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# 数学代写|数学物理代写Mathematical Physics代考|PHYS3260 Adiabatic Limit with Isolated Degenerate Fibres

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## 数学代写|数学物理代写Mathematical Physics代考|Adiabatic Limit with Isolated Degenerate Fibres

Recall that the basic set up of an adiabatic limit corresponds to a fibration of manifolds – a submersion between compact connected manifolds $\phi: X \longrightarrow Y$ with typical fibre $Z$. On the total space $X$ one can consider a family of ‘adiabatic metrics’ $g_t=\phi^* h+t^2 \mu$ where $h$ is a metric on the base (it could also depend on $t$ ) and $\mu$ is a smooth symmetric 2 -cotensor on the fibres which restricts to be positive definite, and hence a metric, on the fibres. In fact I think it is more natural to consider a family of metrics such as $t^{-2} g_t$ where the fibres are of more-or-less constant size and fixed vectors in the base get ‘large’ which means the base is ‘slow’ (hence the term adiabatic). I believe Witten [4] was the first to consider global analysis related to such metrics when he examined the behaviour of the eta invariant for the particular case of a fibration over a circle.

Note that this setting is more general than a Riemannian submersion, which corresponds to the case that $\mu$ has rank exactly equal to the dimension of the fibres at each point and I will mention some other possible generalizations below.

For $t>0$ nothing much is happening, one just has a smooth family of metrics and $t$ is simply a parameter. On the other hand one can view the singular limit at $t=0$ as imposing a ‘geometry’. Then $t$ is no longer a true parameter but should be included in the analysis, so instead of $X$ consider the space $X \times[0,1]$ where the iterval corresponds to $t$. The basic object to consider is the space of smooth vector fields on $X \times[0,1]$
$$\mathcal{V}_{\mathrm{a}}(X)=\left{V ; V t=0, V \text { tangent to } \phi^{-1}(y) \text { at } t=0\right} \text {. }$$

## 数学代写|数学物理代写Mathematical Physics代考|Uniform Degeneracy

Although the ‘main’ solvability issue above appears to be the invertibility of the fibre Laplacian. Really, it is not quite this which is involved, or rather it is a little more than that. Namely what we actually get is a suspended Laplacian. This is the ‘adiabatic symbol’. Note that the ‘semiclassical’ case is when $\phi$ is the identity. Then the adiabatic, or semiclassical, symbol is the ‘full symbol’ of the Laplacian. In case of a general fibration $\phi$ it is a bundle, over $T^* Y$, of operators. For each point in the base we get a differential operator (on some form bundle) over the space $T_y Y \times Z_y$, where $Z_y$ is the fibre above $y$. This is a second order elliptic differential operator and is translation-invariant in $T_y Y$. We can take the Fourier transform and hence get a differential operator on $Z_y$ which is polynomial in $T_y^* Y$. The main issue is then is the invertibility of this ‘suspended’ family (suspended in the topological sense of having some Euclidean variables added). For an adiabatic metric such as described above this turns out to be straightforward, the null space can be decomposed (over forms from the base) in terms of the fibre null spaces which form an ordinary vector bundle, the forms on $Y$ twisted by the flat bundles of fibre harmonic forms.

So for the Laplacian on forms this model operator can never be fully invertible the constants in degree 0 always intervene. For other similar problems it can. For instance if the fibres are manifolds with boundary and one considers the Dirichlet boundary condition, then the adiabatic model operators are fully invertible and in consequence the full Laplacian is also invertible uniformly:-
$$\left|\Delta_{t, \text { Dir }}^{-1}\right|_{L^2} \leq C t^2 \text { as } t \downarrow 0 .$$
Then a more interesting question, touched on below, is the behaviour of the lowest eigenvalue and eigenstate.

# 数学物理代写

## 数学代写|数学物理代写数学物理代考|绝热极限与孤立简并纤维

$$\mathcal{V}_{\mathrm{a}}(X)=\left{V ; V t=0, V \text { tangent to } \phi^{-1}(y) \text { at } t=0\right} \text {. }$$

## 数学代写|数学物理代写数学物理代考|均匀简并

.

$$\left|\Delta_{t, \text { Dir }}^{-1}\right|_{L^2} \leq C t^2 \text { as } t \downarrow 0 .$$

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## MATLAB代写

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