## Spectrum of a Hamiltonian which is a perturbation of Laplacian

Let $$\Delta =\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$$ be the Laplacian on $$\mathbb{R}^3$$. Consider an operator $$H$$ on complex valued functions on $$\mathbb{R}^3$$ $$H\psi=\Delta\psi(x) +i\sum_{p=1}^3A_p(x)\frac{\partial \psi(x)}{\partial x_p} +B(x)\psi(x),$$ where $$A_i,B$$ are smooth real valued functions.

I am looking for a precise result of the following approximate form: (1) if $$A_i$$ and $$B$$ are ‘small’ then the discrete spectrum of $$H$$ is non-positive. (2) If $$A_i,B$$ are ‘large’ then the discrete spectrum of $$H$$ contains necessarily a positive element.

## Solve using singular perturbation method

Solve the following using singular perturbation method:

εy′′ – y = 0, y(0) = 1, y(1) = 0

ε is a small parameter.

## Positive and negative powers of small parameter in perturbation problem

I’d like to perturbatively handle an eigenvalue problem similar to this: $$\lambda f = (\hat{H} + 1/\epsilon^2 \hat{V} + \epsilon {W}) f,$$ where $$f$$ is a function, $$\lambda$$ is an eigenvalue, $$\epsilon$$ is a small parameter, and the rest are linear (differential) operators. The problem is, that if one writes a series for the eigenvalue and the eigenfunction, $$f = f_0 + \epsilon f_1 + \epsilon^2 f_2 + …\ \lambda = \lambda_0 + \epsilon \lambda_1 + \epsilon^2 \lambda_2 + …,$$ one will get e.g. $$\lambda_0 f_0 = \hat{H} f_0 + \hat{V} f_2\ \lambda_1 f_0 + \lambda_0 f_1 = \hat{H} f_1 + \hat{W} f_0 + \hat{V} f_3\ …$$ i.e. the different orders of the series start to mix. Is there a way to develop a systematic perturbation theory for this case?

## What is an equilibrium called when a perturbation only moves it slightly?

In game theory, a equilibrium is:

Stable, if a perturbation form any of the players returns the equilibrium back to it’s original state

Unstable, if a perturbation moves the equilibrium away from the original state

Semi-stable if some of the players perturbations are acceptable.

What do you call an equilibrium if a perturbation merely moves it slightly.

Consider the following game: Two players must state a number. Each of them gets a payoff of \$ 1 if they state the same number, otherwise there is no payoff. (1,1) is an equilibrium. If one of the players chooses a different strategy, say 1.01. Then the Equilibrium is (1.01, 1.01). This does not seem unstable – neither does this seem stable. What is this type of equilibrium called?

## Perturbation with fraction

How to solve using perturbation method for small e?

$$\frac{d}{dx}((1-\frac{ey}{y+1})\frac{dy}{dx})=0$$

$$x=0,y=0$$

$$x=1,y=1$$

## Perturbation of the adiabatic limit

Let $$(M,g_M)$$ be a closed oriented Riemannian manifold that has a fibration structure $$Y \rightarrow M \overset{\pi}{\rightarrow} B$$ where $$(Y,g_Y)$$ and $$(B,g_B)$$ are closed Riemannian manifolds such that $$\pi$$ is a Riemannian submersion.

Now we define $$g_{\epsilon}=\epsilon^{-2}\pi^*g_B+g_Y$$ and $$\tilde g_{\epsilon}=g_{\epsilon}+\alpha_{\epsilon}$$ for some error term $$\alpha_{\epsilon}=O(\epsilon^{\tau-2})$$ for some $$\tau>0$$. If we denote the signature operators on $$M$$ with respect to $$g_{\epsilon}$$ and $$\tilde g_{\epsilon}$$ by $$A_{\epsilon}$$ and $$\tilde A_{\epsilon}$$ respectively. Can we conclude that

$$\lim_{\epsilon \to 0} \eta(A_{\epsilon})= \lim_{\epsilon \to 0} \eta(\tilde A_{\epsilon}),$$ if the former exists?

