Singular Potentials in Quantum Mechanics

• Singular potentials are defined as potentials with non-integrable divergences at isolated points—such as Coulomb, δ, or power-law forms—that challenge standard quantum methods.
• Self-adjoint extensions and rigorous distributional techniques enable accurate treatment of these singularities, ensuring well-defined quantum dynamics.
• Advanced numerical methods and spectral approaches applied to singular potentials yield insights for electronic structures, nano-systems, and field theory.

Singular potentials in quantum mechanics are potentials that display non-integrable divergences at isolated points—typically algebraic (power-law), logarithmic, or distributional (Dirac δ, δ′) behavior. These singularities present fundamental challenges for both the mathematical foundations of quantum theory and for practical computation, requiring advanced techniques in self-adjoint extensions, distributional analysis, and specialized numerical solutions. Singular potentials are essential in realistic modeling of atomic, molecular, and nanoscale systems, as well as in field-theory and geometric applications.

1. Taxonomy and Mathematical Peculiarities of Singular Potentials

Singular potentials are classified by the leading divergence as the spatial argument approaches the singular point. Consider the family $V(x) \sim \alpha / |x|^\beta$ as $|x| \to 0$ for $0<\beta\leq2$; physically relevant cases include the Coulomb potential $(\beta=1)$, the Calogero (inverse-square) potential $(\beta=2)$, point-interaction (δ-function, distributional), and more exotic forms such as logarithmic or highly singular van der Waals potentials ($\beta > 2$ or $n > 3$ in $V(r) \sim r^{-n}$) (Pimentel et al., 2013, Yukalov et al., 7 Mar 2025).

Such potentials challenge standard quantum mechanics because:

• The Schrödinger operator is not essentially self-adjoint; domain specification (choice of boundary or matching conditions) is nontrivial and critical for uniqueness of dynamics.
• For distributional forms (e.g., Dirac δ, δ′), formal manipulation must be replaced by rigorous distributional definitions (Maroun, 2023, Maroun, 2021, Horing, 2014).
• For strong singularities ($\beta \geq 1$ in 1D, $n>d$ in $|x| \to 0$0-dimensional many-body systems), standard mean-field or diagrammatic expansions become divergent (Yukalov et al., 7 Mar 2025).

Table: Principal Types of Singular Potentials and Their Singularities

Potential Type	Functional Form	Singularity Type
Coulomb	$|x| \to 0$1	$|x| \to 0$2
Inverse-square	$|x| \to 0$3	$|x| \to 0$4
Dirac delta	$|x| \to 0$5	Distributional
Delta derivative	$|x| \to 0$6	Distributional
Logarithmic	$|x| \to 0$7	Logarithmic
Power-law $|x| \to 0$8	$|x| \to 0$9	Non-integrable

2. Self-Adjoint Extensions and Distributional Quantum Mechanics

The mathematical structure of singular potentials compels careful specification of the domain for the quantum Hamiltonian. For potentials like $0<\beta\leq2$0 in 1D with $0<\beta\leq2$1, the requirement of Hermiticity and finite expectation values leads to constraints near the origin: wavefunctions must behave as $0<\beta\leq2$2 with $0<\beta\leq2$3 determined by the degree of singularity and the operator's self-adjoint extension (Pimentel et al., 2013). In particular, for the pure inverse-square law $0<\beta\leq2$4, admissibility requires $0<\beta\leq2$5 with $0<\beta\leq2$6.

For distributional potentials (δ, δ′), the operator is defined on $0<\beta\leq2$7 via distributional action. In two and three dimensions, the δ-potential's treatment involves the definition of $0<\beta\leq2$8 as a distribution, absorbing singularities such as $0<\beta\leq2$9 by distributional identities and introducing an "anomalous length scale" $(\beta=1)$0, which breaks naive scale invariance and determines a continuum of spectral values (the so-called C-spectrum) (Maroun, 2023, Maroun, 2021). The three-dimensional case yields the classic bound state energy,

$(\beta=1)$1

where $(\beta=1)$2 for an attractive $(\beta=1)$3 (Maroun, 2021).

