Generalized Levinson's Theorem

• Generalized Levinson's theorem is a family of extensions of classical Levinson results, connecting zeros of L-functions to phase shifts in scattering theory.
• It employs advanced techniques such as mollification, winding numbers, and index theory to address variable resonance behaviors and threshold corrections.
• The theorem has practical implications in counting bound states, analyzing resonance multiplicity, and linking spectral data with topological invariants.

In the literature, “generalized Levinson’s theorem” does not denote a single result. It names several extensions of distinct classical Levinson theorems: the number-theoretic theorem on zeros of the Riemann zeta function, the scattering-theoretic theorem relating phase variation to bound states, Levinson’s log-log theorem in complex analysis, and the asymptotic theorem for perturbed differential systems. The common pattern is an endpoint, winding, or asymptotic identity that survives after additional structure—threshold resonances, varying multiplicity, longer mollifiers, piecewise-constant delay, medium effects, or non-selfadjointness—is introduced (Ray, 8 Nov 2025, Richard, 2015, Logunov et al., 2020, Castillo et al., 2014).

1. Polysemy and classical templates

In analytic number theory, Levinson’s theorem is the statement that at least one-third of the non-trivial zeros of $\zeta(s)$ lie on the critical line, obtained by a mollified second moment argument. In the report “Levinson’s theorem and its generalization for Dirichlet $L$-functions,” the basic quantity is

$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$

and Levinson’s method yields

$$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$

leading to $\kappa\ge \frac13$ for $\zeta(s)$ (Ray, 8 Nov 2025).

In scattering theory, the classical theorem relates phase shift data to the number of bound states. The topological exposition “Levinson’s theorem: an index theorem in scattering theory” formulates the modern version as

$$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$

so that winding data from scattering and threshold operators map to the $K_0$-class of the bound-state projection (Richard, 2015). Closely related formulations use

$\delta(0)-\delta(\infty)=n\pi$

or winding of $\det S(\lambda)$, with threshold corrections when resonances occur (Wergieluk et al., 2012).

A different classical Levinson theorem is Levinson’s log-log theorem for holomorphic functions. Its elliptic analogue replaces subharmonicity by three balls inequalities and retains the borderline condition

$L$0

now for solutions of elliptic equations majorized by $L$1 (Logunov et al., 2020). Yet another classical source is Levinson’s asymptotic theorem for perturbed ODEs; its DEPCAG adaptation preserves the conclusion

$L$2

for differential equations with piecewise constant argument generalized (Castillo et al., 2014).

2. Analytic number theory: from $L$3 to Dirichlet $L$4-functions

In the number-theoretic usage, the generalized Levinson theorem extends the critical-line proportion problem from $L$5 to Dirichlet $L$6-functions. For primitive $L$7, with

$L$8

$L$9

$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$0

the report states the exact theorem

$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$1

for sufficiently large $\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$2 with $\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$3, where

$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$4

This is presented as Wu’s 2018 extension of Levinson’s method, and as a Dirichlet-$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$5-function analogue of Conrey’s two-fifths theorem for $\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$6 (Ray, 8 Nov 2025).

The auxiliary Levinson function is

$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$7

and the decisive mollified second moment is

$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$8

The key inequality is

$\kappa=\liminf_{T\to\infty}\frac{N_0(T)}{N(T)},$9

If $$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$0 is linear, the same framework yields a lower bound for $$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$1, because linear combinations involving $$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$2 discriminate simple zeros from repeated ones (Ray, 8 Nov 2025).

The generalization is not purely formal. The Dirichlet case introduces character twists, the completed $$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$3-function and its functional equation,

$$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$4

the principal character $$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$5, Gauss sums, and a twisted mean square problem. The gain beyond Levinson’s original $$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$6 comes from a longer mollifier $$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$7 with

$$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$8

which is the same Conrey-type length that underlies the two-fifths phenomenon (Ray, 8 Nov 2025).