## Perturbation theory compact operator

Let $$K$$ be a compact self-adjoint operator on a Hilbert space $$H$$ such that for some normalized $$x \in H$$ and $$\lambda \in \mathbb C:$$

$$\Vert Kx-\lambda x \Vert \le \varepsilon.$$

It is well-known that this implies that $$d(\sigma(K),\lambda) \le \varepsilon.$$

However, I am wondering whether this implies also something about $$x.$$

For example, it seems plausible that $$x$$ cannot be orthogonal to the direct sum of eigenspaces of $$K$$ with eigenvalues that are close to $$\lambda.$$

In other words, are there any non-trivial restrictions on $$x$$ coming from the spectral decomposition of $$K$$?

## Resolvent estimate of compact perturbation of self-adjoint operator

Let $$T$$ be a selfadjoint operator on Hilbert space $$H$$. Then we know that there is a resolvent estimation $$\left\lVert (\lambda-T)^{-1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \rho(T).$$ But what if we want to consider the compact perturbation, that is, operator $$A = T+ D,$$
where $$D$$ is a compact operator on $$H$$. Is there any known results on the resolvent estimate? Does anyone know some related references ? Thank you very much.

## Uniform inequality for an analytic perturbation

Let $$T$$ be a bounded linear operator acting on a complex Banach space. Suppose that $$T$$ has spectral radius strictly less than $$1$$. If we introduce an analytic perturbation to $$T$$, $$s\mapsto T_s$$ for $$|s|<\epsilon$$ (with $$T_0 = T$$) then by upper-semicontinuity of the spectrum, assuming that $$\epsilon$$ is sufficiently small, the spectral radius of each $$T_s$$ is less than $$\rho$$ for some $$\rho <1$$. My question is the following:

Is it possible to find $$K>0$$ and $$\rho <\rho ‘<1$$ such that $$\|T_s^n\|\le K (\rho ‘)^n,$$ for all $$|s|<\epsilon$$ and $$n$$?

This certainly seems like it should be true but I can’t find a proof – I think I’m missing something obvious. Any help would be greatly appreciated – cheers!

## Stability of the second-order hyperbolic PDE add a perturbation

Here we assume that $$u=u^{\varepsilon}(x,t)$$ satisfies the following hyperbolic PDE problem,

$$\partial^2_t u^{\varepsilon}-\partial_x \big(a(x)\partial_x u^{\varepsilon} \big)+\varepsilon \partial_x u^{\varepsilon} =0,~(x\in\mathbb{R},t>0),$$ $$u^{\varepsilon}(x,0)=0,\quad \partial_t u^{\varepsilon}(x,0)=F_0(x).$$

Suppose that $$a(x)>0$$ is a smooth even periodic function, i.e., $$a(x+1)=a(x),~a(-x)=a(x)$$. The initial condition $$F_0(x)\in C^{\infty}( \mathbb{R})\cap L^2(\mathbb{R})$$.

If $$\varepsilon=0$$, we denote the above solution as $$u^0(x,t)$$. Our question is when $$\varepsilon>0$$, for any $$t\in[0,\frac1{\varepsilon}]$$, could we get the following estimate under the same initial condition $$\|u^{\varepsilon}(x,t)\|_{L^2(\mathbb{R})}\le C \sup_{\tau\in[0,\frac{1}{\varepsilon}]}\|u^0(x,\tau)\|_{L^2(\mathbb{R})} ~?$$

Where $$C$$ is independent of $$\varepsilon$$. When $$a(x)$$ is a constant, namely $$\mathcal{L}=-\partial_x \big(a(x)\partial_x\big)$$ is Laplacian, we can prove the above result by using Fourier tranform, Plancherel theorem and Gronwall inequality. How about the case of a general elliptic operator, is the result also the same? If so, does Riesz transform work here?