Distributional methods bypass regularization and renormalization, replacing them with a mathematically rigorous identification of products such as $(\beta=1)$4 and the extraction of spectral information purely from distributional equations (Maroun, 2023, Maroun, 2021, Horing, 2014).

3. Quantum Dynamics and Transport Across Singularities

For one-dimensional singular barriers, the transmission and reflection properties are highly sensitive to the strength and nature of the singularity. For $(\beta=1)$5:

• If $(\beta=1)$6 (mildly singular), finite and nonzero transmission is possible even at zero energy. This stems from the cancellation of singularities in the probability current at the origin and leads to unconventional, non-analytic energy dependence for $(\beta=1)$7 (Muradyan, 2021).
• At $(\beta=1)$8 (Coulomb singularity), transmission at zero energy oscillates infinitely rapidly as $(\beta=1)$9 and is suppressed at high energy, with analytic expressions given in terms of confluent hypergeometric functions and Gamow factors.
• For $(\beta=2)$0 (strong singularity), the barrier is completely opaque: no current can pass; $(\beta=2)$1 for all $(\beta=2)$2 (Muradyan, 2021).

The case of combined δ and δ′ interactions produces matching conditions that can fully disconnect the Hilbert space into separate domains, leading to perfect reflection and the lack of bound or scattering states on the far side of the singularity (Horing, 2014, Govindarajan et al., 2016).

4. Numerical and Analytical Approaches for Singular Potentials

Standard discretization and basis-set techniques are poorly suited to singular potentials. Several major methodologies have been developed:

• Spectral and Pseudospectral Methods: The generalized pseudospectral (GPS) method employs a nonuniform grid adapted to the singularity (e.g., $(\beta=2)$3) using mapping to finite intervals and cardinal function approximations. This approach achieves exponential convergence and high-precision computation of both moderate and highly excited states for Hamiltonians with terms such as $(\beta=2)$4, $(\beta=2)$5, $(\beta=2)$6, and more (Roy, 2019, Roy, 2013). Singular terms are handled as diagonal entries in the matrix representation.
• J-Matrix Method: For singular potentials of the Coulombic or screened Coulomb (Yukawa, Hulthén) type, the J-matrix approach separates the Hamiltonian into a reference part containing the singularity (e.g., $(\beta=2)$7) for which exact solutions in a Laguerre basis are available, and a regular, bounded remainder. The reference problem yields a tridiagonal matrix, and the regular part is efficiently handled by Gauss—Laguerre quadrature. Bound and resonance states are found as poles of an algebraic S-matrix (Abdelmonem et al., 2010).
• Wigner Function Approach: The phase-space Wigner function formalism transforms local singular potentials into nonlocal oscillatory convolution kernels in momentum space, smearing the singularity and enabling spectral convergent discretization. For potentials such as δ(x), $(\beta=2)$8, and $(\beta=2)$9 ($\beta > 2$0), the Wigner kernel is smooth or only weakly singular, and nonlocal quantum effects such as interference and geometry-induced scattering are naturally manifest (Shao et al., 2023).
• Green Function and Quantum Impedance: One-dimensional singular oscillator and quantum dot problems are effectively solved by construction of explicit Green functions or transformation to quantum-wave-impedance equations, which reduce the problem to first-order Riccati-type equations and transparent matching rules at singular points (Glasser et al., 2015, Hryhorchak, 2020).

5. Regularization, Physical Realizability, and Alternative Frameworks

Highly singular potentials are non-integrable and lead to divergences in many-body systems, invalidating mean-field treatments such as Hartree or HF. A regularization based on two-body correlation functions—determined from the zero-energy scattering solution—effectively tames the singularity at the Hamiltonian level. All higher-order diagrams are then rendered finite when every bare $\beta > 2$1 is replaced by $\beta > 2$2 (Yukalov et al., 7 Mar 2025).

