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arXiv:2602.07281

The continuous spectrum of bound states in expulsive potentials

H. Sakaguchi, B. A. Malomed, A. C. Aristotelous, E. G. Charalampidis

quant-phcond-mat.quant-gasnlin.PSphysics.opticsSubmitted Feb 7, 2026• Announced Feb 10, 2026
v1 · Feb 7, 2026v2 · Apr 27, 2026
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Abstract

On the contrary to the common intuition that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schrödinger equations, which include expulsive potentials that are \emph{steeper than the quadratic} (anti-harmonic-oscillator) ones, give rise to \emph{normalizable} (effectively localized) eigenstates. These states constitute full continuous spectra in the 1D and 2D cases alike. In 1D, these are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic approximations for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with numerically found counterparts. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability. On the other hand, the quintic self-focusing term, which occurs in the photonic version of the 1D model, initiates the dynamical collapse of states whose norm exceeds a critical value.

Comments
11 pages, 8 figures

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On the contrary to the common intuition that a steep expulsive potential makes quantum states widely delocalized, we demonstrate that one- and two-dimensional (1D and 2D) Schrödinger equations, which include expulsive potentials that are \emph{steeper than the quadratic} (anti-harmonic-oscillator) ones, give rise to \emph{normalizable} (effectively localized) eigenstates. These states constitute full continuous spectra in the 1D and 2D cases alike. In 1D, these are spatially even and odd eigenstates. The 2D states may carry any value of the vorticity (alias magnetic quantum number). Asymptotic approximations for wave functions of the 1D and 2D eigenstates, valid far from the center, are derived analytically, demonstrating excellent agreement with numerically found counterparts. Special exact solutions for vortex states are obtained in the 2D case. These findings suggest an extension of the concept of bound states in the continuum, in quantum mechanics and paraxial photonics. Gross-Pitaevskii equations are considered as the nonlinear extension of the 1D and 2D settings. In 1D, the cubic nonlinearity slightly deforms the eigenstates, maintaining their stability. On the other hand, the quintic self-focusing term, which occurs in the photonic version of the 1D model, initiates the dynamical collapse of states whose norm exceeds a critical value.
We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.
It is known that the attractive singular inverse-square potential gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schroedinger equation. This article summarizes theoretical results which demonstrate suppression of the collapse, caused by this singular potential, and the creation of the otherwise missing ground state (GS) in a 3D gas of bosonic particles, carrying an electric dipole moment, which are pulled to the central electric charge, with repulsive contact interactions between the particles. In the mean-field approximation, the repulsive interactions are represented by the cubic term in the respective Gross-Pitaevskii (GP) equation. In addition to the GS, excited states with angular momentum are briefly considered too. Another topic considered in the article is 1D and 2D bound states in the linear Schroedinger and GP equations with the repulsive potential, which demonstrates a singularity at r --> infinity. A very recent result is that such a potential, growing faster than the negative harmonic-oscillator potential, produces a full spectrum of counter-intuitive normalizable (localized) bound states. The article puts forward perspectives for further studies of linear and nonlinear bound states existing under the action of the potentials with the singularity at r --> 0 or r --> infinity.
The trapping potential is a necessary ingredient of any setup used in experiments with atomic gases in the BEC state (otherwise, the gas would escape). Similarly, the presence of an effective trap is necessary for building various optical and photonic cavities. Therefore, the analysis of models combining the intrinsic nonlinearity acting in multidimensional settings with the trapping potential is a natural step in the theoretical work. The present chapter aims to summarize basic theoretical results produced by the work in this direction. The results are presented for 2D fundamental (zero-voticity) and vortical states trapped in the HO potential and for similar 3D states trapped in the HO potential, which may be anisotropic. Also considered are two-component 2D states with hidden vorticity, i.e., with opposite vorticities if the two components have equal norms. Stability regions are identified for all these settings. The chapter additionally reports findings for two-component 2D fundamental and vortex states in a linearly coupled system, in which the potential acting on one component is trapping, while the other component is subjected to the action of an expulsive potential. As for experimental results demonstrating the creation of 2D or 3D solitons in trapping potentials, very few of them are available. A recent experimental finding that may be relevant in this respect is the creation of quasi-stable effectively two-dimensional Townes solitons in atomic BEC loaded in a trapping potential [Bakkali-Hassani et al., Phys. Rev. Lett. 127, 023603 (2021)].
