Resonance widths for the molecular predissociation (Q459052)

Of interest here are \(2 \times 2\)-matrix Schrödinger operators of the form NEWLINE\[NEWLINEP = -h^2 \Delta + \mathrm{diag}\,(V_1(x),V_2(x)) + hR(x,hD_x), \tag{1}NEWLINE\]NEWLINE for \(x \in \mathbb R^n\), where \(V_1\) and \(V_2\) are bounded, real analytic potentials, \(R(x,hD_x) = (r_{j,k}(x, hD_x))_{1\leq j,k \leq 2}\) is a symmetric matrix of first-order pseudodifferential operators with analytic symbols, and \(h>0\) is a semiclassical parameter.NEWLINENEWLINEThe authors also make the following main assumptions on \(V_1\) and \(V_2\), which describe so-called molecular predissociation in the framework of the Born-Oppenheimer approximation giving rise to operators of the form (1) (we omit here certain other technical assumptions on \(V_1\), \(V_2\), and \(R\)): \(V_1(0)>0\), \(\lim_{|x| \to \infty} V_1(x)\) exists and is negative, \(E=0\) is a non-trapping energy for \(V_1\); \(V_2\) attains a unique, non-degenerate minimum of \(0\) at \(x=0\), \(\mathrm{Hess}\,V_2(0)>0\), and \(\liminf_{|x| \to \infty} V_2 > 0\).NEWLINENEWLINEThe goal is to study the behaviour of the imaginary part of the first resonance \(\rho_1\) of \(P\) in the semiclassical limit, as \(h \to 0\). More precisely, denoting by \(e_1\) the first eigenvalue of the harmonic oscillator \(-\Delta + \langle V_2''(0)x,x\rangle/2\), for the unique resonance \(\rho_1\) of \(P\) satisfying \(\rho_1 = (e_1 + r_{2,2}(0,0))h + \mathcal{O}(h^2)\), the authors obtain a complete asymptotic expansion of the form NEWLINE\[NEWLINE\mathrm{Im}\,\rho_1 \sim -h^{n_0+(1-n_\Gamma)/2} \sum_{0\leq m \leq l} f_{l,m} h^l (-\ln h)^m e^{-S/h}NEWLINE\]NEWLINE as \(h \to 0\), where \(f_{0,0}>0\), \(S>0\) is the so-called Agmon distance, and \(n_0 \geq 1\), \(n_\Gamma \geq 0\) are integers depending on the geometry of the operator in question. This complements and extends results of \textit{M. Klein} [Ann. Phys. 178, 48--73 (1987; Zbl 0649.35024)].NEWLINENEWLINEThe proof uses a WKB construction, the principle difficulty being the crossing of energy levels (i.e., eigenvalues), a problem handled in the case of differential operators by \textit{P. Pettersson} [Asymptotic Anal. 14, No. 1, 1--48 (1997; Zbl 0885.35104)], the principle novelty being that the operators involved are of pseudodifferential type.