Exceptional Points of Infinite Order Giving a Continuous Spectrum

Abstract

The observation in the title made earlier in association with the Pais–Uhlenbeck oscillator with equal frequencies is illustrated for an elementary matrix model. In the limit $N \to \infty$ (N being the order of the exceptional point), an infinity of nontrivial states that do not change their norm during evolution appear. These states have real energies lying in a continuous interval. The norm of the “precursors” of these states at large finite N is not conserved, but the characteristic time scale where this nonconservation shows up grows linearly with N.

Appendix

We explain here why an attempt to construct a sequence of crypto-Hermitian matrix Hamiltonians with the same limit $N \to \infty$ as the limit H ^∗ of the Hamiltonians (5) fails.

Consider the normalized eigenstates of H ^∗, $|\epsilon \rangle : a_{j} = \sqrt {1-\epsilon ^{2}} \, \epsilon ^{j}$ with real 𝜖∈(−1,1). We have

$$\begin{array}{@{}rcl@{}} \langle \epsilon|\mu \rangle = \frac {\sqrt{(1-\epsilon^{2})(1-\mu^{2})}}{1-\epsilon \mu}. \end{array} $$

(A.1)

Consider now a set of N=2M+1N-dimensional unitary vectors |n=−M,…,0,…,M〉 with the inner products

$$\begin{array}{@{}rcl@{}} \langle n | m \rangle \ =\ \frac {\sqrt{(1-{\epsilon_{n}^{2}})(1-{\epsilon_{m}^{2}})}}{1 - \epsilon_{n} \epsilon_{m}}, \epsilon_{n} = \frac n{M+1} \, . \end{array} $$

(A.2)

For example, for M=1, we choose

$$\begin{array}{@{}rcl@{}} |- \rangle = \left( \begin{array}{c} \frac {\sqrt{3}}2 \\ \frac 1{\sqrt{20}} \\ - \frac 1{\sqrt{5}} \end{array} \right), |0 \rangle = \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right), |+ \rangle \ =\ \left( \begin{array}{c} \frac {\sqrt{3}}2 \\ \frac 1{\sqrt{20}} \\ \frac 1{\sqrt{5}} \end{array} \right). \end{array} $$

(A.3)

Obviously, (A.2) represents a discretization of (A.1).

At the next step, we construct the matrix $(\tilde H)_{N}$ having the vectors |n〉 as eigenvectors with eigenvalues 𝜖 _n . This is possible to do as long as the system of N ² equations

$$\begin{array}{@{}rcl@{}} ({\tilde H}_{N})_{ij} a^{(n)}_{j} \ =\ \epsilon_{n} a^{(n)}_{i} \end{array} $$

(A.4)

for the matrix elements $({\tilde H}_{N})_{ij}$ is not degenerate. In other words, as long as the vectors |n〉 are linearly independent and the determinant of the N×N matrix made of the vector components does not vanish. For M=1, this determinant is equal to 0.2. This indicates that, though the vectors (A.3) are not coplanar, they are relatively close to being so. Finally, one can make these vectors orthogonal by redefining the norm in an appropriate way.

However, the larger M and N are, the more difficult is to carry on this program. It is more convenient to study not the determinant of the eigenvector components, but its square, the Gram determinant of the matrix of their scalar products. For M=1, the Gram deteminant Δ₁ of the scalar products (A.2) is equal to 0.04. For M=2, it is equal to 10⁻⁵, meaning that the corresponding five eigenvectors are “almost” linearly dependent. For M=3, it is already 10⁻¹¹. It roughly decays as ${\Delta }_{M} \sim e^{-1.2M^{2}}$.⁵ A matrix with an almost degenerate system of eigenvectors must have very large elements. The limit $N \to \infty$ is singular.