Localized states of the non-Hermitian Schr "o dinger equation with a complex 1/r squared potential: from Efimov states to bipolarons

arXiv:2610.21155 · cond-mat.quant-gas, nucl-th, quant-ph · Submitted 2026-10-15 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Localized states of the non-Hermitian Schr "o dinger equation with a complex 1/r squared potential".

Mira: The gist The study investigates how localized states in non-Hermitian Schrödinger equation with a complex 1/r2 potential bridge Hermitian few-body and non-Hermitian many-body physics by connecting Efimov states to bipolarons…

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're looking at this paper right now, "Localized states of the non-Hermitian Schrödinger equation with a complex one/r squared potential: from Efimov states to bipolarons <ref:2610.21155#pg1,Localized states of the non-Hermitian>." It sounds super technical, but basically, they're taking something that describes three-body physics in a vacuum—those Efimov states—and stretching it out into this new area involving two heavy polarons in some kind of medium.

Mira: Right. The title itself says it bridges Hermitian few-body and non-Hermitian many-body physics, which is the core idea here, linking those clean, real potential systems to the messy stuff that happens when you introduce dissipation or an environment.

Lev: From my side, I see them tackling a problem where a real attractive potential describes Efimov bound states and they are analytically continuing that setup into a purely imaginary one describing interactions between two heavy polarons in this medium.

Kai: Exactly. The paper is doing the math on how those Efimov states behave when you introduce an imaginary part to the potential, which is what happens when you have that medium interaction at large distances, specifically where it looks like a negative inverse square law, one/r squared <ref:2610.21155#pg1>.

Mira: And the implication is that this complex potential lets them study two-body physics in a dissipative environment without needing to model every single bath particle explicitly. They are using this complex potential to reveal how those localized states connect directly to the bipolaron states.

Lev: It’s interesting because they are not just doing a simple calculation; they’re looking at how the medium's induced interaction itself dictates the localization of these states, rather than some external force pulling them in.

Kai: So, it’s about showing that environment-induced dissipation can create localized states that look like bipolarons when you move from real to complex potentials. This whole setup is really important for how we think about quantum systems interacting with something more complicated than just a vacuum.

Mira: And the connection they make is continuous; as they increase the imaginary part of the potential, you see these Efimov state energies start getting imaginary and rotate around in the complex energy plane, which signals that shift into bipolaron behavior.

The paper's summary: Kai: To get into what the paper actually does, they set up this non-Hermitian Schrödinger equation with a complex one/r squared potential and they solve it analytically to find both the localized states and the continuum states <ref:2610.21155#pg1,equation with a complex $1>. They compare these analytical results against numerical solutions using finite-difference methods and also check them against WKB approximations.

Mira: What’s key here is how they show this connection: as the imaginary part of the potential increases, you see that those Efimov bound state energies acquire imaginary parts and rotate counterclockwise in the complex energy plane, which is a clear signature of that transition into bipolaron behavior.

Lev: From a computational standpoint, I’m interested because they use finite-difference methods and WKB approximations to corroborate their analytical expressions for the energies and wave functions of both the localized states and the continuum states. It shows they have checked their math against other ways to solve it.

Kai: They also show that when you solve this using finite differences, every bipolaron state is accompanied by another localized state that oscillates on a scale of the mesh size, which has no direct physical consequence for the continuum physics but is important for the discretization itself.

Mira: That second localized state appearing in the discretized version of Eq. (one) when you use finite differences is a bit strange because it doesn't show up in the continuum Schrödinger equation, so it’s really just reflecting a symmetry of how they set up the discrete Hamiltonian for that purely imaginary potential <ref:2610.21155#pg1>.

Lev: That symmetry they find, where possesses this symmetry and-one = - if and only if the potential is purely imaginary, that’s a really neat structural piece of information about the math they derived <ref:2610.21155#pg3>.

