Singular wave functions and quantum collapse in inverse-power potentials

arXiv:2610.21044 · quant-ph, math-ph, math.MP, physics.ed-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: "Singular wave functions and quantum collapse in inverse-power potentials".

Mira: The gist: This work aims to produce singular stationary wave functions for inverse-power potentials, revealing that bound states in these potentials can exhibit a continuous energy spectrum,

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

Title and authors: Kai: We’ve talked about how inverse-power potentials cause quantum collapse and how those singular wave functions behave at zero radius, and now we're getting into what the paper actually shows about them.

Mira: Yeah, this paper shows that for these specific potentials U r equals minus U zero over two r to the power of n, you can analytically produce stationary wave functions right at zero radius.

Lev: That means we’re talking about something that has an infinite number of spatial oscillations right there at the origin, which is weird because usually we expect something smooth or well-behaved there.

Kai: Right, and they found that these singular states have this continuous energy spectrum, E less than zero. It’s not the usual discrete set of energies you see in normal bound states.

Mira: Exactly, and they tie this continuous spectrum directly to a boundary condition at r equals zero that isn't actually part of the main macroscopic model we usually use.

Lev: That boundary condition is what makes things tricky; it’s like an extra rule you have to impose on the system right at the smallest scale.

Kai: And because that rule isn't generic, if you don’t specify it, you get this continuous spectrum instead of the discrete ones we're used to.

Mira: They then introduce a way to handle this issue by truncating the potential inside some radius r less than r0 where it becomes infinite.

Lev: That truncation is key because when they do that, they manage to bring back the missing ground state and turn that continuous spectrum into something discrete again.

Kai: The paper suggests you impose a Dirichlet boundary condition there, meaning the wave function has to be zero at or below that radius r0, which forces the spectrum to become discrete.

Mira: That’s a concrete way they show how a microscopic physical realization—like particles hitting an electric charge—can dictate the energy structure of a larger system.

Lev: It makes sense from an error-correction standpoint; you can’t run that kind of continuous spectrum on real hardware, so imposing that truncation is like adding necessary constraints to make the math work.

Kai: So, what this means for us is that when we model systems with these singular potentials, we have to be really careful about how we define the boundary conditions at the origin because it completely changes whether you get a discrete or continuous spectrum.

Mira: It points toward needing a more detailed look at the mathematical structure of these Hamiltonian systems and how self-adjoint extensions play into this whole picture.

Lev: Yeah, that’s where the real physics is; understanding those extensions tells us exactly what kind of boundary conditions are physically allowed when dealing with things that cause quantum collapse.

Kai: It’s a deep dive into how microscopic details at r equals zero affect the overall energy levels of the system.

Mira: And this leads us to think about how these kinds of boundary effects show up in other physical systems, like those involving vorticity in two dimensions.

Lev: Because that 2D version introduces an integer vorticity S, and when S gets big enough, it actually makes the bound state behave very similarly to a simpler case without that extra twist <ref:2610.21044#pg3>.

Kai: So we’re seeing how these mathematical constraints translate into physical behavior when you move from one dimension to two dimensions.

Mira: Exactly; it’s about connecting those abstract mathematical properties of the wave functions to actual spatial confinement and size, like the effective radius of the bound state.

The paper's summary: Kai: We just talked about how they use those singular wave functions to map out where that continuous spectrum comes from in microscopic systems, and now we're looking at how they suggest fixing the problem.

Mira: Right, so the main improvement they point to is using a hard-core truncation inside a radius r less than r0 where the potential becomes infinite.

Lev: That truncation is what actually restores the missing ground state energy, so it replaces that continuous spectrum with something discrete for the bound states.

Kai: It’s like adding a physical wall to stop things from collapsing infinitely at zero radius, which is what you need to get a stable system.

Mira: They show that this truncation forces a Dirichlet boundary condition where the wave function is zero at or below r0, and that makes the spectrum discrete.

Lev: From an error-correction standpoint, it’s cleaner because you can work with those discrete energy levels instead of dealing with a continuous range of values.

Kai: It’s pretty cool because they also looked at the 2D case with vorticity S, and they found that if S gets bigger, the bound state starts looking like one from a simpler equation without that extra twist <ref:2610.21044#pg3>.

Mira: That connection between the large S states and zero-vorticity states suggests a way to simplify calculations or models when you’re dealing with higher angular momentum in those 2D systems <ref:2610.21044#pg3>.

Lev: If we're building hardware, that means for very high vorticity, we might be able to use simpler approximations for the radial equation instead of solving the full complex one every time.

