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arXiv:2605.28756

Critique of Breit-Wigner resonance scattering

Philip D. Mannheim

Published May 27, 2026• 0 citations
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Abstract

In the standard Breit-Wigner approach to scattering the phase shift is to have a form $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ at a real energy resonance. This leads to complex energy poles in the scattering amplitude at $E_{\rm BW}=E_1-iΓ_1$, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting $\tanδ_{\rm BW} =Γ_1/(E_1-E)$ is not always a good description of the real energy scattering amplitude, that $Γ_1$ can be negative, that $E_{\rm BW}$ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear $PT$ symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs $E_{\mp}=E_2\mp i Γ_2$ with $E_- \neq E_{\rm BW}$, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.

DOI
10.48550/arxiv.2605.28756

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In the standard Breit-Wigner approach to scattering the phase shift is to have a form tan⁡δBW=Γ1/(E1−E)\tanδ_{\rm BW} =Γ_1/(E_1-E)tanδBW​=Γ1​/(E1​−E) at a real energy resonance. This leads to complex energy poles in the scattering amplitude at EBW=E1−iΓ1E_{\rm BW}=E_1-iΓ_1EBW​=E1​−iΓ1​, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting tan⁡δBW=Γ1/(E1−E)\tanδ_{\rm BW} =Γ_1/(E_1-E)tanδBW​=Γ1​/(E1​−E) is not always a good description of the real energy scattering amplitude, that Γ1Γ_1Γ1​ can be negative, that EBWE_{\rm BW}EBW​ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear PTPTPT symmetry solutions to the square well Schrödinger equation appear in complex conjugate energy pairs E∓=E2∓iΓ2E_{\mp}=E_2\mp i Γ_2E∓​=E2​∓iΓ2​ with E−≠EBWE_- \neq E_{\rm BW}E−​=EBW​, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.
We reexamine the problem of particle decay within the Hamiltonian formalism. By deforming contours of integration, the survival amplitude is expressed as a sum of purely exponential contributions arising from the simple poles of the resolvent on the second sheet plus a background integral along a complex contour $\ensuremath{\Gamma}$ running below the location of the poles. We observe that the time dependence of the survival amplitude in the small-time region is strongly correlated to the asymptotic behavior of the energy spectrum of the system; we compute the small-time behavior of the survival amplitude for a wide variety of asymptotic behaviors. In the special case of the Lee model, using a formal procedure of analytic continuation, we show that a complete set of complex energy eigenvectors of the Hamiltonian can be associated with the poles of the resolvent and the background contour $\ensuremath{\Gamma}$. These poles and points along $\ensuremath{\Gamma}$ correspond to the discrete and the continuum states, respectively. In this context, each unstable particle is associated with a well-defined object, which is a discrete generalized eigenstate of the Hamiltonian having a complex eigenvalue, with its real and negative imaginary parts being the mass and half-width of the particle, respectively. Finally, we briefly discuss the analytic continuation of the scattering amplitude within this generalized scheme, and note the appearance of "redundant poles" which do not correspond to discrete solutions of the modified eigenvalue problem.
The Hermitian properties of the Hamiltonian are related not only to the operator itself but also to the functions on which it acts. Hermitian Hamiltonians operate on functions in the L 2 Hilbert space which correspond to boundary conditions which vanish at infinity. In this chapter, in order to move into the non-Hermitian domain, we will impose on the solutions to the time-independent Schrödinger equation (TISE) different boundary conditions which lead to solutions which can be associated with different types of the complex poles of the scattering matrix. These solutions will contain information which was not available within the scope of functions in L 2 . By imposing outgoing boundary conditions on the eigenfunctions of the timeindependent Hamiltonian complex eigenvalues, E res = ε − (i/2)Г, are obtained. These complex energies are associated with decaying resonance states which were discussed in the previous two chapters. The bound states (if they exist) appear as real eigenvalues since they result from exactly such outgoing boundary conditions which appear under the threshold energy. When incoming boundary conditions are imposed two kind of solution are obtained. One type of solution is the complex conjugates of the decay resonance solutions mentioned above. In scattering theory text books (see Taylor for example) the physical resonance solutions are associated with the poles of the scattering matrix which are embedded in the lower half of the complex energy plane. However, in nuclear physics the complex poles embedded in the upper half of the complex energy plane, so-called virtual states , are denoted as capture resonances .
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