Cornell-type Double-Well Potential

• Cornell-type double-well potentials combine Coulombic short-range attraction and linear long-range confinement with a two-minima structure achieved via modification or state hybridization.
• They employ mechanisms like explicit hybridization, effective barrier formation by orthogonality, and formal ansatz representations to yield exponentially small tunnel splitting and robust state localization.
• Analytical, numerical, and many-body studies validate these models by revealing key spectral features, tunneling dynamics, and connections to effective Bose–Hubbard dynamics in confined systems.

A Cornell-type double-well potential is not a single standardized potential in the literature but a composite notion that combines Cornell-inspired short-range attraction and long-range confinement with the spectral, localization, and tunneling structure of a double well. In the strict quarkonium sense, the standard Cornell form $V(r)=-\kappa/r+\sigma r+C$ is not itself a double well: for the usual parameter regime it is monotonic on $r>0$, so it has no pair of local minima separated by a barrier (Pathak et al., 2020). The expression therefore appears in two distinct ways. In one usage, it denotes a modified Cornell interaction that has been altered enough to support two-well physics. In another, it denotes a hybrid construction in which Cornell-inspired localized states or confinement enter the wavefunction ansatz, while the actual Schrödinger problem is solved in an explicit double-well potential, such as a quartic barrier between two minima (Pathak, 30 Oct 2025).

1. Terminology and scope

The canonical Cornell potential used in heavy-quark phenomenology is

$V(r)=-\frac{4\alpha_s}{3r}+br+c,$

with $\alpha_s$ controlling the Coulombic short-distance term, $b$ the linear confining slope, and $c$ an additive constant (Pathak et al., 2020). A closely related exact Cornell form also appears in a modified Lee–Wick-inspired Abelian model,

$$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$

which reproduces Coulomb behavior at short distance and linear confinement at long distance, but again does not generate a double well (Smailagic et al., 2020).

The term “Cornell-type” is also used more broadly in relativistic scalar-potential settings. In cosmic-string and Kaluza–Klein formulations, the scalar interaction is introduced as

$$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$

respectively, through the replacement $m\to m+S$ in the Klein–Gordon equation (Ahmed, 2020, Leite et al., 2019). These are Cornell-type in the sense of combining Coulomb-like and linear terms, but they are not double wells.

For that reason, “Cornell-type double-well potential” is best understood as a research shorthand for models that import Cornell-like localization or confinement into a bona fide two-well setting. The phrase does not identify a universally accepted analytic potential in the way that “Cornell potential” or “quartic double well” does.

2. The standard Cornell potential and the absence of intrinsic two-well structure

For the standard Cornell form with $\alpha_s>0$ and $r>0$0,

$r>0$1

the derivative is

$r>0$2

so the potential is strictly increasing on the positive radial half-line (Pathak et al., 2020). It tends to $r>0$3 as $r>0$4 because of the Coulomb singularity and to $r>0$5 as $r>0$6 because of the linear term. The constant $r>0$7 shifts the graph vertically and changes the zero-crossing radius $r>0$8, but it does not affect the force and cannot create local extrema (Pathak et al., 2020).

This monotonicity is not an incidental detail; it is the main obstruction to identifying the unmodified Cornell interaction with a double well. In the heavy-light meson analysis of the Cornell model, the principal issue is not multi-minimum structure but the consistency of the perturbative split between Coulomb and confinement pieces. Under the assumptions $r>0$9, constituent masses $V(r)=-\frac{4\alpha_s}{3r}+br+c,$0, $V(r)=-\frac{4\alpha_s}{3r}+br+c,$1, $V(r)=-\frac{4\alpha_s}{3r}+br+c,$2, $V(r)=-\frac{4\alpha_s}{3r}+br+c,$3, and a Coulomb-parent Dalgarno treatment, the quoted allowed domain is

$V(r)=-\frac{4\alpha_s}{3r}+br+c,$4

with $V(r)=-\frac{4\alpha_s}{3r}+br+c,$5 required for the reality of the Dirac factor $V(r)=-\frac{4\alpha_s}{3r}+br+c,$6 (Pathak et al., 2020). None of these parameter choices changes the monotonic character of the radial potential.

