and equals $2/R^2$ at $\ln(\delta_0/2R) = -1$, so for $\delta_0 > 2R/e$ the lowest positive eigenvalue of the sector is $2/R^2$ and simple, while at $\delta_0 = 2R/e$ it remains $2/R^2$ with the deformed root coincident. $\square$

Every deformed antisymmetric eigenfunction has $\mathcal{N} \ne 0$, hence diverges like $\mathcal{N}\ln(\lvert\delta\rvert/\delta_0)$ at the cone; in particular the bound state at $\lambda_b$ is nodeless on each open half (it is the lowest eigenfunction of the half-interval problem, hence positive there by oscillation theory [Weidmann]), with opposite signs on the two halves and a logarithmic divergence at $p_c$. It vanishes nowhere on the smooth locus. The constant $-4e^{-2\gamma}$ in (4.13) is the standard two-dimensional one-center point-interaction binding energy [AGHH], with $\delta_0$ in the role of the point-interaction renormalization length; the bridging family realizes this point interaction inside the Boscain-Prandi conic framework.

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5. Classification of the bottom of the spectrum

Theorem 5.1 (Bottom of the spectrum across realizations). Let $A$ be any self-adjoint extension of the minimal twisted Laplacian on $M(W)$, $0 < W < \pi R$. Then:

(i) $\inf\sigma(A) \le 0$, with no self-adjoint extension admitting a strictly positive ground state.

(ii) $\inf\sigma(A) = 0$ if and only if $A$ is nonnegative; the Friedrichs extension is the maximal such realization, with $\inf\sigma(\Delta_F) = 0$ attained by the discontinuous zero mode $\phi_0$.

(iii) For the bridging extension at renormalization length $\delta_0$, $\inf\sigma = \lambda_b(\delta_0) < 0$ of (4.13), a single cone-localized bound state; zero lies in the resolvent set.

For the Friedrichs and bridging extensions the symmetric zonal tower (4.4) is unchanged, and the difference between these natural realizations lives entirely in the antisymmetric subsector; a general parity-mixing (Robin-type) cone condition can instead deform the symmetric subsector, but part (i) is unaffected, as it follows from maximality independently of the parity decomposition. Parts (ii)–(iii) give the explicit bottom for the nonnegative and bridging realizations; part (i) holds for every self-adjoint extension, including the additional non-finite-energy realizations that the wide-band odd sector admits.

Proof. Part (i) is Lemma 3.3(iii), which bounds $\inf\sigma(A) \le \inf\sigma(\Delta_F) = 0$ for nonnegative $A$ and is immediate for non-nonnegative $A$; the argument is independent of the parity decomposition and so applies to parity-mixing extensions as well. Part (ii): if $A$ is nonnegative then $\inf\sigma(A) \ge 0$; combined with (i) this forces $\inf\sigma(A) = 0$, and Lemma 3.3(i) identifies Friedrichs as the maximal nonnegative realization, with the bottom attained by $\phi_0$ (Lemma 4.1). Part (iii) is Proposition 4.4. The robustness of (4.4) for the Friedrichs and bridging extensions is §4.2; a general cone condition is not claimed robust. $\square$

So the cone lets the bottom escape the positivity the twist would otherwise force, and the form of the escape is set by the extension: the Friedrichs ground state $\phi_0$ is bounded but discontinuous at $p_c$, while the bridging ground state at $\lambda_b$ is continuous in regularized value but logarithmically divergent there. Each evades the obstruction only through its singular behavior at the cone, where elliptic regularity fails. No realization raises the bottom above zero.

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6. Nonconstant sectors and the first positive eigenvalue

The first positive eigenvalue is governed by the competition between the symmetric zonal mode at $2/R^2$ and the nonconstant sectors, where all extensions agree. We record the nonconstant sector spectra and then prove Theorem 1.2.

6.1 The odd (periodic) sectors

The odd transverse modes ($\Phi_j^{\mathrm{o}}$, $j = 0, 1, 2, \ldots$) produce periodic longitudinal functions. The reduced eigenequation (3.6) with $\mu_j = \bigl((2j+1)\pi/(2W)\bigr)^2$ has, in Schrödinger form, an effective potential proportional to $\sec^2(y/R)$, a symmetric Pöschl-Teller potential.

Claim 6.1. The function $u = \lvert\cos(y/R)\rvert^{\alpha}$ with $\alpha = (2j+1)\pi R/(2W)$ satisfies (3.6) with eigenvalue $\lambda = \alpha(\alpha+1)/R^2$.

