  We consider equal-mass quantum Toda lattice with balanced loss-gain for two
and three particles. The two-particle Toda lattice is integrable and two
integrals of motion which are in involution have been found. The bound-state
energy and the corresponding eigenfunctions have been obtained numerically for
a few low-lying states. The three-particle quantum Toda lattice with balanced
loss-gain and velocity mediated coupling admits mixed phases of integrability
and chaos depending on the value of the loss-gain parameter. We have obtained
analytic expressions for two integrals of motion which are in involution.
Although an analytic expression for the third integral has not been found, the
numerical investigation suggests integrability below a critical value of the
loss-gain strength and chaos above this critical value. The level spacing
distribution changes from the Wigner-Dyson to the Poisson distribution as the
loss-gain parameter passes through this critical value and approaches zero. An
identical behaviour is seen in terms of the gap-ratio distribution of the
energy levels. The existence of mixed phases of quantum integrability and chaos
in the specified ranges of the loss-gain parameter has also been confirmed
independently via the study of level repulsion and complexity in higher order
excited states.
