One-Centre Point Interactions

• One-centre point interactions are quantum systems with interactions localized at a single point, defined by singular boundary conditions and self-adjoint extension parameters.
• They are modeled using operator extension theory, transfer matrices, and distributional approaches to handle the challenges of zero-range or singular potentials.
• These interactions underpin phenomena such as resonant tunnelling and renormalization, making them crucial in scattering theory, spectral analysis, and effective field approaches.

Searching arXiv for recent and foundational work on one-centre point interactions to support a comprehensive synthesis. One-centre point interactions are quantum-mechanical interactions concentrated at a single spatial point and encoded either by singular boundary conditions, by self-adjoint extension parameters of punctured operators, or by zero-range limits of regularized short-range structures. In one dimension they are commonly represented by transfer or matching matrices at $x=0$; in higher dimensions they appear as rank-one singular perturbations of Laplacians or of Hamiltonians with central backgrounds such as Coulomb-like or inverse-square potentials. The subject connects operator extension theory, distribution theory, resonance tunnelling, renormalization-group analysis, and exact scattering theory, and it includes both self-adjoint and non-self-adjoint regimes depending on the physical model (Calçada et al., 2014, Duclos et al., 2010, Noja et al., 5 Jul 2026).

1. Operator-theoretic formulation

A standard one-dimensional parametrization relates the two-sided boundary values of the wave function and its derivative at the singular point by a $2\times 2$ matrix $M$ with unit determinant,

$$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$

with $M_{11}M_{22}-M_{12}M_{21}=1$. This form subsumes pure $\delta$, pure $\delta'$, mixed $\delta+\delta'$, and diagonal $\delta'$-like interactions, and it yields left/right reflection and transmission amplitudes as explicit algebraic functions of the matrix entries and of the asymptotic wave numbers $k_L,k_R$ (Zolotaryuk et al., 2019).

In higher dimensions, the same structural role is played by self-adjoint extensions of a symmetric operator obtained by removing the interaction centre from the domain. For the Dirichlet Laplacian on an unbounded domain $2\times 2$0, $2\times 2$1, one restricts $2\times 2$2 to functions vanishing at the interaction point $2\times 2$3, obtaining a closed symmetric operator with deficiency indices $2\times 2$4. Its self-adjoint extensions are parametrized by $2\times 2$5, and the resolvent has the rank-one Kreĭn form

$2\times 2$6

where $2\times 2$7 is the Weyl function extracted from the local singular expansion of the Green function (Noja et al., 5 Jul 2026).

A closely related construction appears for the two-dimensional Coulomb-like Hamiltonian $2\times 2$8. After separation of variables, all angular-momentum channels with $2\times 2$9 are essentially self-adjoint, whereas the $M$0 channel has deficiency indices $M$1. The one-parameter family of extensions $M$2 is characterized by the short-distance asymptotic

$M$3

so the point interaction is again encoded by a scalar relation between singular and regular parts (Duclos et al., 2010).

2. One-dimensional local models and the $M$4 problem

Within the distributional approach, the interaction term in the stationary Schrödinger equation is not treated as the ill-defined product $M$5, but as a distribution $M$6 supported at $M$7. Imposing support at the origin, a bound on singular order, and conservation of probability current yields the general non-separated matching law

$M$8

with $M$9, $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$0, and $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$1. The separated subfamily is given by Robin-type conditions $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$2 (Calçada et al., 2014).

Two distinguished one-parameter subclasses are the ordinary $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$3-interaction and the diagonal $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$4-like interaction. The former has continuous $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$5 and a jump in $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$6. The latter, in Kurasov’s formulation on discontinuous test functions, has diagonal transfer matrix

$$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$7

equivalently

$$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$8

In this parametrization, $$\psi(+0)=M_{11}\psi(-0)+M_{12}\psi'(-0),\qquad \psi'(+0)=M_{21}\psi(-0)+M_{22}\psi'(-0),$$9 corresponds to a symmetric barrier-well dipole, whereas $M_{11}M_{22}-M_{12}M_{21}=1$0 corresponds to a double-well resonator (Zolotaryuk, 2018).

A persistent controversy concerns the meaning of $M_{11}M_{22}-M_{12}M_{21}=1$1. In the distributional analysis, the naive product $M_{11}M_{22}-M_{12}M_{21}=1$2 is not well defined in Schwartz theory unless $M_{11}M_{22}-M_{12}M_{21}=1$3 and $M_{11}M_{22}-M_{12}M_{21}=1$4 are both continuous at the singular point. The resulting admissible pure $M_{11}M_{22}-M_{12}M_{21}=1$5 point interaction is therefore not an “odd” contact potential; rather, it is an even interaction under parity. The same analysis concludes that no genuine nontrivial odd point interaction exists in one dimension (Calçada et al., 2014).

