Localisation in coupled quartic oscillator (Q2734904)

The authors study the localisation of eigenfunctions in two different model Hamiltonian systems representing coupled quartic oscillators. The Hamiltonian has the form \(H(x,y,p_x,p_y;\alpha)\) \(=\) \((2m)^{-1}(p_x^2+p_y^2)+V(x,y;\alpha)\), where \(\alpha \) is a parameter. The classical dynamics can be regular or chaotic depending on \(\alpha \). In particular, the authors investigate the coupled quartic oscillator given by the homogeneous potential \(V(x,y;\alpha)=x^4+y^4+\alpha x^2y^2 \) and \(m=1/2\). The term ``localisation'' is used for those particular class of wavefunctions of quantum systems which show selective probability density enhancements and exhibit, roughly spoken, the non-ergodic behavior of quantum wavefunctions. Scarring localisation, as it is known, is attributed to the influence of certain isolated least unstable classical periodic orbits in the system. Scarred states which have strong systematization and are scarred by short periodic orbits are called here localised states. It is shown that the states are exponentially localised in the unperturbed basis. The degree of localisation is strongly correlated with the classical stability and with the local structure of scarring orbits.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00047].