This concise review aims to provide a summary of the most relevant recent experimental and theoretical results for solitons, i.e., self-trapped bound states of nonlinear waves, in two- and three-dimensional (2D and 3D) media. In comparison with commonly known one-dimensional solitons, which are, normally, stable modes, a challenging problem is the propensity of 2D and 3D solitons to instability, caused by the occurrence of the critical or supercritical wave collapse (catastrophic self-compression) in the same spatial dimension. A remarkable feature of multidimensional solitons is their ability to carry vorticity; however, 2D vortex rings and 3D vortex tori are subject to strong splitting instability. Therefore, it is natural to categorize the basic results according to physically relevant settings which make it possible to maintain stability of fundamental (non-topological) and vortex solitons against the collapse and splitting, respectively. The present review is focused on schemes that were recently elaborated in terms of Bose-Einstein condensates and similar photonic setups. These are two-component systems with spin-orbit coupling, and ones stabilized by the beyond-mean-field Lee-Huang-Yang effect. The latter setting has been implemented experimentally, giving rise to stable self-trapped quasi-2D and 3D "quantum droplets".
We construct exact solutions of the Gross-Pitaevskii equation for solitary vortices, and approximate ones for fundamental solitons, in 2D models of Bose-Einstein condensates with a spatially modulated nonlinearity of either sign and a harmonic trapping potential. The number of vortex-soliton (VS) modes is determined by the discrete energy spectrum of a related linear Schrödinger equation. The VS families in the system with the attractive and repulsive nonlinearity are mutually complementary. \emph{% Stable} VSs with vorticity S≥2S\geq 2S≥2 and those corresponding to higher-order radial states are reported for the first time, in the case of the attraction and repulsion, respectively.
I Plenary talks.- An adiabatic theorem without a gap condition.- Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum.- Spectral problems in the theory of photonic crystals.- Optimal eigenvalues for some Laplacians and Schroedinger operators depending on curvature.- The spectral shift operator.- On the scattering operator for the Schroedinger equation with a time-dependent potential.- H_2-construction and some applications.- Scattering with time periodic potentials and cyclic states.- Some geometry related to decay properties of the resolvent of a class of symmetric operators.- On some asymptotic formulas in the strong localization regime of the theory of disordered systems.- Spectral measures and category.- Quantum dots. A survey of rigorous results.- II Session talks.- A simple model of concentrated nonlinearity.- Anomalous electron trapping by magnetic flux tubes and electric current vortices.- On the absolutely continuous energy distribution of a quantum mechanical system in a bounded domain.- Some aspects of generalized contact interaction in one-dimensional quantum mechanics.- Traces and trace norms for semigroup differences...- About a resolvent formula.- The determinant anomaly in low-dimensional quantum systems.- Linear Boltzmann equation as the weak coupling limit of the random Schroedinger equation.- Coexistence of different spectral types for almost periodic Schroedinger equations in dimension one.- Dynamical localization for random Schroedinger operators and an application to the almost Mathieu operator.- On fractal structure of the spectrum for periodic point perturbations of the Schroedinger operator with a uniform magnetic field.- A Weyl-Berry formula for the scattering operator associated to self-similar potentials on the line.- Localization and Lifshitz tails for random quantum waveguides.- Birman-Schwinger analysis for bound states in a pair of parallel quantum waveguides with a semitransparent boundary.- On the absolute continuity of spectra of periodic elliptic operators.- Hardy inequalities for magnetic Dirichlet forms.- Adiabatic curvature, chaos and the deformation of Riemann surfaces.- Operator interpretation of resonances arising in spectral problems for 2 x 2 matrix Hamiltonians.- On the operator-norm convergence of the Trotter-Kato product formula.- A particular case of the inverse problem for the Sturm-Liouville equation with parameter dependent potential.- One-dimensional Schroedinger operators with decaying potentials.- Stability of limiting absorption under singular perturbations.- Existence of averaging integrals for self-adjoint operators.- Monotonicity versus non-monotonicity in random operators.- A model in perturbation theory.- Band gap of the spectrum in periodically curved quantum waveguides.- A list of other talks.- A list of participants.