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In this contribution, we review the most important physics presentedoriginally in our recent publications. Some new analyses, insights andperspectives are also provided. We showed recently that the symmetry energy$E_{sym}(\rho)$ and its density slope $L(\rho)$ at an arbitrary density $\rho$can be expressed analytically in terms of the magnitude and momentum dependenceof the single-nucleon potentials using the Hugenholtz-Van Hove (HVH) theorem.These relationships provide new insights about the fundamental physicsgoverning the density dependence of nuclear symmetry energy. Using the isospinand momentum (k) dependent MDI interaction as an example, the contribution ofdifferent terms in the single-nucleon potential to the $E_{sym}(\rho)$ and$L(\rho)$ are analyzed in detail at different densities. It is shown that thebehavior of $E_{sym}(\rho)$ is mainly determined by the first-order symmetrypotential $U_{sym,1}(\rho,k)$ of the single-nucleon potential. The densityslope $L(\rho)$ depends not only on the first-order symmetry potential$U_{sym,1}(\rho,k)$ but also the second-order one $U_{sym,2}(\rho,k)$. Both the$U_{sym,1}(\rho,k)$ and $U_{sym,2}(\rho,k)$ at normal density $\rho_0$ areconstrained by the isospin and momentum dependent nucleon optical potentialextracted from the available nucleon-nucleus scattering data. The$U_{sym,2}(\rho,k)$ especially at high density and momentum affectssignificantly the $L(\rho)$, but it is theoretically poorly understood andcurrently there is almost no experimental constraints known.

Relationship between the symmetry energy and the single-nucleon potential in isospin-asymmetric nucleonic matter