Kai: It helps explain why those two localized states appear symmetrically about a certain line in their energy spectrum when beta is less than beta(epsilon p), and then they collide exactly on that line at the critical value beta = beta(epsilon p).

Mira: So, they’re using this symmetry to explain an exceptional point, which is where things get really interesting in non-Hermitian systems. This isn't just any exceptional point; it’s a specific type of one that arises from this symmetry in the discretized Hamiltonian.

The paper's improvements: Kai: One major improvement they highlight is showing how this complex one/r squared potential bridges the gap between the few-body Hermitian physics and many-body non-Hermitian physics, which is a big conceptual step <ref:2610.21155#pg1>. They show how to analytically continue the Efimov states into localized bipolaron states arising from environment-induced dissipation.

Mira: That’s a powerful statement because it suggests that what we usually study as isolated three-body systems can be analytically continued into localized two-body physics governed by medium interactions, which is a direct link between vacuum physics and dissipative many-body physics.

Lev: From an engineering standpoint, this analytical continuation is useful because they provide the analytical expressions for the energies and wave functions of both the localized states and the continuum states as well as their corroboration with both WKB approximation and numerical solutions. That gives researchers a solid way to test their models against other methods.

Kai: They also point out that the real and imaginary parts of these energies are accessible, which means you could potentially access them through things like radio frequency spectroscopy or damped oscillations in Ramsey spectroscopy if you properly subtract the single-polaron contributions.

Mira: That potential to measure those real and imaginary parts is really significant because it means we aren't just looking at one side of the physics; we can probe the dissipative nature of these states directly by measuring how they shift in energy.

Lev: And regarding their limitations, they mention that the complex potential itself is derived microscopically, but they admit that the consequences for the two-polaron problem remain largely unexplored, which is a fair limitation. They are solving a specific setup defined by this potential rather than making a universal statement about all dissipative media.

Kai: So while they’ve made this strong analytical link, they’ve also clearly stated that applying these results universally to any environment is still something that needs further work and exploration.

Conclusion: Mira: So wrapping up the paper "Localized states of the non-Hermitian Schrödinger equation with a complex one/r squared potential: from Efimov states to bipolarons," they confirm that this complex one/r squared potential problem successfully bridges few-body Hermitian physics and many-body non-Hermitian physics by analytically continuing Efimov states into localized bipolaron states arising from environment-induced dissipation <ref:2610.21155#pg1,Localized states of the non-Hermitian>.

Kai: It shows that a paradigmatic example of quantum three-body physics can be analytically continued into a localized bipolaron state arising from environment-induced dissipation, which is a really neat way to connect those two worlds.

Lev: From an error correction perspective, the fact that they've derived these analytical expressions for the energies and wave functions, and corroborated them with WKB approximation and finite-difference methods, gives us a concrete benchmark for what we can expect when building real hardware.

Mira: And that potential to measure the real and imaginary parts of those energies through spectroscopy is a big part of it because it opens up a way to probe the dissipative nature of these states directly by measuring how they shift in energy.

Kai: It’s a solid piece of work, showing that environment-induced interactions can lead to localized states described in this non-Hermitian framework, and we should keep an eye on how this connects to other many-body systems.

Lev: I think the ability to precisely locate those exceptional points as the strength of the purely imaginary potential is varied is a key finding for understanding how these symmetries manifest in these types of non-Hermitian systems.

Kyoka Tsunoda, Kazuki Yamamoto, Shimpei Endo

Department of Engineering Science, University of Electro-Communications · Research Institute for Innovation and Co-Creation, Osaka Metropolitan University · Department of Physics, Osaka Metropolitan University · Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University · Institute for Advanced Science, University of Electro-Communications

cond-mat.quant-gas, nucl-th, quant-ph

Submitted: 2026-10-15

Updated: 2026-10-15

Comments: 13 pages, 8 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist The study investigates how localized states in non-Hermitian Schrödinger equation with a complex 1/r2 potential bridge Hermitian few-body and non-Hermitian many-body physics by connecting

