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Since $P(G)\propto{\rm exp}\bigl(-\frac{1}{2}{\rm Tr}\,GG^{\rm T}\bigr)$ is invariant under orthogonal transformations, if $A$ is real Hermitian you can work in a basis where $A$ is diagonal, with the …

This is first order perturbation theory: a perturbation $\delta A$ to a Hermitian matrix $A$ gives to first order a correction $\delta \lambda$ to an eigenvalue $\lambda$ (with corresponding eigenvec …

Here is a counterexample: consider the real symmetric matrix $$A=\begin{pmatrix} \cos\phi&\sin\phi\\ \sin\phi&-\cos\phi \end{pmatrix}.$$ The eigenvectors $v_\pm$ for the eigenvalues $\pm 1$ are $$v_\p …

The functions $\lambda(x)$ and $w(x)$ obey a set of coupled first-order differential equations in $x$, derived in Structure of trajectories of complex matrix eigenvalues (Bohigas, De Carvalho, and Pat …

Kato's theory has been extended to include degenerate eigenvalues by Hunziker and Pillet, Degenerate asymptotic perturbation theory (1983). These beautiful results [of Kato and others] are not qui …

Since the eigenvalues of a real matrix $A(\theta)$ come in complex conjugate pairs, an eigenvalue on the real axis with multiplicity 1 cannot move off the real axis when $\theta$ is varied over a smal …

since you want $f$ to be the same for all $A$, let's first take the special case $A=0$; then the eigenvalues of $A+\epsilon G$ have an average spacing of order $\epsilon/\sqrt{n}$, but the fraction of …

Let me consider the case of the Schrödinger equation, $L=-\nabla^2+V(x)$. Then the operator $L+ikx$ has PT-symmetry, meaning that it is invariant under the combined action of inversion $x\mapsto-x$ (p …

The Fourier transform $F_\epsilon(x,t)$ has a closed-form expression in terms of hypergeometric functions, $$F_\epsilon(x,t)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx}e^{i\epsilon^4 tk^4}\,dk=$$ $$\q …

Theorem (1.1) of Perturbation theory for normal operators is likely what you are looking for. See also Differentiable perturbation of unbounded operators. If you would order the eigenvalues by their m …

Here is a log-log plot of $$\delta I=I_{\text{appr}}-\int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx,$$ as a function of $\mu$, with $$I_{\text{appr}} …

First order perturbation theory gives you $$\frac{1}{\lambda_{\rm min}(A+\epsilon B)}=\frac{1}{\lambda_{\rm min}(A)}-\frac{\epsilon}{\lambda_{\rm min}(A)^2}\langle v_0|B|v_0\rangle+{\cal O}(\epsilon^2 …

If $H$ is fully degenerate, all eigenvalues are identical, it means that $H$ is proportional to the unit matrix. The zeroth order eigenstates can be chosen as any orthonormal basis. Degenerate perturb …