The same conclusion holds for the exact Cornell form derived in Lee–Wick-inspired electrodynamics. There the point-source potential is singular at the origin, Coulombic at short distance, and linearly confining at large distance; Gaussian smearing regularizes $V(r)=-\frac{4\alpha_s}{3r}+br+c,$7 while preserving the long-distance linear term, but still does not introduce a barrier separating two minima (Smailagic et al., 2020). A literal Cornell-type double well therefore requires added structure beyond the standard $V(r)=-\frac{4\alpha_s}{3r}+br+c,$8 ansatz.

3. Mechanisms by which Cornell-like ingredients enter double-well physics

One mechanism is explicit hybridization. In the proton-transfer model of hydrogen bonds, the “Cornell-type” ingredient is a localized wavefunction ansatz rather than the potential actually diagonalized. The ansatz is written in Cornell-inspired Coulomb-plus-confinement form,

$V(r)=-\frac{4\alpha_s}{3r}+br+c,$9

with $\alpha_s$0, $\alpha_s$1, $\alpha_s$2, $\alpha_s$3, $\alpha_s$4, and reduced mass $\alpha_s$5 controlling length scale, confinement, short-range correction, and normalization (Pathak, 30 Oct 2025). A two-well picture is then created by shifting this localized state to the left and right,

$\alpha_s$6

and estimating the tunnel splitting through the overlap

$\alpha_s$7

The explicit numerical Schrödinger equation, however, is solved in the quartic double well

$\alpha_s$8

not in a Cornell potential (Pathak, 30 Oct 2025).

A second mechanism is effective barrier formation by orthogonality. In the one-dimensional infinite square well split by a central $\alpha_s$9-function, a repulsive wall and an attractive moat both generate nearly identical low-energy two-well physics. For the attractive case, a localized bound state forms at the center; the positive-energy scattering states must remain orthogonal to it, and this orthogonality suppresses their amplitude near the center. In an orthogonalized basis, the effective pseudopotential becomes

$b$0

with nonlocal kernel

$b$1

which is repulsive in the projected subspace because $b$2 (Ibrahim et al., 2017). This suggests that in a Cornell-inspired environment, a strongly attractive localized core inside a broader confining region can act as an effective separator between left and right sectors without an explicit repulsive hump.

A third mechanism is formal rather than dynamical: one may start from a genuine double well and use Cornell-type functions only as a semianalytical representation of localized states. This is the sense in which the hydrogen-bond tunneling model is Cornell-type. The phrase then describes the ansatz layer, not the Hamiltonian itself (Pathak, 30 Oct 2025).

4. Spectral structure, tunnel splitting, and asymmetry

Once a genuine double-well geometry is present, the low-energy sector typically organizes into a near-degenerate even/odd pair. In the quartic potential

$b$3

the minima are at $b$4, the central barrier is at $b$5, and $b$6 (Pathak, 30 Oct 2025). Near either minimum, the potential is locally harmonic with curvature

$b$7

so the low-lying states may be viewed as two weakly coupled local oscillators (Pathak, 30 Oct 2025).

The tunnel splitting is controlled by barrier width, barrier height, and mass. In the same model, the forbidden-region tail is taken as

$b$8

which gives

$b$9

The WKB form is

$c$0

For $c$1 Å and $c$2 eV, the reported splittings are

$c$3

$c$4

showing the expected exponential isotope suppression as $c$5 increases (Pathak, 30 Oct 2025).

A small asymmetry can qualitatively reorganize this two-state sector. In the square double-well toy model with effective Hamiltonian

$c$6

the eigenvalues are

$c$7

and the ground-state weights are

$c$8

The relevant control parameter is $c$9, not the asymmetry relative to the barrier height (Dauphinee et al., 2015). In a case with $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$0, $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$1, $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$2, and a tiny perturbation $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$3, the fitted tunnel coupling is only $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$4, so the ground state localizes almost entirely in the right well despite the perturbation being only more than $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$5 part in $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$6 relative to the barrier scale (Dauphinee et al., 2015). For any Cornell-type double well with exponentially small splitting, the same asymmetry logic should apply.

