Computation and Verification of Spectra for Non-Hermitian Systems

TL;DR

The work identifies two fundamental obstacles to computing spectra for broad non-Hermitian operators and introduces locally trivial pseudospectra (LTP) to regain computability. It proves impossibility results under general conditions, then constructs a practical, verifiable approach using rectangular truncations and resolvent bounds that yields error-controlled eigenvalues and eigenfunctions. The authors demonstrate the method by obtaining verified eigenpairs for the imaginary cubic oscillator H_B = p^2 + i x^3, including the 100th eigenvalue, while avoiding spurious modes induced by truncation. The framework generalizes to a wide class of non-Hermitian operators, offering a rigorous bridge between quantum mechanics and computation with broad potential applications in PT-symmetric and nonlocal problems.

Abstract

We establish a connection between quantum mechanics and computation, revealing fundamental limitations for algorithms computing spectra, especially in non-Hermitian settings. Introducing the concept of locally trivial pseudospectra (LTP), we show such assumptions are necessary for spectral computation. LTP adapts dynamically to system energies, enabling spectral analysis across a broad class of challenging non-Hermitian problems. Exploiting this framework, we overcome a longstanding obstacle by computing the eigenvalues and eigenfunctions of the imaginary cubic oscillator $H_{\mathrm{B}} = p^2 + i x^3$ with error bounds and no spurious modes -- yielding, to our knowledge, the first such error-controlled result. We confirm, for instance, the 100th eigenvalue as $627.6947122484365113526737029011536\ldots$. Here, truncation-induced $\mathcal{PT}$-symmetry breaking causes spurious eigenvalues -- a pitfall our method avoids, highlighting the link between truncation and physics. Finally, we illustrate the approach's generality via spectral computations for a range of physically relevant operators. This letter provides a rigorous framework linking computational theory to quantum mechanics and offers a precise tool for spectral calculations with error bounds.

Figures (11)

• Figure 1: (a): Illustration of statement (A). We construct direct sums of Hermitian matrices with eigenvalues $\{0,2\}$ and long-range terms. Block sizes are chosen to 'trick' the sequence of algorithms into approximating an eigenvalue $1$. The red box shows the information read by the algorithm when making the approximation. (b): Illustration of statement (B). We use direct sums of Jordan matrices whose pseudospectra grow rapidly with size (top panel). By varying block sizes, the algorithm is forced to oscillate (green arrow) between $\{0\}$ (when sizes are bounded) and the unit disk (otherwise).
• Figure 2: (a): Pseudospectra approximations via truncated Hermite expansions (square truncations), which break $\mathcal{PT}$-symmetry and produce spurious eigenvalues (yellow crosses). (b,c): Rectangular truncations preserve $\mathcal{PT}$-symmetry, yield convergent approximations, and remain within the true pseudospectrum. Color indicates $\epsilon=\|(H_{\mathrm{B}}-zI)^{-1}\|^{-1}$. (d,e): Eigenfunction oscillations. (f): Exponential instability of eigenvalues, with $\lim_{n\rightarrow\infty}\kappa_n\exp(-n\pi/\sqrt{3})n^{1/4}\approx 0.176$.
• Figure 3: Left: Square truncations of $H_{\mathrm{B}}$'s infinite matrix (blue dots indicate nonzero entries) omit important, physically relevant interactions. A direct check shows that the truncations have broken $\mathcal{PT}$-symmetry. Right: Rectangular truncations capture these terms, enabling verified spectra and pseudospectra. Properties of $H_\mathrm{B}$ (including unbroken $\mathcal{PT}$-symmetry) are preserved under rectangular truncations when we restrict to the subspace (range of $\mathcal{P}_N$).
• Figure 4: The proof of Eq. \ref{['eqmain_theorem']} (S.M.) establishes LTP on vertical strips (shaded gray) between eigenvalues. To the left of each strip, we bound eigenvalue contributions using pole bounds $\kappa_m/|\lambda_m-z|$ and truncated operators $\mathcal{C}_m$. To the right, we bound the semigroup generated by $-H_{\mathrm{B}}$. This approach extends to other operators with LTP.
• Figure 5: (a): Approximations of $\|(H_{\mathrm{B}}-xI)^{-1}\|^{-1}$ using $\mathrm{Sp}_\epsilon(\mathcal{P}_{N+3}H_{\mathrm{B}}\mathcal{P}_N)$, which converge from above as $N\rightarrow\infty$, free of spurious eigenvalues. (b): Rescaled $\|(H_{\mathrm{B}}-xI)^{-1}\|^{-1}$, which are upper bounds on $\mathrm{dist}(x,\mathrm{Sp}(H_{\mathrm{B}}))$ used to compute $\mathrm{Sp}(H_{\mathrm{B}})$ with error bounds.

Theorems & Definitions (18)

• Definition 1: Computational problem
• Example 1
• Definition 2: General algorithm
• Remark 1
• Theorem A
• proof
• Remark 2: Extension to approximate eigenvectors
• Example 2: Instability of spectrum of Jordan blocks
• Theorem B
• proof