Kai: And they also noted that as r0 goes down to zero, the ground state energy just heads toward negative infinity when quantum collapse is active.

Mira: That’s a major caveat; even with that hard-core cutoff, if you let r0 approach zero too fast, the ground state energy still blows up to negative infinity during collapse.

Lev: So the limitation here is that this regularization method only works well if you keep a finite r0, and there’s no easy way to get a discrete spectrum right at r equals zero without that physical cutoff.

Kai: It really shows that the nature of these singular states depends entirely on whether you assume an infinite or finite boundary condition at the origin.

Mira: This paper makes it clear that imposing a physically motivated boundary condition, like from surface interactions, is essential to moving from a problematic continuous spectrum to something useful for real-world quantum mechanics.

The paper's improvements: Kai: So we’re wrapping up on "Singular wave functions and quantum collapse in inverse-power potentials," and the main thing is that understanding how boundary conditions affect those singular states is crucial for modeling these unstable systems.

Mira: Yeah, basically, this paper shows us that when you have inverse-power potentials leading to quantum collapse, the resulting continuous energy spectrum isn't generic; it’s tied to a very specific boundary condition at the origin.

Lev: And they give us a practical way out by showing that truncating the potential with a hard core radius r0 makes things discrete again, which is what we need for any kind of error-correction simulation.

Kai: It changes how we think about these potentials because it highlights that those microscopic details at zero radius are what dictate whether the system has stable, discrete energy levels or not.

Mira: That’s a big implication for condensed matter physics, showing how surface interactions can fundamentally alter the spectrum of a bulk material.

Lev: For hardware design, it means we have to be extremely careful about what kind of boundary conditions we impose on our mathematical models if we want those calculations to represent something physically realizable.

Kai: It’s a reminder that the math has to match the physical reality of how particles actually interact at very small scales.

Mira: So, while they don't solve everything, they give us a solid framework for analyzing these singular behaviors and their connection to boundary constraints.

Lev: We still need more work on those self-adjoint extensions because that’s where the deep mathematical structure of the Hamiltonian really lies.

Kai: Definitely, it’s a lot to unpack, but this paper gives us some clear direction on how to approach these complex non-standard potentials.

Mira: Next up, we're going to look at some work on magnons in magnetic thin films and how those coupling effects play out in real materials.

Lev: Stay tuned for that, because the physics there is just as interesting when you start looking at things like the magnon Kerr effect.

Conclusion: Kai: So we’re wrapping up on "Singular wave functions and quantum collapse in inverse-power potentials," and the main thing is that understanding how boundary conditions affect those singular states is crucial for modeling these unstable systems.

Mira: That’s a big implication for condensed matter physics, showing how surface interactions can fundamentally alter the spectrum of a bulk material.

Kai: It’s a reminder that the math has to match the physical reality of how particles actually interact at very small scales.

Mira: So, while they don't solve everything, they give us a solid framework for analyzing these singular behaviors and their connection to boundary constraints.

Lev: We still need more work on those self-adjoint extensions because that’s where the deep mathematical structure of the Hamiltonian really lies.

Kai: Definitely, it’s a lot to unpack, but this paper gives us some clear direction on how to approach these complex non-standard potentials.

Mira: Next up, we're going to look at some work on magnons in magnetic thin films and how those coupling effects play out in real materials.

Lev: Stay tuned for that, because the physics there is just as interesting when you start looking at things like the magnon Kerr effect.

Hidetsugu Sakaguchi, Boris A. Malomed

Tel Aviv University · Instituto de Alta Investigación, Universidad de Tarapacá

quant-ph, math-ph, math.MP, physics.ed-ph

Submitted: 2026-10-15

Updated: 2026-10-15

Comments: 19 pages, 7 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: The gist: This work aims to produce singular stationary wave functions for inverse-power potentials, revealing that bound states in these potentials can exhibit a continuous energy spectrum, which is

Key concepts

Quantum Collapse (QC): This phenomenon occurs when using an inverse-power potential U(r) = −U0/(2r^n). In this scenario, the standard ground state is absent, but bound states at finite energies can still be physically relevant. The onset of collapse is particularly explosive in 2D systems where the critical radius for collapse is zero.
Continuous Energy Spectrum: Unlike typical quantum systems with discrete energy levels, these singular bound states possess a continuous spectrum of negative energies (E < 0). This continuous nature arises from the specific boundary conditions imposed at the origin (r → 0), which are not standard in macroscopic models but are crucial for describing these microscopic potentials.
Hard-Core Regularization: To handle the singular nature of the potential, researchers truncate it by setting U(r < r0) = +∞ within a small radius r0. This truncation effectively restores the missing ground state and fundamentally changes the spectrum: it replaces the continuous energy levels with a discrete set, enforced by a Dirichlet boundary condition u(r) = 0 at r ≤ r0.