3. Squeezing limits, resonance sets, and resonant tunnelling

A major realization mechanism for one-centre interactions is the squeezing of thin layered heterostructures. For an $M_{11}M_{22}-M_{12}M_{21}=1$6-layer piecewise-constant potential, the total transfer matrix $M_{11}M_{22}-M_{12}M_{21}=1$7 may converge, as all widths $M_{11}M_{22}-M_{12}M_{21}=1$8, to

$M_{11}M_{22}-M_{12}M_{21}=1$9

leading to the matching conditions

$\delta$0

The existence of a nontrivial limit depends on path-dependent cancellations of divergences in the $\delta$1 entry. If the resonance equations are not satisfied, the limit is typically the separated Dirichlet interaction $\delta$2; if they are satisfied, one obtains resonant $\delta$3-, $\delta$4-, or mixed $\delta$5-$\delta$6-type point interactions with nonzero tunnelling (Zolotaryuk et al., 2021).

For regularized two- and three-$\delta$7 systems, the squeezing exponents $\delta$8 and $\delta$9 govern the limiting interaction. In the regime $\delta'$0, single-valued resonance sets appear: for $\delta'$1,

$\delta'$2

while at the critical boundary $\delta'$3 they furcate into countably many transcendental branches $\delta'$4 and $\delta'$5; analogous surface relations $\delta'$6 hold for $\delta'$7. The terminology “furcation” refers precisely to this passage from single-valued algebraic resonance sets to multi-valued countable families as the squeezing path approaches the critical exponent (Zolotaryuk, 2016).

In the two- and three-layer models with $\delta'$8, the $\delta'$9-plane is partitioned into opaque, $\delta+\delta'$0-resonant, diagonal $\delta+\delta'$1-resonant, and reflectionless sectors. For $\delta+\delta'$2, one gets Kurasov-type diagonal $\delta+\delta'$3 interactions on nonlinear resonance sets $\delta+\delta'$4; for $\delta+\delta'$5, one gets ordinary $\delta+\delta'$6-interactions on linear resonance sets $\delta+\delta'$7; for intermediate $\delta+\delta'$8 the limit is opaque; and for larger $\delta+\delta'$9 it is reflectionless only on the corresponding resonance set. As $\delta'$0, the resonance curves and surfaces split into countable families $\delta'$1 and $\delta'$2 (Zolotaryuk, 2017).

A particularly explicit two-layer realization uses

$\delta'$3

For $\delta'$4 and $\delta'$5, the model yields four limiting regimes: a separated Dirichlet interaction in $\delta'$6, a Kurasov $\delta'$7 interaction on the plane $\delta'$8, a Šeba $\delta'$9 interaction on $k_L,k_R$0, and a unit-matrix reflectionless interaction $k_L,k_R$1 in $k_L,k_R$2. On the open $k_L,k_R$3-plane, the first-type resonance condition is the curve

$k_L,k_R$4

while on the boundary sets $k_L,k_R$5 this single Kurasov curve splits into countably many continuous curves $k_L,k_R$6 defined by trigonometric equations involving $k_L,k_R$7 and $k_L,k_R$8. This resolves the apparent discrepancy between continuously parametrized diagonal $k_L,k_R$9 interactions and discrete resonant transparency by showing that the latter arise as a splitting of resonance sets in the squeezing limit (Zolotaryuk, 2018).

External bias potentials preserve the same zero-range logic. In biased two- and three-terminal heterostructures, the matrix element $2\times 2$00 generically diverges as $2\times 2$01, but a secular equation $2\times 2$02 selects a countable resonance set $2\times 2$03 of bias values for which the limit matrix is finite and transmission is nonzero. Off $2\times 2$04, the interaction acts as a perfect reflector, and the same formalism yields “point” transistor models built from $2\times 2$05- and $2\times 2$06-like limits (Zolotaryuk et al., 2019).

4. Spectral and geometric structure in higher dimensions

For a bounded smooth domain $2\times 2$07, $2\times 2$08, with a point interaction at $2\times 2$09, the self-adjoint Hamiltonian $2\times 2$10 has a quadratic-form domain

$2\times 2$11

and near the centre its elements satisfy the Bethe-Peierls contact condition

$2\times 2$12

The operator is nonnegative iff $2\times 2$13, and below the first Dirichlet eigenvalue $2\times 2$14 it has a single simple eigenvalue $2\times 2$15, strictly monotone in $2\times 2$16. Among all bounded $2\times 2$17 domains of fixed volume, and all centres $2\times 2$18, the Faber-Krahn inequality takes the sharp form

$2\times 2$19

so the optimizer is the ball with the interaction at its centre (Lotoreichik et al., 2020).

For unbounded exterior or special Lipschitz domains, the Dirichlet point interaction $2\times 2$20 remains a rank-one perturbation of $2\times 2$21. The negative spectrum is completely characterized by the scalar equation

$2\times 2$22

which has at most one positive root $2\times 2$23 and hence at most one simple negative eigenvalue with eigenfunction $2\times 2$24. The critical coupling

$2\times 2$25

separates the binding and non-binding regimes: there is one negative bound state for $2\times 2$26 and none for $2\times 2$27 (Noja et al., 5 Jul 2026).

