Quasimodes of the two-dimensional quartic oscillator (Q1282539)

The article treats the eigenvalue problem for the Schrödinger equation \[ \left(-{\hbar^2\over 2}\Delta+ V_\varepsilon \right) \psi=E\psi \] in \(L^2(\mathbb R^2)\), where \(V_\varepsilon (x_1,x_2)= {x^2_1x^2_2\over 2}+ {\varepsilon\over 4} (x^4_1+ x_2^4)\) is the quartic potential. The authors obtain explicit formulas for the quasimodes of this spectral problem. These formulas are uniform in the widest range of the parameter \(\varepsilon\). Proofs are omitted.