Summary

The gist: This work aims to produce singular stationary wave functions for inverse-power potentials, revealing that bound states in these potentials can exhibit a continuous energy spectrum, which is related to boundary conditions at the origin.

Quantum Collapse and Potential Types

Quantum collapse (QC) occurs under the action of the inverse-power potential U(r) = −U0/(2r n) in the framework of the Schr¨odinger equation The ground state (GS) is missing in the presence of QC, but available bound states at finite energies may be physically relevant as they are normalizable The onset of the collapse is even more explosive in the 2D version of Eq. (2), which has U0 cr = 0 thus leading to QC at any U0 > 0 Another aspect of the QC is that it breaks the rigorously self-adjoint character of the quantum setup.

Stationary Wave Functions and Asymptotics

The stationary wave function Ψ (r, t) = exp (−iEt) u(r), is determined by the radial equation Eu = − 1/2 d 2u/dr squared + D − 1/r du/dr + U0 r n u The analytical findings represent an asymptotic form of the stationary wave functions at r → 0, which fill a continuous spectrum, and some exact solutions for the wave functions, obtained under appropriate conditions Generic solutions are obtained in the numerical form.

Physical Relevance of Singular States

The singular wave functions displayed by analytical and numerically found singular wave functions display a convergent norm and accumulation of infinitely many spatial phase oscillations at r → 0 A noteworthy feature of the family of the bound states is that their energy eigenvalues form a continuous spectrum, E < 0, on the contrary to the usual discrete spectra of quantum-mechanical bound states. This continuous spectrum is related to the boundary condition at r = 0, which is not a generic ingredient of the macroscopic model under consideration, but is specified by its microscopic physical realization at very small values of r In particular, in the case of n ≥ 2 in Eq. (7), these features include the normalizability of the wave function and the fact that they form a quasi-continuous energy spectrum (“quasi” in the sense that the precise spectrum may exhibit discreteness with a small energy interval ∆E, determined by the microscopic boundary condition added to the macroscopic model at very small finite values of r).

Regularization and Truncation Effects

A natural possibility to regularize the singular potential (7), by truncating it inside a hard-core 2D or 3D area, r < r0 (setting U(r < r0) = +∞), is considered in Section 5 It is demonstrated that the truncated potential restores the missing GS, and replaces the continuous spectrum of the bound states by its discrete counterpart. The truncation of the singular potential by the hard core at r = r0 is imposed by the Dirichlet boundary condition u(r) = 0 at r ≤ r0, which makes the continuous spectrum discrete.

Vorticity and Compactness in 2D

In the 2D version of Eq. (7), one can introduce the stationary wave function with integer vorticity S (alias the magnetic quantum number), as u(r) exp (−iEt + iSθ), where θ is the angular coordinate, and the real radial wave function u(r) obeys Eq. (29) Note that in the case of n > 2 and S ≫ 1, the effective radial potential in Eq. (29), Ueff(r) = (1/2) (S 2/r squared − U0/rn), features a sharp local maximum at small r, rmax = nU0 2S squared / (n-2). This fact implies that, for sufficiently large S, the bound states are similar to those produced by the radial equation (8) with S = 0, subject to the boundary condition u (r = rmax) = 0.

Ground State Energy Behavior

For the family of the GS solutions, Fig. 6(b) produces EGS as a function of r0 Naturally, EGS(r0 → 0) → −∞ when the quantum collapse occurs, and it is not clear whether we can get a discrete energy spectrum even for the Dirichlet boundary condition u(r) = 0 at r = 0. The ground state energy seems to approach the potential energy at r = r0, which goes to −∞ for r0 → 0.

Conclusion on Boundary Conditions

The continuous spectrum is related to the loose boundary condition at r = 0 (the wave function vanishes at r → 0, while its derivative diverges) A specific boundary condition can be added to the present macroscopic model as a microscopic ingredient – e.g., as a condition for the wave function of particles at the surface of a central electric charge which induces the singular potential. The truncation of the singular potential by the hard core at r = r0 is imposed by the Dirichlet boundary condition u(r) = 0 at r ≤ r0, which makes the continuous spectrum discrete. If r0 → 0, the GS energy goes to −∞ when the quantum collapse occurs, and it is not clear whether we can get a discrete energy spectrum even for the Dirichlet boundary condition u(r) = 0 at r = 0. The topic addressed in this work calls for more detailed studies of its mathematical structure based on the self-adjoint extensions of Hamiltonian systems.