3. Selfadjoint scattering: threshold corrections, spectral shift, and resonance multiplicity

In scattering theory, generalized Levinson theorems are usually threshold-corrected identities. On quantum star graphs with Kirchhoff coupling, the central object is the perturbation determinant

$$\kappa \ge 1-\frac1R\log\!\left(\frac1T\int_1^T |V\psi(\sigma_0+it)|^2\,dt\right), \qquad \sigma_0=\frac12-\frac{R}{\log T},$$9

and the paper proves the low-energy asymptotic

$\kappa\ge \frac13$0

where $\kappa\ge \frac13$1 is the multiplicity of the zero-energy resonance. The resulting Levinson-type theorem is

$\kappa\ge \frac13$2

so the threshold correction is $\kappa\ge \frac13$3, not the usual scalar half-bound-state term (Demirel, 2012).

For two-dimensional Schrödinger operators, the threshold structure is more singular. The topological theorem

$\kappa\ge \frac13$4

shows that each $\kappa\ge \frac13$5-resonance contributes $\kappa\ge \frac13$6, while an $\kappa\ge \frac13$7-wave resonance contributes nothing. The same paper derives

$\kappa\ge \frac13$8

so the zero-energy value of the spectral shift function includes bound states and $\kappa\ge \frac13$9-resonances, but not $\zeta(s)$0-resonances (Alexander et al., 2023).

For potentials with critical inverse-square decay,

$\zeta(s)$1

the threshold correction is no longer a fixed half-integer. The generalized residue at $\zeta(s)$2 is

$\zeta(s)$3

and the final Levinson identity is

$\zeta(s)$4

Here the classical half-bound-state term is replaced by a weighted resonance sum $\zeta(s)$5 (Jia et al., 2010).

4. Topological and index-theoretic formulations

A major line of generalization recasts Levinson’s theorem as an index theorem for wave operators. In the $\zeta(s)$6-algebraic framework of Richard, the abstract identity

$\zeta(s)$7

states that the $\zeta(s)$8-class of the quotient image of the wave operator maps to minus the $\zeta(s)$9-class of the bound-state projection. In concrete models this becomes a winding-number theorem, with threshold corrections appearing as additional boundary components of the quotient image rather than as ad hoc terms (Richard, 2015).

For generic Euclidean Schrödinger scattering with $$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$0, the wave operator has the universal form

$$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$1

and the generalized Levinson theorem is the index pairing

$$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$2

Here the scattering operator determines the number of bound states through the $$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$3-theory class of $$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$4 and the $$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$5-homology class of the generator of dilations (Alexander et al., 2023).

The spectral-flow formulation pushes this further. For loops of unitaries $$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$6 with $$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$7, the paper “Analytic spectral flow formula for unitaries and Levinson’s theorem” proves

$$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$8

for $$\operatorname{ind}([q(W_-)]_1)=-[E_{\mathrm p}(H)]_0,$$9, and equivalent regularized formulas with $K_0$0. Applied to scattering matrices,

$K_0$1

so Levinson’s theorem becomes an equality among spectral flow, regularized winding number, regularized determinant integral, and bound-state count (Alexander et al., 24 Apr 2026).

The same topological strategy survives even when the scattering matrix is only piecewise continuous and changes size across thresholds. For a family of discrete Schrödinger operators with embedded thresholds and varying multiplicity, the quotient image of the wave operator lives on an “upside down comb” with $K_0$2 teeth, and the numerical formula becomes

$K_0$3

This is a topological Levinson theorem in the presence of embedded thresholds, resonances, and discontinuities of the scattering matrix (Austen et al., 2024).

5. Discrete, matrix, and finite-dimensional scattering models

For the selfadjoint matrix Schrödinger operator on the half-line with the general selfadjoint boundary condition

$K_0$4

Levinson’s theorem is expressed through the Jost matrix $K_0$5 and the matrix scattering operator

$K_0$6

If $K_0$7 is the total number of bound states, $K_0$8 is the multiplicity of the eigenvalue $K_0$9 of $\delta(0)-\delta(\infty)=n\pi$0, and $\delta(0)-\delta(\infty)=n\pi$1 count mixed and Neumann boundary channels, then

$\delta(0)-\delta(\infty)=n\pi$2

This is a multichannel half-line generalization with both threshold and boundary-condition corrections (Aktosun et al., 2012).