Key concepts

Efimov States: These are special quantum states in few-body systems characterized by a discrete scale invariance where energy levels follow a specific geometric ratio. They exist when the potential is real and negative, representing fundamental three-body physics in vacuum.
Bipolarons: These are localized two-body states that emerge specifically due to the presence of an external dissipative medium. Unlike standard bound states, these bipolarons are localized solely by the imaginary interaction induced by the environment, not by a real attractive potential.
Exceptional Points (EP): In non-Hermitian systems, Exceptional Points are points where two or more eigenvalues and their corresponding eigenvectors coalesce. This paper identifies a specific EP arising from the symmetry of the discretized Hamiltonian when the potential is purely imaginary, marking a transition point in state behavior.

Summary

The gist The study investigates how localized states in non-Hermitian Schrödinger equation with a complex 1/r2 potential bridge Hermitian few-body and non-Hermitian many-body physics by connecting Efimov states to bipolarons through the medium's induced interaction.

Model Setup and Potential Behavior

The research solves the non-Hermitian Schrödinger equation with a complex parameter s squared in Eq. (1) to study two heavy polarons immersed in a quantum degenerate bosonic or fermionic medium. For real negative s squared, the equation supports Efimov states, which are characterized by discrete scale invariance where En+1/En = e-2π/s. The Hamiltonian in Eq. (1) is Hermitian for real s squared but becomes non-Hermitian once s squared acquires an imaginary part. Inverse-square interactions have also been studied in other non-Hermitian contexts, including a Calogero–Sutherland model with a complex interaction describing dissipative Tomonaga–Luttinger liquids. The imaginary part of the potential between the two heavy impurities behaves as-1/r2 at large distances in finite-temperature bosonic and fermionic media. This effective description justifies studying the low-energy (i.e., large-distance) behavior of the two heavy polarons via the dissipative potential in Eq. (1) with Im[s 2] < 0.

Connection Between Efimov States and Bipolarons

The paper demonstrates a continuous connection between few-body physics in vacuum and two-body physics in a dissipative medium through the complex potential. As the imaginary part of the potential is increased, the energies of the Efimov bound states acquire imaginary parts and rotate counterclockwise in the complex energy plane. On approaching a purely imaginary potential, excited states disappear sequentially, while the Efimov ground state is smoothly connected to a bipolaron state localized at short distance. Such a bipolaron is localized solely by the medium-induced imaginary interaction, without any attractive real potential.

Localized States in Discretized Formulation

When solving by finite differences, each bipolaron state is necessarily accompanied by another localized state that oscillates on the scale of the mesh size. These two states can form an exceptional point when the potential is purely imaginary, a consequence of a symmetry of the discretized Hamiltonian. The second localized state (green filled circle) in Fig. 3(a) does not appear in the continuum Schrödinger equation (1) and hence has no direct physical consequence for the bipolaron physics, but it reflects a fundamental symmetry of the discretized Hamiltonian for the purely imaginary potential.

Symmetry and Exceptional Points

The second localized state's behavior is governed by a symmetry of the discretized Hamiltonian, where Cˆ possesses the symmetry (ST) Cˆ (ST)-1 = -C if and only if the potential is purely imaginary. This symmetry implies that iCˆ commutes with ST: (ST)iCˆ (ST)-1 = iCˆ, leading to energy eigenvalues of the discretized Hamiltonian Hˆ either lying on the line Re[2µE] = 2/h squared, or forming pairs symmetric about this line when the ST symmetry is spontaneously broken. The two localized states appear symmetrically about this line for β < β(ep), and they collide on the line Re[2µE] = 2/h squared at the critical value β = β(ep) (see Appendix D). This symmetry-enabled exceptional point is of a different origin from those recently reported in Ref. [98], which occur between a resonance and the rotated continuum.