Geometric regularization is another powerful paradigm: a singular potential in flat space can be mapped onto free motion on a suitably curved manifold. For example, the conical geometry regularizes the $\beta > 2$3 singularity via short-distance (apex) curvature, preventing "collapse to the center" and rendering the Hamiltonian self-adjoint without ad hoc cutoff (Curtright et al., 2021).

In discrete and phase-space formulations, either configuration-space discretization or exact discrete phase-space representations remove Coulomb-type singularities entirely, replacing $\beta > 2$4 with finite, smooth kernels or difference operators invariant under the appropriate symmetry group (Das et al., 2015).

6. Physical Consequences and Applications

Singular potentials appear ubiquitously:

• Electronic Structure: The Coulomb singularity at electron-nucleus coalescence dictates cusp conditions, determining the correct asymptotic expansions for atomic and molecular wavefunctions. The explicit Green operator encodes these asymptotics and allows systematic construction of optimized basis sets for high-accuracy calculations (Flad et al., 2010).
• Nano-Systems and Quantum Dots: Modelings such as double-barrier quantum dots, quantum wires with junctions, and electron transport in sharply confined geometries depend on δ and δ′ potentials. The transport is extremely sensitive; e.g., all transmission is blocked by a $\beta > 2$5 barrier (Horing, 2014).
• Quantum Fluids and Rydberg Matter: Strong dipolar and higher-order interactions often behave as $\beta > 2$6 with $\beta > 2$7, making regularization essential for the theoretical description of collective quantum phenomena in cold gases and condensed matter (Yukalov et al., 7 Mar 2025).
• Black Hole Analogs and Field Theory: The introduction of δ and δ' potentials models boundary effects and horizon dynamics in field-theoretic and gravitational analogs, permitting the study of edge modes, Robin-like boundary effects, and horizon-induced quasinormal spectra (Govindarajan et al., 2016).

7. Emergent Nonlocal Quantum Phenomena

Singular potentials induce nonclassical effects mediated by the regularization inherent to the mathematical or numerical formalism:

• Tunneling Through Point Barriers: Even an infinitely strong δ-barrier does not block tunneling in quantum mechanics; finite nonzero probability remains due to nonlocality, and attempts to model the singularity with smooth approximations fail to reproduce genuine point-charge behavior (Shao et al., 2023).
• Nonlocalization in Wigner Space: Wigner formalism reveals oscillatory fine structure, interference patterns, and geometry-dependent scattering even for singular potentials, all captured with spectral accuracy (Shao et al., 2023).
• Localization by Geometry: Singular geometric junctions in configuration or phase space (e.g., $\beta > 2$8 intersecting 1D or 2D sheets) generate bound states and spatial localization purely via kinetic energy, with no explicit potential. The equivalent effective potential is singular and can be mapped to a δ-well of corresponding strength (He et al., 2023).

• (Shao et al., 2023) Nonlocalization of singular potentials in quantum dynamics
• (Abdelmonem et al., 2010) Singular Short Range Potentials in the J-Matrix Approach
• (Pimentel et al., 2013) A brief discussion on the possible bound states for a class of singular potentials
• (Roy, 2019, Roy, 2013) Generalized pseudospectral method for singular central potentials
• (Maroun, 2023, Maroun, 2021, Horing, 2014) Distributional and Green-function treatments of singular point interactions
• (Yukalov et al., 7 Mar 2025) Quantum systems of atoms with highly singular interaction potentials
• (Muradyan, 2021) Quantum tunneling of a singular potential
• (Glasser et al., 2015, Hryhorchak, 2020) Green function and impedance approaches for singular perturbations
• (Flad et al., 2010) Explicit Green operators for quantum mechanical Hamiltonians
• (Govindarajan et al., 2016) Singular potentials, self-adjoint extension, and black-hole analogs
• (Das et al., 2015) Discrete phase space and non-singular potential functions
• (Curtright et al., 2021) Potentials versus geometry: geometric regularization and mapping

These developments delineate both the technical challenges and the broad applicability of singular potentials in quantum theory, highlighting their continued centrality in both fundamental and applied quantum research.