Proof. Set $r = \lvert\cos(y/R)\rvert$, so $\lvert f\rvert = r$ and $u = r^\alpha$. On the smooth locus, $R^2(r')^2 = 1 - r^2$ and $r'' = -r/R^2$. Then $\lvert f\rvert\,u' = \alpha\,r^\alpha r'$ and

$(\lvert f\rvert\,u')' = \frac{\alpha}{R^2}\,r^{\alpha-1}\bigl[\alpha(1 - r^2) - r^2\bigr] = \frac{\alpha}{R^2}\,r^{\alpha-1}\bigl[\alpha - (\alpha+1)r^2\bigr],$

so $-(\lvert f\rvert\,u')' + (\mu_j/\lvert f\rvert)\,u = r^{\alpha-1}[\alpha(\alpha+1)R^{-2}r^2 - \alpha^2 R^{-2} + \mu_j]$. With $\mu_j = \alpha^2/R^2$ the constant terms cancel, leaving $\alpha(\alpha+1)R^{-2}\,\lvert f\rvert\,u$. $\square$

Under the substitution $x = \sin(y/R)$, $\lvert\cos(y/R)\rvert^\alpha = (1-x^2)^{\alpha/2}$ is the branch regular at $x = 1$ of the associated Legendre equation of degree and order $\alpha$; for integer $\alpha$ it is proportional to the sectoral harmonic $P_\alpha^\alpha$, and at $\alpha = 1$ it reduces to $Y_1^1$ on the covering $S^2$. At both seams $\lvert\cos\rvert^\alpha = 1$ with vanishing derivative, so periodicity holds for all $\alpha$, and the spectral parameter runs continuously with $W$. Near the cone $\lvert\cos(y/R)\rvert^\alpha \sim \lvert\delta/R\rvert^\alpha \to 0$; this is the regular Frobenius branch, with finite Dirichlet energy for all $\alpha > 0$, so the Friedrichs and every finite-energy extension agree. A standard ground-state transform ($u = hv$ with $h = \lvert\cos(y/R)\rvert^\alpha$) shows $\lambda = \alpha(\alpha+1)/R^2$ is the sector ground eigenvalue, unique up to scaling. Since $\alpha_j = (2j+1)\pi R/(2W)$ is increasing in $j$, the odd-sector minimum is at $j = 0$:

\lambda_0^{\mathrm{odd}}(W) = \frac{\alpha_0(\alpha_0+1)}{R^2}, \qquad \alpha_0 = \frac{\pi R}{2W}, \qquad \psi_0^{\mathrm{odd}} = \lvert\cos(y/R)\rvert^{\alpha_0}\sin(\pi w/2W). \tag{6.1}

This is decreasing in $W$: above $2/R^2$ for $W < \pi R/2$, equal at $W = \pi R/2$, and below for $W > \pi R/2$.

6.2 The even nonconstant sectors

The even nonconstant modes ($\Phi_k^{\mathrm{e}}$, $k \ge 1$) produce anti-periodic longitudinal functions. The sector ground state is obtained by a sign flip across the cone,

u_k(y) = \operatorname{sgn}(\delta)\,\lvert\cos(y/R)\rvert^{\alpha_k}, \qquad \alpha_k = \frac{k\pi R}{W}, \tag{6.2}

which is anti-periodic at the seams and continuous at the cone (the sign flip multiplies a vanishing factor), with vanishing weighted flux and no distributional contribution; it lies in every finite-energy extension's domain. On each half, $\lvert\cos(y/R)\rvert^{\alpha_k}$ is positive and nodeless, so by the Sturm oscillation theorem for the separated half-interval problem (Neumann at the seam, regular Frobenius at the cone) it is the half-interval ground state, with eigenvalue $\alpha_k(\alpha_k+1)/R^2$. Since $\alpha_k = k\pi R/W \ge \pi R/W > 1$ for all $k \ge 1$ and $W < \pi R$, every even nonconstant sector lies strictly above $2/R^2$. The sharp $+1$ in $\alpha_k(\alpha_k+1)$ matters: a potential-only bound gives $\alpha_k^2/R^2$, which fails to clear $2/R^2$ for $\alpha_k \in (1, \sqrt{2})$, whereas $\alpha_k(\alpha_k+1)/R^2 > 2/R^2$ for all $\alpha_k > 1$.

6.3 The transition

Proof of Theorem 1.2. Because $\cos^2(y/R)$ multiplies $\psi_{ww}$ and depends on $y$ alone, the twisted Laplacian commutes with the transverse Neumann Laplacian, and $L^2(M, dA)$ decomposes as a Hilbert-space direct sum over transverse sectors; the cone conditions are imposed on the longitudinal factor in each sector, so every finite-energy extension respects the decomposition. Each reduced one-dimensional operator has compact resolvent (a finite interval, integrable weight, and a limit-circle or limit-point endpoint), and the transverse sector bottoms diverge, so the direct sum has compact resolvent and the spectrum is discrete. The first positive eigenvalue is the minimum of the first positive sector entries. By §4.2 the constant sector contributes $2/R^2$ as its first positive value, shared by the Friedrichs extension and, by Proposition 4.4, by every bridging extension with $\delta_0 > 2R/e$. By (6.1) the odd sectors contribute $\alpha_0(\alpha_0+1)/R^2$ with $\alpha_0 = \pi R/(2W)$, and by §6.2 the even nonconstant sectors lie strictly above $2/R^2$. The function $\alpha \mapsto \alpha(\alpha+1)$ is increasing, and $\alpha_0$ is decreasing in $W$, so the first positive eigenvalue is $\min(2/R^2,\,\alpha_0(\alpha_0+1)/R^2)$: it equals $2/R^2$ when $\alpha_0 \ge 1$, i.e. $W \le \pi R/2$, and $\alpha_0(\alpha_0+1)/R^2$ when $W > \pi R/2$. At $W = \pi R/2$ both branches give $2/R^2$. $\square$