For discrete graph scattering with one lead, the reflection coefficient has the analytic continuation

$\delta(0)-\delta(\infty)=n\pi$3

and the theorem reads

$\delta(0)-\delta(\infty)=n\pi$4

Thus ordinary bound states, confined bound states, and half-bound states all enter the count, with $\delta(0)-\delta(\infty)=n\pi$5 as the threshold correction (Childs et al., 2011). For $\delta(0)-\delta(\infty)=n\pi$6 leads and an $\delta(0)-\delta(\infty)=n\pi$7 scattering matrix $\delta(0)-\delta(\infty)=n\pi$8, the exact multichannel analogue is

$\delta(0)-\delta(\infty)=n\pi$9

together with a completeness theorem for scattering states plus bound states (Childs et al., 2012).

A different discrete reformulation appears on finite momentum grids evolved by the Similarity Renormalization Group. There the paper uses the grid version

$\det S(\lambda)$0

but shows that an isospectral energy-shift definition satisfies Levinson’s theorem only after a correct ordering of the diagonal eigenvalues in the infrared SRG limit. The Wilson generator induces an ascending ordering incompatible with Levinson’s theorem in the presence of bound states, while the Wegner generator gives a much better ordering by decoupling the bound state at an interior momentum scale (Arriola et al., 2014).

6. Other generalized frameworks

In hot PNJL quark matter, Levinson’s theorem functions as a consistency condition for the generalized Beth-Uhlenbeck representation. The total mesonic phase shift $\det S(\lambda)$1 must satisfy

$\det S(\lambda)$2

with $\det S(\lambda)$3 the number of bound states below the in-medium threshold $\det S(\lambda)$4. The paper shows that the resonant phase shift $\det S(\lambda)$5 alone violates this identity, so the continuum scattering contribution $\det S(\lambda)$6 is indispensable for the thermodynamics of pion dissociation (Wergieluk et al., 2012).

The elliptic adaptation of Levinson’s log-log theorem replaces holomorphic methods by propagation of smallness. If $\det S(\lambda)$7 in $\det S(\lambda)$8, $\det S(\lambda)$9, and

$L$00

then, under a suitable three balls inequality or its “wild set” version,

$L$01

for every compact $L$02. This preserves the classical log-log threshold in an elliptic PDE setting (Logunov et al., 2020).

For DEPCAGs, the generalized Levinson theorem is an asymptotic-mode existence result. Under the scalar-mode relation

$L$03

projection estimates relative to $L$04, and the weighted smallness condition

$L$05

there exists a solution of the perturbed DEPCAG such that

$L$06

The proof is by a Banach fixed point argument based on an adapted variation-of-constants formula (Castillo et al., 2014).

In dissipative three-dimensional Schrödinger scattering, the theorem counts “asymptotically disappearing states.” For

$L$07

with sufficiently strong decay and no positive-energy spectral singularities,

$L$08

Here the counted states coincide with the algebraic multiplicity of the discrete complex spectrum (Alexander et al., 16 Sep 2025).

For particle form factors, the theorem becomes a statement about the asymptotic phase on the upper edge of the time-like cut. If

$L$09

then the contour argument yields

$L$10

For hadronic form factors $L$11 with asymptotic power law $L$12, the final identity is

$L$13

This transfers Levinson’s endpoint-phase logic from scattering amplitudes to analytic form factors (Rosini et al., 10 Apr 2026).

Across these literatures, the phrase “generalized Levinson’s theorem” designates a family of extensions rather than a unique theorem. This suggests a durable template: endpoint phase, winding, spectral-shift, or asymptotic data continue to encode discrete information—critical-line zeros, bound states, resonance multiplicities, Chern numbers, disappearing states, or analytic zero counts—even after the classical setting is replaced by Dirichlet $L$14-functions, singular thresholds, variable channel multiplicity, non-selfadjoint dynamics, elliptic propagation-of-smallness, or hybrid differential-functional equations.