Adiabatic limits, Theta functions, and geometric quantization

Let $\pi\colon (M,\omega)\to B$ be a (non-singular) Lagrangian torus fibration on a compact, complete base $B$ with prequantum line bundle $(L,\nabla^L)\to (M,\omega)$. For a positive integer $N$ and a compatible almost complex structure $J$ on $(M,\omega)$ invariant along the fiber of $\pi$, let $D$ be the associated Spin${}^c$ Dirac operator with coefficients in $L^{\otimes N}$. Then, we give an orthogonal family $\{ {\tilde \vartheta}_b\}_{b\in B_{BS}}$ of sections of $L^{\otimes N}$ indexed by the Bohr-Sommerfeld points $B_{BS}$, and show that each ${\tilde \vartheta}_b$ converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber $\pi^{-1}(b)$ and the $L^2$-norm of $D{\tilde \vartheta}_b$ converges to $0$ by the adiabatic(-type) limit. Moreover, if $J$ is integrable, we also give an orthogonal basis $\{ \vartheta_b\}_{b\in B_{BS}}$ of the space of holomorphic sections of $L^{\otimes N}$ indexed by $B_{BS}$, and show that each $\vartheta_b$ converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber $\pi^{-1}(b)$ by the adiabatic(-type) limit. We also explain the relation of $\vartheta_b$ with Jacobi's theta functions when $(M,\omega)$ is $T^{2n}$.Comment: 41 page

JAPAN ALPS, its Physical Profiles and the Beginnings of People's Mountaineerings under the outward Looks of Worship

Article人文科学論集. 人間情報学科編 41: 171-183(2007)departmental bulletin pape

Land Use System Transformation with the Development of Tourism and some Environmental Problems in Shinshu District, an Inland Area of Central Japan

Article内陸文化研究 1: 81-90(2001)departmental bulletin pape

Appearance of similar triangles by certain operations on triangles

In this paper, a theorem about similar triangles is proved. It shows that two small and four large triangles similar to the original triangle can appear if we choose well among several intersections of the perpendicular bisectors of the sides with perpendicular lines of sides passing through the vertices of the triangle

From photoelectron detachment spectra of BrHBr−, BrDBr− and IHI−, IDI− to vibrational bonding of BrMuBr and IMuI

Photoelectron detachment XLX−(0000) + hν → XLX(vib) + e − + KER (X = Br or I, L = H or D) at sufficiently low temperatures photoionizes linear dihalogen anions XLX− in the vibrational ground state (v 1 v 2 l v 3 = 0000) and prepares the neutral radicals XLX(vib) in vibrational states (vib). At the same time, part of the photon energy (hν) is converted into kinetic energy release (KER) of the electron [R. B. Metz, S. E. Bradforth, and D. M. Neumark, Adv. Chem. Phys. 81, 1 (1992)]. The process may be described approximately in terms of a Franck-Condon type transfer of the vibrational wavefunction representing XLX−(0000) from the domain close to the minimum of its potential energy surface (PES) to the domain close to the linear transition state of the PES of the neutral XLX. As a consequence, prominent peaks of the photoelectron detachment spectra (pds) correlate with the vibrational energies E XLX,vib of states XLX(vib) which are centered at linear transition state. The corresponding vibrational quantum numbers may be labeled vib = (v 1 v 2 l v 3) = (000 v 3). Accordingly, the related most prominent peaks in the pds are labeled v 3. We construct a model PES which mimics the “true” PES in the domain of transition state such that it supports vibrational states with energies E XLX,pds,000v3 close to the peaks of the pds labeled v 3 = 0, 2, and 4. Subsequently, the same model PES is also used to calculate approximate values of the energies E XMuX,0000 of the isotopomers XMuX(0000). For the heavy isotopomers XHX and XDX, it turns out that all energies E XLX,000 v 3 are above the threshold for dissociation, which means that all heavy XLX(000 v 3) with wavefunctions centered at the transition state are unstable resonances with finite lifetimes. Turning the table, bound states of the heavy XLX are van der Waals (vdW) bonded. In contrast, the energies E XMuX,0000 of the light isotopomers XMuX(0000) are below the threshold for dissociation, with wavefunctions centered at the transition state. This means that XMuX(0000) are vibrationally bonded. This implies a fundamental change of the nature of chemical bonding, from vdW bonding of the heavy XHX, XDX to vibrational bonding of XMuX. For BrMuBr, the present results derived from experimental pds of BrHBr− and BrDBr− confirm the recent discovery of vibrational bonding based on quantum chemical ab initio calculations [D. G. Fleming, J. Manz, K. Sato, and T. Takayanagi, Angew. Chem., Int. Ed. 53, 13706 (2014)]. The extension from BrLBr to ILI means the discovery of a new example of vibrational bonding. These empirical results for the vibrational bonding of IMuI, derived from the photoelectron spectra of IHI− and IDI−, are supported by ab initio simulations of the spectra and of the wavefunction representing vibrational bonding of IMuI