References

[1] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Nauka publishers, Moscow, 1974).

[2] D. Bohm, Quantum Theory (Dover, New York, 1989).

[3] K. S. Gupta and S. G. Rajeev, Renormalization in quantum mechanics. Phys. Rev. D 48, 5940-5945 (1993).

[4] H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A. Garc´ıa Canal, Renormalization of the Inverse Square Potential. Phys Rev Lett 85, 1590 (2000).

[5] M Avila-Aoki, C Cisneros R P Martinez-y Romero H N Nu´˜nez-Yepez and A L SalasBrito Classical and quantum motion in an inverse square potential. Phys Lett A 373, 418-421 (2009).

[6] M Olshanii, H Perrin, and V Lorent Example of a quantum anomaly in the physics of ultracold gases. Phys Rev Lett 105, 095302 (2010).

[7] H Sakaguchi and B A Malomed Suppression of the quantum-mechanical collapse by repulsive interactions in a quantum gas. Phys Rev A 83, 013607 (2011).

[8] H Sakaguchi and B A Malomed Suppression of the quantum collapse in binary bosonic gases. Phys Rev A 88, 043638 (2013).

[9] S Sundaram, C P Burgess, and D H J O’Dell Duality between the quantum inverted harmonic oscillator and inverse square potentials. New J Phys 26, 053023 (2024).

[10] Y Zang, Y Gu, and S Jiang Detecting Quantum Anomalies in Open Systems. Phys Rev Lett 133, 106503 (2024).

[11] M I Tribelsky Exact solutions to fall of particle to singular potential: classical versus quantumtum cases. Proc R Soc A 479, 20230366 (2023).

[18] T F¨ul¨op Singular Potentials in Quantum Mechanics and Ambiguity in the Self-Adjoint Hamiltonian. SIGMA 3, 107 (2007).

[19] J Deiglmayr, A Grochola, M Repp, K M¨ortlbauer C Gluck J Lange O Dulieu R Wester and M Weidem¨uller Formation of ultracold polar molecules in the rovibrational ground state. Phys Rev Lett 101, 133004 (2008).

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Improvements for AI systems

Integral Singular Wave Function Generation for Quantum Collapse Analysis: The AI can now analytically produce singular stationary wave functions at r → 0, which feature an infinite number of spatial oscillations, allowing for a deeper investigation into the pre-collapse dynamics described by Eq. (8).
Model Parameter Optimization via Spectral Analysis: The system can use the derived asymptotic forms to relate bound states to the 1D SE with an exponential potential and determine relationships between energy eigenvalues and boundary conditions, such as how "the energy eigenvalues form a continuous spectrum, E < 0."
Non-Generic Boundary Condition Sensitivity: The AI can analyze the critical difference between the continuous spectrum and discrete spectra by examining the role of boundary conditions, noting that The continuous spectrum is related to the loose boundary condition at r = 0 (the wave function vanishes at r → 0, while its derivative diverges).
Finite Regularization Strategy Implementation: The AI can implement a regularization scheme where it "truncates it inside a hard-core 2D or 3D area, r < r0 (setting U(r < r0) = +∞), to demonstrate how this process restores the missing GS, and replaces the continuous spectrum of the bound states by its discrete counterpart."
Vorticity-Dependent Bound State Characterization: The AI can analyze 2D bound states carrying vorticity by solving Eq. (29), enabling it to predict how larger S makes the bound state more compact, connecting this finding to established quantum mechanical results regarding the radial size ¯r of the bound state.
Ground State Energy Prediction under Hard-Core Constraints: The system can calculate ground state energies, such as EGS = -959.604 for the potential (7), truncated by the hard core at r0 = 0.1, and predict that the GS energy seems to approach the potential energy at r = r0, which goes to −∞ for r0 → 0.

Abstract

Quantum collapse (QC) occurs under the action of the inverse-power potential U(r)=-U 0/(2r n) in the framework of the Schroedinger equation. The corresponding stationary wave functions feature an infinite number of spatial oscillations at r to 0. The ground state (GS) is missing in the presence of the QC, but the available bound states at finite energies may be physically relevant, as they are normalizable. This work aims to produce singular stationary wave functions. The analytical findings represent an asymptotic form of the stationary wave functions at r to 0, which fill a continuous spectrum, and some exact solutions for the wave functions, obtained under appropriate conditions. As a regularized version, we also consider the potential truncated at a finite difference from the center. The regularized wave functions feature a finite number of spatial oscillations, the missing GS reappears, and the spectrum of the bound states is discrete.