--- title: Cornell-type Double-Well Potential url: https://www.emergentmind.com/topics/cornell-type-double-well-potential type: topic --- # Cornell-type Double-Well Potential 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 [2010.00832]. 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 [2511.01899]. ## 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 [2010.00832]. 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 [2001.07530]. 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 [2007.03439, 1908.05076]. 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 \(b>0\), \[ V(r)=-\frac{4\alpha_s}{3r}+br+c, \] the derivative is \[ \frac{dV}{dr}=\frac{4\alpha_s}{3r^2}+b>0 \qquad (r>0), \] so the potential is strictly increasing on the positive radial half-line [2010.00832]. It tends to \(-\infty\) as \(r\to 0^+\) because of the Coulomb singularity and to \(+\infty\) as \(r\to\infty\) because of the linear term. The constant \(c\) shifts the graph vertically and changes the zero-crossing radius \(r_0\), but it does not affect the force and cannot create local extrema [2010.00832]. 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 \(b=0.183\ \mathrm{GeV}^2\), constituent masses \(m_{u/d}=0.33\ \mathrm{GeV}\), \(m_s=0.483\ \mathrm{GeV}\), \(m_c=1.55\ \mathrm{GeV}\), \(m_b=4.97\ \mathrm{GeV}\), and a Coulomb-parent Dalgarno treatment, the quoted allowed domain is \[ 0.20 \le \alpha_s \le 0.64,\qquad -1.2 \le c \le -0.66, \] with \(\alpha_s\le 0.75\) required for the reality of the Dirac factor \(\epsilon\) [2010.00832]. 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 \(r=0\) while preserving the long-distance linear term, but still does not introduce a barrier separating two minima [2001.07530]. A literal Cornell-type double well therefore requires added structure beyond the standard \(-\kappa/r+\sigma r+C\) 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, \[ \psi_{rel+conf}(r)\sim N' e^{-r/a_0}\left(C' - \frac{\mu b a_0}{2}r^2\right)\left(\frac{r}{a_0}\right)^{-\epsilon}, \] with \(a_0\), \(b\), \(C'\), \(\epsilon\), \(N'\), and reduced mass \(\mu\) controlling length scale, confinement, short-range correction, and normalization [2511.01899]. A two-well picture is then created by shifting this localized state to the left and right, \[ \psi_L(x)=\psi_{rel+conf}(|x+d/2|),\qquad \psi_R(x)=\psi_{rel+conf}(|x-d/2|), \] and estimating the tunnel splitting through the overlap \[ S(d)=\int_{-\infty}^{\infty}\psi_L(x)\psi_R(x)\,dx,\qquad \Delta E\approx 2S(d)E_0. \] The explicit numerical Schrödinger equation, however, is solved in the quartic double well \[ V(x)=V_0\frac{(x^2-a^2)^2}{a^4},\qquad a=\frac d2, \] not in a Cornell potential [2511.01899]. A second mechanism is effective barrier formation by orthogonality. In the one-dimensional infinite square well split by a central \(\delta\)-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 \[ V_{\rm ps}=V+(E-E_b)|\phi_b\rangle\langle\phi_b|, \] with nonlocal kernel \[ V_{\rm nl}(x,x')=(E-E_b)\phi_b(x)\phi_b(x'), \] which is repulsive in the projected subspace because \(E_b<0\) [1801.00648]. 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 [2511.01899]. ## 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 \[ V(x)=V_0\frac{(x^2-a^2)^2}{a^4}, \] the minima are at \(x=\pm a=\pm d/2\), the central barrier is at \(x=0\), and \(V(0)=V_0\) [2511.01899]. Near either minimum, the potential is locally harmonic with curvature \[ V''(\pm a)=\frac{8V_0}{a^2}, \qquad \omega=\sqrt{\frac{8V_0}{\mu a^2}}, \] so the low-lying states may be viewed as two weakly coupled local oscillators [2511.01899]. The tunnel splitting is controlled by barrier width, barrier height, and mass. In the same model, the forbidden-region tail is taken as \[ \psi(r)\sim A e^{-\kappa r},\qquad \kappa=\sqrt{\frac{2\mu(V_b-E)}{\hbar^2}}, \] which gives \[ S(d)\approx \tilde A e^{-\kappa d}, \qquad \Delta E \sim e^{-d\sqrt{2\mu(V_b-E)}/\hbar}. \] The WKB form is \[ \Delta E \propto E_0 \exp\Big[-\frac{2}{\hbar}\int_{-x_1}^{x_1}\sqrt{2\mu(V(x)-E)}\,dx\Big]. \] For \(d=2.7\) Å and \(V_0=0.05,0.10,0.15\) eV, the reported splittings are \[ \Delta E_H = 1.2\times 10^{-3},\ 5.0\times 10^{-5},\ 1.2\times 10^{-7}\ \text{eV}, \] \[ \Delta E_D = 1.5\times 10^{-4},\ 2.0\times 10^{-6},\ 1.8\times 10^{-10}\ \text{eV}, \] showing the expected exponential isotope suppression as \(\mu\) increases [2511.01899]. A small asymmetry can qualitatively reorganize this two-state sector. In the square double-well toy model with effective Hamiltonian \[ H_{\rm eff}= \begin{pmatrix} E_L & -t\\ -t & E_R \end{pmatrix}, \qquad \delta=\frac{E_L-E_R}{2}, \] the eigenvalues are \[ E_{\pm}=\frac{E_L+E_R}{2}\pm\sqrt{\delta^2+t^2}, \] and the ground-state weights are \[ |c_L|^2=\frac12\left(1-\frac{\delta}{\sqrt{\delta^2+t^2}}\right),\qquad |c_R|^2=\frac12\left(1+\frac{\delta}{\sqrt{\delta^2+t^2}}\right). \] The relevant control parameter is \(\delta/t\), not the asymmetry relative to the barrier height [1505.03164]. In a case with \(V_0/E_1^0=500\), \(b/a=0.2\), \(V_L=0\), and a tiny perturbation \(V_R=-10^{-5}E_1^0\), the fitted tunnel coupling is only \(t\approx 6.84\times 10^{-7}E_1^0\), so the ground state localizes almost entirely in the right well despite the perturbation being only more than \(1\) part in \(50{,}000{,}000\) relative to the barrier scale [1505.03164]. 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 \[ V_D(x)=\min[(x+d)^2,(x-d)^2], \] with minima at \(x=\pm d\) and central barrier \(V_D(0)=d^2\) [2209.09445]. Because each half-line is a shifted oscillator, the eigenfunctions are piecewise square-integrable \(U\)-type combinations of confluent hypergeometric functions, and the even/odd energies are given exactly as zeros of corresponding matching conditions. As \(d\) increases, the spectrum forms the familiar quasi-degenerate even/odd pairs. For example, at \(d=3\), \[ E_0^e=0.999551,\qquad E_0^o=1.00039, \] \[ E_1^e=2.99252,\qquad E_1^o=3.00604, \] which explicitly resolves tunneling-induced splitting in an analytically tractable model [2209.09445]. 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 \[ \psi_n(x)=\sqrt{\frac{2}{a}}\sin\!\left(\frac{n\pi x}{a}\right), \] leading to \[ \sum_m H_{nm}c_m=Ec_n, \qquad H_{nm}=\delta_{nm}E_n^{(0)}+\frac{2}{a}\int_0^a dx\,\sin\!\left(\frac{n\pi x}{a}\right)V(x)\sin\!\left(\frac{m\pi x}{a}\right). \] For smooth double wells, truncation at \(N=20\) already gives ground-state energies accurate to better than \(0.2\%\), whereas the square double well requires about \(50\) basis states; the main spectrum in that study uses \(N=200\) [1209.2521]. The same work shows that WKB is remarkably accurate for tunnel-split doublets and remains good for excited states [1209.2521]. For the quartic hydrogen-bond model, the explicit numerical solution uses finite differences and ARPACK for \[ \left[-\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2}+V_0\frac{(x^2-a^2)^2}{a^4}\right]\psi_n(x)=E_n\psi_n(x), \] with convergence checks at \(V_0=0.10\) eV showing splittings \(5.2\times10^{-5}\), \(5.0\times10^{-5}\), \(4.98\times10^{-5}\), \(4.96\times10^{-5}\), and \(4.98\times10^{-5}\) eV for increasingly refined \((N,L)\) choices [2511.01899]. 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 \(B_c\) study solves \[ -\frac{1}{2\mu}\frac{d^2u_{n\ell}}{dr^2} +\left[\frac{\ell(\ell+1)}{2\mu r^2}+V(r)\right]u_{n\ell} =E_{n\ell}u_{n\ell}, \] and combines VMC trial-state optimization with fixed-node GFMC projection. Because the formalism depends on the input \(V(r)\) rather than the Cornell form alone, it is structurally adaptable to modified radial Cornell-type potentials; that study adopts \(r_{\max}=15\ \mathrm{GeV}^{-1}\) and \(N_r=10^3\) after plateau tests, and removes short-time bias by fitting \[ E_{\mathrm{mix}}(\Delta\tau)=E_0+a\,\Delta\tau+b\,(\Delta\tau)^2 \] across a ladder of time steps [2511.10986]. 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 \[ H_N := \sum_{j=1}^N\bigl(-\Delta_j + V_{\mathrm{DW}}(x_j)\bigr) +\frac{\lambda}{N-1}\sum_{1\le i