Narrow Results

We study the semiclassical distribution of resonances of a $2\times 2$ matrix Schrödinger operator (1.1) near a fixed energy level where the underlying classical trajectories ${\mathrm{\Gamma }}_1$ of $P_1$ and ${\mathrm{\Gamma }}_2$ of $P_2$ are, respectively, periodic and non-trapping. The aim is to compute the imaginary part of the resonances appearing near the eigenvalues created by $P_1$ when ${\mathrm{\Gamma }}_1$ intersects ${\mathrm{\Gamma }}_2$ with finite contact order $m$. Recent results [M. Assal, S. Fujiié and K. Higuchi, Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories, Int. Math. Res. Not.2024 (2024) 6879–6905] and [M. Assal, S. Fujiié and K. Higuchi, Transition of the semiclassical resonance widths across a tangential crossing energy-level, J. Math. Pures Appl.191 (2024) 103634] assert that the width of resonances is of polynomial order in the semiclassical parameter with exponent $1+2/(m+1)$, if the interaction $U=r_0(x)+ih{r}_1(x)D_x$ is elliptic at the crossing point. Here, we remove this ellipticity assumption and prove that the exponent is $1+2(k+1)/(m+1)$, where $k$ is a “vanishing order” of the interaction at the crossing points, suitably defined in terms of the vanishing orders of $r_0$ and $r_1$ depending on whether the crossing point is a turning point or not.

We study the time-dependent Schrödinger equation with matrix-valued potential presenting a generic crossing of type B, I, J or K in Hagedorn's classification. We use two-scale Wigner measures for describing the Landau–Zener energy transfer which occurs at the crossing. In particular, in the case of multiplicity 2 eigenvalues, we calculate precisely the change of polarization at the crossing. Our method provides a unified framework in which codimension 2, 3 or 5 crossings can be discussed. We recover Hagedorn's result for wave packets, from Wigner measure point of view, and extend them to any data uniformly bounded in L². The proof is based on a normal form theorem which reduces the problem to an operator-valued Landau–Zener formula.

We propose an alternative to the usual time-independent Born–Oppenheimer approximation that is specifically designed to describe molecules with non-symmetrical hydrogen bonds. In our approach, the masses of the hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. As a result, the different vibrational modes appear at different orders of approximation.

Although we develop a general theory, our analysis is motivated by an examination of the FHCl^- ion. We describe our results for it in detail.

We prove the existence of quasimodes and quasienergies for the nuclear vibrational and rotational motion to arbitrary order in the Born–Oppenheimer parameter ∊. When the electronic motion is also included, we provide simple formulas for the quasienergies up to order ∊³ that compare well with experiment and numerical results.

We consider a $2\times 2$ system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order $h^{\nu }$ in the off-diagonal part, where $h$ is a semiclassical parameter and $\nu$ is a constant larger than $1/2$. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by $Mhlog(1/h)$ and the coefficient $M$ is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.

Through the Born-Oppenheimer approximation and One-Pion-Exchange model, We have explored four doubly heavy tri-meson systems DD^*K, $D{D}^{*}K,D{\overline{D}}^{*}K,B{B}^{*}\overline{K}$ and $B{\overline{B}}^{*}\overline{K}$. After solving the three-body Schrödinger equation with considering the coupled-channel effects and S-D wave mixing, we find loosely bound molecular states for the four doubly heavy tri-meson systems. For instance, With an estimated error, the mass of the molecular states for the four doubly heavy tri-meson systems. For instance, With an estimated error, the mass of the DD^*K or $B{B}^{*}\overline{K}$ molecule is ${4317.92}_{-4.32}^{+3.66}$ MeV or ${11013.65}_{-9.02}^{+8.68}$ MeV. We also extend our investigation to the triply heavy tri-meson system BBB^* via the Born-Oppenheimer Potential method. By sharing one virtual pion, the BBB^* system has a loosely bound state as long as the molecular states of its two-body subsystem BB^* exist.

The following sections are included:

• Introduction
• Theoretical Background
• Avoided Crossing versus Conical Intersection
• Ultrafast Processes Involving Conical Intersections
• Ultrafast Dynamics through Avoided-Crossing of NaI
• Ultrafast Dynamics Through Conical Intersection of Ethylene
• References

We study non-adiabatic scattering transitions in the Born-Oppenheimer limit for a molecular Schrödinger operator in which the nuclei have one degree of freedom and the electron Hamiltonian is a 2 × 2 matrix.