[Paper Review] Pseudospectra of the Schroedinger operator with a discontinuous complex potential

Raphaël Henry, David Krejčiřı́k|arXiv (Cornell University)|Mar 9, 2015

Spectral Theory in Mathematical Physics26 references3 citations

TL;DR

This paper investigates the pseudospectral properties of the Schrödinger operator with a discontinuous imaginary potential $ i\,\mathop{\mathrm{sgn}}\nolimits(x) $ on $ \mathbb{R} $. Despite the potential being bounded and the spectrum lying in a half-strip, the resolvent norm blows up at infinity, leading to highly non-trivial pseudospectra—demonstrating strong spectral instability under small perturbations, in stark contrast to self-adjoint or weakly coupled systems.

ABSTRACT

We study spectral properties of the Schroedinger operator with an imaginary sign potential on the real line. By constructing the resolvent kernel, we show that the pseudospectra of this operator are highly non-trivial, because of a blow-up of the resolvent at infinity. Furthermore, we derive estimates on the location of eigenvalues of the operator perturbed by complex potentials. The overall analysis demonstrates striking differences with respect to the weak-coupling behaviour of the Laplacian.

Motivation & Objective

• To analyze the spectral instability of a non-self-adjoint Schrödinger operator with a discontinuous imaginary potential.
• To investigate the structure of pseudospectra in the absence of smoothness and in the presence of essential spectrum.
• To contrast the behavior of this operator with weak-coupling regimes of self-adjoint or smooth non-self-adjoint operators.
• To establish rigorous estimates on eigenvalue locations under complex perturbations using the Birman-Schwinger principle.
• To demonstrate that spectral instability arises even for bounded, discontinuous potentials, challenging classical expectations.

Proposed method

• Constructing the resolvent kernel explicitly for the Schrödinger operator $ H = -\frac{d^2}{dx^2} + i\,\mathop{\mathrm{sgn}}\nolimits(x) $ on $ L^2(\mathbb{R}) $, using solutions to the inhomogeneous equation.
• Analyzing the operator norm of the resolvent $ \|(H - z)^{-1}\| $ to characterize pseudospectra via the definition $ \sigma_\varepsilon(H) = \{ z \in \mathbb{C} : \|(H - z)^{-1}\| > \varepsilon^{-1} \} $.
• Using the Dirichlet boundary condition at $ x = 0 $ to decompose the solution into left and right half-line components, enabling explicit computation of the resolvent kernel.
• Applying the Birman-Schwinger principle to study eigenvalue perturbations under complex potentials $ V \in L^1(\mathbb{R}, (1+x^2)dx) $, by analyzing the Hilbert-Schmidt norm of the operator $ K_z^D = |V|^{1/2}(H^D - z)^{-1}|V|^{1/2} $.
• Proving uniform bounds on the resolvent kernel $ \mathcal{R}_z^D(x,y) $, showing $ |\mathcal{R}_z^D(x,y)| \leq C(1 + |x| + |y|) $, which implies uniform boundedness of the Hilbert-Schmidt norm.
• Comparing the pseudospectra of $ H $ with those of its Dirichlet-regularized version $ H^D $, showing that $ H^D $ has trivial pseudospectra while $ H $ does not.

Experimental results

Research questions

• RQ1 How does the pseudospectrum of a Schrödinger operator with a discontinuous imaginary potential differ from that of self-adjoint or smooth non-self-adjoint operators?
• RQ2 Can spectral instability (as measured by pseudospectra) occur even when the potential is bounded and the spectrum is contained in a half-strip?
• RQ3 What is the role of discontinuity in the potential in causing blow-up of the resolvent norm at infinity, leading to non-trivial pseudospectra?
• RQ4 To what extent do weakly coupled complex perturbations generate eigenvalues outside the spectrum, and can this be quantified?
• RQ5 Why does the Dirichlet-regularized version $ H^D $ have trivial pseudospectra despite sharing the same spectrum as $ H $, and what does this imply about the role of boundary conditions?

Key findings

• The pseudospectrum of the Schrödinger operator $ H = -\frac{d^2}{dx^2} + i\,\mathop{\mathrm{sgn}}\nolimits(x) $ is non-trivial, with the resolvent norm $ \|(H - z)^{-1}\| $ becoming arbitrarily large outside any fixed neighborhood of the spectrum $ \sigma(H) \subset \overline{\mathcal{S}} = [0,\infty) + i(-1,1) $.
• Despite the potential being bounded and the spectrum lying in a half-strip, the operator exhibits strong spectral instability, as the pseudospectrum extends far beyond the spectrum due to resolvent blow-up at infinity.
• The Dirichlet-regularized operator $ H^D $, which shares the same spectrum $ \sigma(H^D) = \mathbb{R}_+ + i\{ -1, +1 \} $, has trivial pseudospectra, highlighting that the non-trivial pseudospectrum of $ H $ arises from the lack of boundary conditions at $ x = 0 $.
• For small $ \varepsilon > 0 $, the perturbed operator $ H^D_\varepsilon = H^D + \varepsilon V $ with $ V \in L^1(\mathbb{R}, (1+x^2)dx) $ has spectrum unchanged: $ \sigma(H^D_\varepsilon) = \mathbb{R}_+ + i\{ -1, +1 \} $, indicating absence of weakly coupled eigenvalues.
• The Hilbert-Schmidt norm of the Birman-Schwinger operator $ K_z^D $ is uniformly bounded: $ \|K_z^D\|_{\mathrm{HS}} \leq C \int_{\mathbb{R}} (1+x^2)|V(x)|\,dx $, which ensures stability of the spectrum under small $ L^1 $-perturbations with polynomial weight.
• The result establishes a non-trivial analog of a Hardy inequality for non-self-adjoint operators, showing that no virtual bound states (i.e., weakly coupled eigenvalues) emerge in the Dirichlet-regularized case.