We study a two-degree-of-freedom model with a velocity-coupling term and an inverse-square interaction. The classical dynamics becomes transparent after a change of variables that separates a conserved “drift-like” quantity from an Ermakov–Pinney-type radial equation. Quantization in the same variables leads to a solvable singular-oscillator equation whose normalizability requires contour (Stokes-wedge) boundary conditions rather than the real axis.

Classical Dynamics and Hamiltonian Structure

Start from the Lagrangian

\[L=\dot{x}\dot{y}+\omega(x\dot{y}-y\dot{x})-\omega^2xy-\frac{\epsilon}{2}(x^2+y^2)-\frac{g}{2(x-y)^2}.\]

The conjugate momenta are

\[P_x=\frac{\partial L}{\partial \dot{x}}=\dot{y}-\omega y,\qquad P_y=\frac{\partial L}{\partial \dot{y}}=\dot{x}+\omega x.\]

Eliminating $\dot{x},\dot{y}$ gives the Hamiltonian

\[H=P_xP_y+\omega(yP_y-xP_x)+(\omega^2-\epsilon)xy+\frac{g}{2(x-y)^2}+\frac{\epsilon}{2}(x^2+y^2).\]

The Euler–Lagrange equations yield coupled second-order dynamics with a singular inverse-cube force:

\[\ddot{x}+2\omega\dot{x}+\omega^2x+\epsilon y+\frac{g}{(x-y)^3}=0,\] \[\ddot{y}-2\omega\dot{y}+\omega^2y+\epsilon x-\frac{g}{(x-y)^3}=0.\]

Introduce symmetric/antisymmetric variables

\[z_1=x+y,\qquad z_2=x-y.\]

Then the equations become

\[\ddot{z}_1+(\omega^2+\epsilon)z_1+2\omega\dot{z}_2=0,\] \[\ddot{z}_2+(\omega^2-\epsilon)z_2+2\omega\dot{z}_1+\frac{2g}{z_2^3}=0.\]

A key gain is conceptual: $z_2$ carries the singular interaction, while $z_1$ couples only through first derivatives, so conserved combinations can emerge under special parameter choices.

Rational Calogero Reduction and Integrability

For the rational Calogero case,

\[\epsilon=-\omega^2,\]

the system simplifies to

\[\ddot{z}_1+2\omega\dot{z}_2=0,\] \[\ddot{z}_2+2\omega^2 z_2+2\omega\dot{z}_1+\frac{2g}{z_2^3}=0.\]

The first equation integrates once to a conserved quantity

\[\Pi=\dot{z}_1+2\omega z_2=\text{constant}.\]

Using $\dot{z}_1=\Pi-2\omega z_2$ in the second equation gives a single forced Ermakov–Pinney-type equation for $z_2$:

\[\ddot{z}_2+\Omega^2 z_2+\frac{2g}{z_2^3}=-2\omega\Pi,\]

with $\Omega$ an effective frequency determined by the reduced coefficients in the $z_2$ equation. In particular, when

\[\Pi=0,\]

one obtains the standard Ermakov–Pinney form

\[\ddot{z}_2+\Omega^2 z_2+\frac{2g}{z_2^3}=0.\]

A convenient explicit solution can be written in the amplitude-squared form

\[z_2(t)=\left[\frac{1}{b^2\Omega^2}\left((a^2b^2-2g)\sin^2(\Omega t)+\frac{2ab}{\Omega}\sin(\Omega t)\cos(\Omega t)+b^2\cos^2(\Omega t)\right)\right]^{\frac{1}{2}}.\]

Once $z_2(t)$ is known, $z_1(t)$ follows from the conserved $\Pi$:

\[z_1(t)=\int\left(\Pi-2\omega z_2(t)\right)\,dt+I.\]

For $\Pi=0$, the integral typically reduces to an elliptic integral of the second kind, so $z_1(t)$ is naturally expressed through an elliptic function $E(\phi,k^2)$. The classical system is therefore integrable in the Liouville sense with conserved quantities $H$ and $\Pi$, and the reduction shows explicitly how the singular interaction enters only through the $z_2$ sector.

Quantum Reduction, Spectrum, and Stokes-Wedge Normalization

Quantization in the $(z_1,z_2)$ variables is imposed via the canonical commutators

\[[z_1,P_{z_1}]=i,\qquad [z_2,P_{z_2}]=i,\]

with the Schrödinger representation

\[P_{z_1}=-i\partial_{z_1},\qquad P_{z_2}=-i\partial_{z_2}.\]

The Hamiltonian operator takes the form

\[\hat{H}=-(\partial_{z_1}^2-\partial_{z_2}^2)+i\omega(z_1\partial_{z_2}+z_2\partial_{z_1})-\frac{\omega^2}{2}z_2^2+\frac{g}{2z_2^2}-\frac{\omega^2}{4}(z_1^2-z_2^2).\]

A conserved operator corresponding to the classical constant $\Pi$ is

\[\hat{\Pi}=-2i\frac{\partial}{\partial z_1}+\omega z_2.\]

To separate variables, take $\psi$ to be an eigenfunction of $\hat{\Pi}$ with eigenvalue $k$:

\[\psi=\exp\left[\frac{i}{2}z_1(k-\omega z_2)\right]\tilde{\phi}(z_2).\]

Substitution into the Schrödinger equation $\hat{H}\psi=E\psi$ reduces the problem to a single ordinary differential equation for $\tilde{\phi}(z_2)$:

\[\tilde{\phi}''-\frac{1}{4}\Omega^2 z_2^2\tilde{\phi}+\frac{g}{2z_2^2}\tilde{\phi}+\frac{k}{4}(k-2\omega z_2)\tilde{\phi}=E\tilde{\phi}.\]

For the simplest sector $k=0$, this becomes a singular-oscillator equation

\[\tilde{\phi}''-\frac{1}{4}\Omega^2 z_2^2\tilde{\phi}+\frac{g}{2z_2^2}\tilde{\phi}=E\tilde{\phi}.\]

A ground-state ansatz consistent with the inverse-square singularity and Gaussian asymptotics is

\[\tilde{\phi}_0=z_2^\lambda e^{-C z_2^2},\]

leading to

\[E_0=-(2+4\lambda)C,\]

with parameter constraints

\[\lambda(\lambda-1)=-\frac{g}{2},\qquad C=\pm\frac{\Omega}{4}.\]

A systematic series solution is obtained by

\[\tilde{\phi}(z_2)=z_2^\lambda e^{-\frac{1}{4}\Omega z_2^2}\sum_{n=0}^{\infty}a_n z_2^n,\]

with recursion relation

\[a_{n+2}=\frac{(E+2C+4\lambda C)+4Cn}{(n+2)(n+1+2\lambda)}a_n.\]

Polynomial termination of the series yields a discrete spectrum

\[E=-2C(2n+1+2\lambda).\]

A crucial physical point is that the resulting wavefunctions are generally not normalizable on the real axis. Instead, normalization is defined along complex contours (Stokes wedges) chosen so that:

• the exponential factor decays at infinity,
• the physical states remain bounded,
• the inner product becomes well-defined in the selected complex domain.