5. Analytical and numerical formulations

Several benchmark double-well models clarify how Cornell-type constructions may be analyzed even when the Cornell ingredient itself is not the explicit barrier profile. An exactly solvable symmetric piecewise-harmonic example is

$$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$7

with minima at $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$8 and central barrier $$V(r)= -\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r} +\frac{e^2m^2}{8\pi\varepsilon_0}\,r,$$9 (Sasaki, 2022). Because each half-line is a shifted oscillator, the eigenfunctions are piecewise square-integrable $$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$0-type combinations of confluent hypergeometric functions, and the even/odd energies are given exactly as zeros of corresponding matching conditions. As $$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$1 increases, the spectrum forms the familiar quasi-degenerate even/odd pairs. For example, at $$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$2,

$$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$3

$$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$4

which explicitly resolves tunneling-induced splitting in an analytically tractable model (Sasaki, 2022).

A numerically exact basis-expansion route is provided by the infinite-square-well embedding method for one-dimensional symmetric double wells. The wavefunction is expanded in

$$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$5

leading to

$$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$6

For smooth double wells, truncation at $$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$7 already gives ground-state energies accurate to better than $$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$8, whereas the square double well requires about $$S(r)=\frac{\eta_c}{r}+\eta_L r \quad\text{or}\quad S(\rho)=\frac{a}{\rho}+b\rho,$$9 basis states; the main spectrum in that study uses $m\to m+S$0 (Jelic et al., 2012). The same work shows that WKB is remarkably accurate for tunnel-split doublets and remains good for excited states (Jelic et al., 2012).

For the quartic hydrogen-bond model, the explicit numerical solution uses finite differences and ARPACK for

$m\to m+S$1

with convergence checks at $m\to m+S$2 eV showing splittings $m\to m+S$3, $m\to m+S$4, $m\to m+S$5, $m\to m+S$6, and $m\to m+S$7 eV for increasingly refined $m\to m+S$8 choices (Pathak, 30 Oct 2025). The stability of the lowest doublet is thus directly verified.

In radial Cornell problems, even when double wells are absent, the numerical framework can still be transferable. The nonrelativistic $m\to m+S$9 study solves

$\alpha_s>0$0

and combines VMC trial-state optimization with fixed-node GFMC projection. Because the formalism depends on the input $\alpha_s>0$1 rather than the Cornell form alone, it is structurally adaptable to modified radial Cornell-type potentials; that study adopts $\alpha_s>0$2 and $\alpha_s>0$3 after plateau tests, and removes short-time bias by fitting

$\alpha_s>0$4

across a ladder of time steps (Akan, 14 Nov 2025). For a radial Cornell-type double well, the main extra difficulty would be trial-state design and resolution of near-degenerate wells.

6. Many-body generalizations and present interpretive boundaries

The many-body extension of double-well physics is best developed for generic symmetric double wells rather than for Cornell-specific forms. In the bosonic setting with

$\alpha_s>0$5

and

$\alpha_s>0$6

the rigorous large-$\alpha_s>0$7, large-separation reduction yields left/right localized orbitals

$\alpha_s>0$8

and an effective two-site Bose–Hubbard Hamiltonian

$\alpha_s>0$9

with tunneling scale $r>0$00 and number-squeezing result

$r>0$01

(Olgiati et al., 2021). This does not constitute a Cornell-type double-well theory, but it identifies the structural prerequisites any such theory would need: two localized one-body modes, exponentially small tunneling, and a gap to higher excitations.

The present literature therefore supports a precise boundary statement. A standard Cornell potential is monotonic and not a double well (Pathak et al., 2020). Several exact or effective Cornell-type constructions reproduce Coulomb-plus-linear confinement, but still lack two-well topology (Smailagic et al., 2020, Ahmed, 2020). The phrase “Cornell-type double-well” is most defensible when it refers either to a modified Cornell interaction that genuinely develops two minima, or to a hybrid model in which Cornell-inspired localized states are embedded in an explicit double-well Hamiltonian, as in the hydrogen-bond tunneling framework (Pathak, 30 Oct 2025). Beyond that, much of the operative theory currently comes from generic double-well analysis: two-state reduction, exponentially small tunnel splitting, extreme sensitivity to tiny asymmetry, pseudopotential barriers generated by orthogonality, and, in the many-body regime, effective Bose–Hubbard dynamics (Dauphinee et al., 2015, Ibrahim et al., 2017, Olgiati et al., 2021).