Conclusion

The complex 1/r2 potential problem bridges few-body Hermitian physics and many-body nonHermitian physics by analytically continuing Efimov states into localized bipolaron states arising from environment-induced dissipation. This behavior demonstrates that a paradigmatic example of quantum three-body physics is analytically continued in the complex plane into a localized bipolaron state arising from environment-induced dissipation. The real and imaginary parts of its energy would be accessible from RF spectroscopy or damped oscillations in Ramsey spectroscopy by properly subtracting singlepolaron contributions.

Acknowledgments

The authors thank Yukinao Akamatsu, Yasuyuki Hatsuda, Masaru Hongo, Tetsuo Hyodo, Masaya Nakagawa, Shoya Ogawa, Hiroyuki Tajima, and Shion Yamashika for fruitful discussions. SE acknowledges support from Matsuo Foundation and JSPS KAKENHI Grant Numbers JP23H01174 and JP25K00217. KY acknowledges support from JSPS Pro-gram for Forming Japan’s Peak Research Universities (J-PEAKS) Grant No. JPJS00420230008, JSPS KAKENHI Grant No. JP25K17327, Hirose Foundation, Fujikura Foundation, Toyota Riken Scholar Program, and Support Center for Advanced Telecommunications Technology Research. KY used Claude Opus 5 and 5.5 (Anthropic) to polish the English, and Claude Fable 5 (Anthropic) to suggest a candidate symmetry of the matrix in Eq. (9), which we then proved analytically. The authors have carefully checked all the AI-assisted outputs.

Appendix A

The derivation of analytical results involves transforming Eq. (1) into the Bessel differential equation and using asymptotic forms of Hankel functions to obtain the analytical expression for complex energies as E = 1/2µ squared rmin squared exp (πi + 1/s ln-Γ(s) Γ(-s)) exp (-2πi/s n). The analytical expression for the continuum state energy is obtained from Eq. (6) which is E = 1/2µ squared π/4 rmax squared (4n - 2s + 3) squared.

Appendix B

The WKB approximation yields energy expressions for localized states as E = 1/2µ squared rmin squared exp (-2n + 3/2 πi s - 2 + 2 ln (s)). The continuum state energy obtained by the WKB approximation exactly agrees with Eq. (6).

Appendix C

The eigenvalues of iCˆ are either real or appear as complex conjugate pairs, following the argument for PT-symmetric Hamiltonians in Ref. [95]. The critical strengths β(ep)n are found to coalesce at an exceptional point at the corresponding critical strength β(ep) n for each n.

Improvements for AI systems

textbf Stronger few-body/many-body bridge capability for quantum simulation. The system can analytically continue Efimov bound states (real potential) into localized bipolaron states arising from medium-induced imaginary interactions, providing a direct link between few-body Hermitian physics and many-body nonHermitian physics.
textbf Precise prediction of dissipation effects in heavy polaron systems. The model allows for the study of how the imaginary part of the potential between the two heavy impurities behaves as −1/r2 at large distances in finite-temperature bosonic and fermionic media, enabling predictions for the decay rate of the two-polaron state due to collisions with the bath particles.
textbf Identification and handling of exceptional points in non-Hermitian systems. The system can identify when each bipolaron state is necessarily accompanied by another localized state that oscillates on the scale of the mesh size and hence has no counterpart in the continuum limit and precisely locate where these states coalesce at an exceptional point as the strength of the purely imaginary potential is varied.
textbf Enhanced spectral analysis for complex potentials. The system can predict how the energies of the Efimov bound states acquire imaginary parts and rotate counterclockwise in the complex energy plane, allowing researchers to map out these rotational dynamics as a function of increasing Im[s 2].
textbf Discretized Hamiltonian analysis for emergent symmetries. The system can analyze the discretized Hamiltonian to show that this behavior originates from a symmetry of the discretized Schr¨odinger equation that holds for any purely imaginary potential, which is crucial for understanding the origin of exceptional points in non-Hermitian systems.