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# 数学代写|数学物理代写Mathematical Physics代考|PHZ3113 Degenerate Fibres: Analytic Case

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## 数学代写|数学物理代写Mathematical Physics代考|Degenerate Fibres: Analytic Case

One natural question is what happens to the adiabatic limit if the form of the metric is generalized. This is related to issues below and there are several levels of ‘relaxation’ of the conditions. One might first consider perturbations of first order in $t$ which include cross terms between base and fibre such as $t d y \otimes d z$. Note that it does not really make sense to want these to have coefficients which only depend on the base variables. The effect of adding such a term can be quite dramatic, since it can change the invertibility properties of the adiabatic model operators although this is still an elliptic suspended family – the null spaces may no longer form a bundle, as they may not be smooth over the base. If the failure to be smooth is itself reasonably smooth, as discussed below, then something can be done. I don’t really know what happens to the null space in the general case – if anyone wants to try to work it out they are welcome to try and I am interested to discuss it!

Another possible ‘perturbation’ is to replace $h$ by a basic tensor – that is, to let its coefficients vary on the fibre. This is actually less of a problem than the introduction of cross terms and leads to a very similar structure, but the details have not been written up as far as I know.

Next consider the effect of adding a potential, for simplicity in the case of the Laplacian on functions. The simplest case is when the potential is real and nonnegative. If it is non-zero on any fibre then the model on that fibre is invertible. The opposite extreme to uniform degeneracy is when $V$ vanishes on isolated fibres and has non-zero Hessian (in the base variables) at every point on those fibres

Then $\Delta_t+V$ has an inverse with a weaker (optimal) bound than in a case such as (4) when the model operators are invertible
$$\left|\left(\Delta_t+V\right)^{-1}\right|_{L^2} \leq C t$$
In this case there are additional model operators at the singular fibres which are harmonic oscillators.

## 数学代写|数学物理代写Mathematical Physics代考|Eigenvalues for Triangles

One geometric setting closely related to the perturbation by a potential with isolated minima is the vertical collapse of a manifold with boundary. For example, if one takes a region in the plane between two $2 \pi$-periodic smooth curves, $\Omega=\left{(x, y) \in \mathbb{R}^2 ; L(x)<y<U(x)\right}$, and ‘collapses’ it by vertical scaling to $\Omega_t=\left{(x, y) \in \mathbb{R}^2 ; L(x)<y / t<U(x)\right}, 0<t \leq 1$, then consider the behaviour of the eigenvalues for the Dirichlet, or Neumann, problem. For the Dirichlet problem the smallest eigenvalue has an asymptotic expansion related to that above, in particular there are harmonic oscillator models, provided $U(x)-L(x)$ only has non-degenerated maxima.

The general problem of the behaviour of the eigenvalues for the Dirichlet problem for triangles as functions on moduli space remains open and certainly there is behaviour similar to this under vertical collapse, except that the harmonic oscillator is replaced by its ‘one-sided’ cousion, Airy’s operator.

# 数学物理代写

## 数学代写数学物理代写Mathematical Physics代考|Degenerate Fibres: Analytic Case

$$\left|\left(\Delta_t+V\right)^{-1}\right|_{L^2} \leq C t$$

## 数学代写数学物理代写Mathematical Physics代考|Eigenvalues for Triangles

\eft 的分隔符缺失或无法识别，然后考虑 Dirichlet 或 Neumann 问题的特征值的行为。对于狄利克雷问

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