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We prove that the spectrum of the linear delay differential equation$x'(t)=A_{0}x(t)+A_{1}x(t-\tau_{1})+\ldots+A_{n}x(t-\tau_{n})$ with multiplehierarchical large delays $1\ll\tau_{1}\ll\tau_{2}\ll\ldots\ll\tau_{n}$ splitsinto two distinct parts: the strong spectrum and the pseudo-continuousspectrum. As the delays tend to infinity, the strong spectrum converges tospecific eigenvalues of $A_{0}$, the so-called asymptotic strong spectrum.Eigenvalues in the pseudo-continuous spectrum however, converge to theimaginary axis. We show that after rescaling, the pseudo-continuous spectrumexhibits a hierarchical structure corresponding to the time-scales$\tau_{1},\tau_{2},\ldots,\tau_{n}.$ Each level of this hierarchy isapproximated by spectral manifolds that can be easily computed. The set ofspectral manifolds comprises the so-called asymptotic continuous spectrum. Itis shown that the position of the asymptotic strong spectrum and asymptoticcontinuous spectrum with respect to the imaginary axis completely determinesstability. In particular, a generic destabilization is mediated by the crossingof an $n$-dimensional spectral manifold corresponding to the timescale$\tau_{n}$.

The Spectrum of Delay Differential Equations with Multiple Hierarchical Large Delays