Corollary 6.2 (Degeneracy at the transition). For the Friedrichs extension and for bridging with $\delta_0 > 2R/e$, at $W = \pi R/2$ the first-positive eigenspace is two-dimensional, spanned by $\psi_1 = \sin(y/R)$ and $\psi_2 = \lvert\cos(y/R)\rvert\,\sin(w/R)$; the next contributions ($\ell = 3$ zonal, $j = 1$ odd, $k = 1$ even) are $12/R^2$, $12/R^2$, $6/R^2$, all above. On the covering $S^2$, $\psi_1$ restricts from $Y_1^0$ while $\psi_2$ does not lift to a smooth function (the absolute value is required by equivariance, not a coordinate artifact). Away from the critical width the first positive eigenvalue is simple.

Corollary 6.3 (Sign-changing first positive eigenfunction). For $W > \pi R/2$ the first positive eigenfunction $\lvert\cos(y/R)\rvert^{\alpha_0}\sin(\pi w/2W)$ changes sign across $w = 0$ (the core loop); for $W < \pi R/2$ the eigenfunction $\sin(y/R)$ is positive on the interior but vanishes at the seam, where the Möbius identification imposes the sign reversal. Because $\mathcal{L}$ is nontrivial, no section admits a globally positive representative, so the Perron-Frobenius nodeless-ground argument does not apply.

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7. Discussion

This model separates two spectral effects. The first positive eigenvalue is the robust one: in the narrow regime it equals $2/R^2$ for the Friedrichs extension and for bridging with $\delta_0 > 2R/e$, with the width transition at $W = \pi R/2$. The bottom of the spectrum, by contrast, depends entirely on the extension: it is $0$ for every nonnegative realization (including the Friedrichs and Krein-von Neumann extensions [AlonsoSimon]) and strictly negative for every other, in particular the bridging family at finite $\delta_0$. So anti-periodicity does not force a positive ground state: Friedrichs maximality (Lemma 3.3) caps the bottom at $0$, and bridging replaces the zero mode by a logarithmic bound state.

The nodal pattern also changes at the transition. For $W > \pi R/2$ the first positive eigenfunction changes sign across the core loop; for $W < \pi R/2$ it is positive in the interior and vanishes at the seam, where the twist imposes the sign reversal. On an orientable surface Perron-Frobenius and Courant's nodal domain theorem [CourantHilbert] would give a positive ground eigenfunction, but on $M(W)$ a nowhere-vanishing section would trivialize $\mathcal{L}$. This is the line-bundle analogue of Shigekawa's magnetic spectral-positivity criterion [Shigekawa]; the cone goes further, letting the bottom itself fall to zero or below.

The renormalization length $\delta_0$ is the one quantity in the model that the intrinsic geometry leaves free. The Friedrichs extension needs no such scale and is the standard conic choice [Cheeger1979, Cheeger1983]; under it the topology is spectrally invisible (Remark 4.2) and the bottom is the zero mode. At $\alpha_{\mathrm{BP}} = -1$ it is moreover the unique Markovian realization (Theorem 1.13(ii) of [BP]), the bridging extensions being non-Markovian, and together with the generic decoupling of two-dimensional point interactions this makes Friedrichs the default; a transmitting limit would survive only if a smoothing of the cone were tuned to a zero-energy resonance. Whether a geometric regularization selects a canonical $\delta_0$, for instance as a norm-resolvent limit of smoothed-cone Laplacians [AGHH], remains open. Until then $\lambda_b$ is an extension artifact rather than an intrinsic invariant, and its leading form (4.13) carries $O(\delta_0^2/R^2)$ corrections that are not small at moderate $\delta_0$.

Two directions seem immediate. A Lichnerowicz-type lower bound [Lichnerowicz, Obata] adapted to $M(W)$ would give a geometric reading of the narrow-regime value $2/R^2 = R_{\mathrm{sm}}$; and the higher sector spectra, the thin-strip limit ($W \to 0$), and the flat limit ($R \to \infty$) should connect the closed forms here with the asymptotic formulas of [KKZ].

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