The same framework yields strong geometric information. If $2\times 2$28 and $2\times 2$29, then $2\times 2$30, hence $2\times 2$31; the bound state, when it exists, is more negative in the larger domain. Near a uniformly $2\times 2$32 boundary,

$2\times 2$33

with $2\times 2$34. Thus, for fixed coupling, nonpositive spectrum disappears as the interaction centre approaches the Dirichlet boundary. At threshold, the zero-energy state depends on geometry at infinity: 3D exterior domains exhibit monopole resonances, domains contained in a 3D half-space exhibit true threshold eigenvalues, the half-plane exhibits a $2\times 2$35-wave resonance, and wedges exhibit aperture-dependent threshold states (Noja et al., 5 Jul 2026).

5. Central backgrounds, renormalization, and non-self-adjoint regimes

Adding a point interaction to a two-dimensional Coulomb-like potential produces a hybrid singular problem with a complete extension theory in the $2\times 2$36 partial wave. The resolvent of $2\times 2$37 is given by a Kreĭn formula involving a defect vector proportional to a Whittaker function and a scalar coefficient $2\times 2$38. Negative eigenvalues are determined by the transcendental equation

$2\times 2$39

while in the scattering sector the $2\times 2$40 phase shift satisfies

$2\times 2$41

The coordinate-space parameter $2\times 2$42 is equivalent to a momentum-space parameter $2\times 2$43, with

$2\times 2$44

and the pure Coulomb Hamiltonian arises as the norm-resolvent limit of the 3D hydrogen atom confined to a planar slab of width $2\times 2$45 (Duclos et al., 2010).

A different singular background is the attractive inverse-square potential $2\times 2$46. Here the small-$2\times 2$47 ambiguity of the radial equation is controlled by a point-particle EFT boundary action. The running boundary coupling obeys the exact RG equation

$2\times 2$48

with a critical coupling

$2\times 2$49

For $2\times 2$50, $2\times 2$51 and the flow has two real fixed points $2\times 2$52, corresponding to self-adjoint boundary conditions. For $2\times 2$53, $2\times 2$54 is imaginary, the real fixed points disappear, two complex fixed points $2\times 2$55 emerge, and generic real trajectories execute a limit cycle. The fixed point $2\times 2$56 is the perfect absorber boundary condition and is explicitly not self-adjoint; $2\times 2$57 is the perfect emitter. The formalism predicts anomalous breaking of scale invariance, a tower of scales $2\times 2$58, Efimov-like quasi-bound states in the supercritical regime, and log-periodic absorptive observables (Plestid et al., 2018).

6. Generalizations and dynamical extensions

One-centre interactions extend beyond local scalar Schrödinger models. In one dimension, nonlocal one-point interactions are described by an integral kernel

$2\times 2$59

with $2\times 2$60. Using boundary triplets, all proper extensions are parametrized by a $2\times 2$61 matrix $2\times 2$62 through $2\times 2$63, and the resolvent has the Kreĭn form

$2\times 2$64

The discrete spectrum is determined by

$2\times 2$65

and the same framework captures exceptional points and spectral singularities in manifestly non-Hermitian settings (Kuzhel et al., 2016).

For the three-dimensional Laplacian with a point interaction at the origin, the operator $2\times 2$66 may be defined either as a self-adjoint extension of $2\times 2$67 or as a norm-resolvent limit of scaled short-range potentials. Its domain is characterized by

$2\times 2$68

For the associated heat equation, there exists for each $2\times 2$69 and $2\times 2$70 a probability law on path space and a normalizing function $2\times 2$71 such that

$2\times 2$72

This provides a Feynman-Kac-type representation for the heat semigroup with a one-centre point interaction in $2\times 2$73 (Nakashima, 10 Jun 2026).

The notion also extends to multicomponent first-order systems. For the one-dimensional pseudospin-one Hamiltonian, the zero-width limit of a rectangular three-component potential reduces the matching problem to the two-component field $2\times 2$74, with general self-adjoint point interaction

$2\times 2$75

Four characteristic spectral families, denoted $2\times 2$76, are obtained from different scalings of the rectangular profile, with explicit one-parameter matrices $2\times 2$77 and closed-form or transcendental bound-state equations. This shows that the one-centre framework is not confined to scalar second-order Hamiltonians but persists in pseudospin models with algebraic constraints among components (Zolotaryuk et al., 2023).

Across these settings, one-centre point interactions retain a common core: a singularity concentrated at one point, a finite set of boundary data tied together by a small number of extension parameters, and exact control of scattering, spectral thresholds, and zero-range limits. What varies from model to model is the geometry of the ambient space, the background singularity, the admissible self-adjoint or non-self-adjoint boundary laws, and the mechanism—regularization, extension theory, or renormalization—by which the point interaction is selected.