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I am currently looking at systems converging into quantum dynamics and I would like to understand this transition better. For simplicity I illustrate this on the Hilbert space $mathcal H:=ell^2(mathbb Z)^2.$

(For some very nice physical intuition about my problem, please see the explanation below by Carlo Beenakker).

Quantum evolution is usually of the form $T_t:=e^{Wt}$ where $W$ is a skew-adjoint operator. Usually in quantum dyanmics $W=iH$ where $H$ is then called the self-adjoint Hamiltonian.

The simplest way to realize a skew-adjoint operator is to choose a self-adjoint operator $H$ and define $W_H:=left(begin{matrix} 0 & H \
-H & 0 end{matrix}right).$ It is easy to verify that $sigma(W_H)=pm isigma(H).$

Now, if $Hu=-Delta u+u$ let's say, where $Delta$ is the positive Laplacian on $ell^2(mathbb Z),$ then one can define $H_N:=Hvert_{ell^2({-N,-N+1,...,N})}$ with some suitable choice of boundary conditions (let's say Dirichlet).

We then define an approximation of the operator $W_H$ by $A_{N}: ell^2(mathbb Z)^2 rightarrow ell^2(mathbb Z)^2$as above:

$$A_{N}(u,v)(n_1,n_2):= ((-Hu)(n_1)delta_{vert n_1vertle N}+v(n_2)delta_{n_2,N},(Hv)(n_2)delta_{vert n_2vertle N}))$$

The above choice of $A_N$, corresponds to matrices $W_{N}:=left(begin{matrix} 0 & H_N \
-H_N & E_{N} end{matrix}right)$ on $ell^2({-N,-N+1,...,N})^2$ with $E_N=(0,0,.,0,-1)$, that are just extended by zero to the entire space, so that everything is defined on a single space $ell^2(mathbb Z^2)$ rather than different ones.

In particular $sigma(W_N)=sigma(A_N)backslash {0}.$
The nice property of the above operators $W_N$ is that they are dissipative, i.e. $text{Real part}(sigma(W_N))<0.$

By strong convergence $A_{H_N}x rightarrow W_Hx$ we know that the dissipative dynamics converges strongly to the quantum dynamics $T_N(t)x:=e^{A_Nt}x rightarrow T(t)x:=e^{W_Ht}x.$

What this means is, that if we fix a state and evolve it under $T_N$ then the dynamics will become more and more quantum.

I would like to know whether in each dissipative system, there is actually an eigenmode that approximates the quantum dynamics, i.e. is it true that the spectrum approximates the imaginary axis $$lim_{N rightarrow infty}d(sigma(W_N),imathbb R)=0?$$

I should say that for this very particular example I did some simple numerics with MATLAB and the answer should be yes. Actually there is a lot of spectrum approximating the imaginary axis.

EDIT

According to Carlo Beenakker's answer, we'd even expect that
$$d(sigma(W_N),imathbb R)le C frac{1}{N}!$$

However, I'd even be happy to know whether for the above system, we converge at all.

This problem of "non-Hermitian" quantum mechanics has been studied in the context of random-matrix theory (RMT), see for example the review Random matrix approaches to open quantum systems. The particular case pointed to in the OP is when the Hermitian matrix ${cal H}=iW_{H}$ of high rank $N$ is perturbed by a positive-definite matrix $-ipi WW^{t}$ of low rank $delta Nll N$. The eigenvalues $E_n-iGamma_n$ of the effective Hamiltonian ${cal H}_{text{eff}}={cal H}-ipi WW^t$ describe resonant scattering from a heavy atom or quantum dot (with $E_n$ the center of the resonance and $Gamma_n$ its width). The special case in the OP where ${cal H}$ is block-off-diagonal is referred to as the case of "chiral symmetry" in the context of random-matrix theory. (The appropriate ensemble for real ${cal H}$ is the socalled "chiral orthogonal ensemble".)

Now the question in the OP is the distribution of the $Gamma_n$'s in the limit $Nrightarrowinfty$ at fixed $delta N$. The "universal" result of random-matrix theory (see section IV.A in the cited review) is that for $delta Nll N$ all $Gamma_n$'s are greater than a minimal value $Gamma_{text{min}}simeqdelta N/N$ and they accumulate near that value.
_{Note that the OP refers to $i$ times the eigenvalues of ${cal H}$, so this approach to the real axis corresponds to an approach to the imaginary axis in the OP.}

Hence my answer to the final question of the OP, "does the spectrum approach the imaginary axis in the large-$N$ limit" is yes, it does, if the non-Hermitian perturbation has a rank $delta N$ that remains small compared to the rank of the Hermitian part. There remains, however, for any finite $N$ a gap of order $delta N/N$ that separates the "dissipative dynamics" from the unitary quantum mechanical evolution.

I show a plot that illustrates the clustering of the eigenvalues $E_n-iGamma_n$ of ${cal H}_{text{eff}}$ near the $Gamma=0$ axis, with a sharp threshold (blue horizontal line) and a gap.
_{In the plot $N=500$ and $delta N=50$. There is no chiral symmetry, but that would only introduce a $pm E_n$ symmetry, it would not affect the gap in the $Gamma$'s.}

source: arXiv:1405.6896