The ( ε -)pseudospectrum σ ε ( H ) (with positive ε ) of an operator H in a Hilbert space is the union of the spectrum σ ( H ) of H and all those complex numbers λ from the resolvent set ρ ( H ) of H for which ‖ H − λ − 1 ‖ > 1 ε . Equivalently, σ ε ( H ) comprises the spectrum of H and λ ∈ C (pseudoeigenvalues) for which there exists a vector ψ (pseudomode) in the domain of H such that ‖ H − λ ψ ‖ < ε   ‖ ψ ‖ . If H is self-adjoint (or, more generally, normal), the ε -pseudospectrum is trivial in the sense that it is just the ε -tubular neighbourhood of the spectrum of H . In general, however, the pseudoeigenvalues can lie outside the ε -tubular neighbourhood and their location is important to correctly seize various properties of H , see [1–3].

The goal of this brief research report is to explain in a succinct way the approach in Krejčiřík and Siegl [4] to locate pseudoeigenvalues of (non-semiclassical) Schrödinger operators − d 2 d x 2 + V x in L 2 R , (1) where V : R → C is at least locally square-integrable and R V ≥ 0 . In such a case, there exists a unique m-accretive extension H V of Equation 1 initially defined on C 0 ∞ ( R ) , see ([5], Thm. VII.2.6). Since our constructed pseudomodes are compactly supported and at least twice weakly differentiable, they belong to the domain of H V .

The operator H V is self-adjoint (respectively, normal) if, and only if, V is real-valued (respectively, I V is constant). To ensure non-trivial pseudospectra, we shall therefore adopt the standing hypothesis lim sup x → − ∞ I V x < 0 < lim inf x → + ∞ I V x , (2) where the limits are allowed to be infinite. The assumption (Equation 2) can be interpreted as a “global” version of the Davies’ condition I V ′ ≠ 0 , see [6] and also [7].

To simplify the presentation, the potential V will be assumed to be smooth and imaginary-valued. Typical examples to keep in mind are as follows: V 1 x ≔ i ⁡ arctan x , V 2 x ≔ i x m with m  odd , V 3 x ≔ i ⁡ sinh x , (3) or their imaginary shifts. In particular, V 2 with m = 3 is the celebrated imaginary cubic (or Bender’s) oscillator (with purely real and discrete spectrum, see Figure 1), which was made popular in the context of the so-called P T -symmetric quantum mechanics in [8].

The objective is to develop a systematic construction of pseudomodes ensuring that, for any diminishing ε → 0 , there is a complex number λ with large magnitude | λ | → ∞ such that λ ∈ σ ε ( H V ) . The results are particularly striking whenever this set of pseudoeigenvalues lie outside (in fact, “very far” from) the ε -tubular neighbourhood of σ ( H ) . This is particularly the case of the imaginary cubic oscillator, for which the analysis below show that for an arbitrarily small ε there exists a pseudoeigenvalue λ with an arbitrarily large imaginary part, despite the fact that the spectrum is purely real (see Figure 1 for a numerical quantification of the pseudospectrum level lines). This property implies the lack of Riesz basis for the eigenfunctions, challenging in the spirit of [9] the physical relevance of the L 2 ( R ) -realisation of the Bender’s oscillator. The follow-up [4] summarised in this report can be considered as a methodical and more advanced study of not necessarily polynomial potentials.

The feature of the approach of [4] is that it does not rely on semiclassical methods developed in [6, 7, 10]. In fact, we are able to construct large-energy pseudomodes for potentials (like of exponential type, see V 3 of Equation 3) which cannot be reduced (by scaling) to a small Planck’s constant included in the kinetic energy. On the contrary, the known claims in the semiclassical setting follow immediately from our approach.

Our strategy of the construction of pseudomodes is based on the Liouville–Green approximation, also known as the JWKB method in mathematical physics. The key idea is that, if V were constant, exact solutions of the differential equation associated with H V g = λ g would be the two non-integrable functions g ± x ≔ exp ± i ∫ 0 x λ − V t   d t .

The starting point of the approximation scheme is to use the same ansatz for variable V as well. More specifically, we choose g 0 ≔ g − for it is exponentially decaying under the hypothesis (Equation 2), whenever I λ is small with respect to the limits of I V at ± ∞ . A direct computation yields H V − λ g 0 = r 0   g 0 with r 0 ≔ − i 2 V ′ λ − V . (4) Recalling the simplifying hypothesis that R V = 0 and assuming in addition that I λ = 0 and R λ > 0 (typically large), one has the estimate ‖ r 0 ‖ ∞ ≤ 1 R λ 1 − δ | V ′ | 2 | V | δ / 2 ∞ (5) for every δ ∈ [ 0,1 ) . It follows that large real energies always lie in the pseudospectrum, namely, for every positive ε , λ ∈ C : R λ 1 − δ > 1 ε | V ′ | 2 | V | δ / 2 ∞ ⊂ σ ε H V . Of course, this result is interesting only if the supremum norm is bounded. From examples (Equation 3), relevant potentials are thus V 1 and V 2 with m = 1 , in which case we can take δ = 0 and obtain thus a pseudomode satisfying the decay ‖ ( H V − λ ) g 0 ‖ = O ( ( R λ ) − 1 / 2 )   ‖ g 0 ‖ as R λ → ∞ . The latter is particularly interesting because the spectrum of the imaginary Airy operator is empty, see, e.g., ([3, 11], Section VII.A) or more generally [12], where the last reference includes also an elementary proof of the optimal resolvent norm estimate for the Airy operator.

It is not difficult to modify the exponentially decaying pseudomode g 0 to a compactly supported pseudomode f 0 , while still keeping the same decay ‖ ( H V − λ ) f 0 ‖ = O ( ( R λ ) − 1 / 2 )   ‖ f 0 ‖ as R λ → ∞ . Indeed, let ξ 1 : R → [ 0,1 ] be a smooth function such that ξ 1 = 1 on [ − 1,1 ] and ξ 1 = 0 outside [ − 2,2 ] . Given any positive number l , let us define the rescaled cut-off function ξ l ( x ) ≔ ξ 1 ( x / l ) . Then f 0 ≔ ξ l   g 0 is compactly supported and one has H V − λ f 0 = ξ l   H V g 0 + − ξ l ′ ′ + 2 i λ − V ξ l ′ g 0 . Using that ξ l → 1 pointwise as l → ∞ , while one gains one l − 1 by each derivative, it is possible to verify the desired decay by the λ -dependent choice l ≔ R λ .

To cover a larger class of potentials, let us consider a modified ansatz g 1 ≔ g 0   exp ( − ψ 0 ) , where ψ 0 is a function to be chosen later. A direct computation yields H V − λ g 1 = r 0 − 2 i λ − V ψ 0 ′ + ψ 0 ″ − ψ 0 ′ 2 g 1 . Now we choose ψ 0 to annihilate the error term r 0 from Equation 4, by solving the first-order linear differential equation r 0 − 2 i λ − V ψ 0 ′ = 0 , namely, ψ 0 ≔ log λ − V 4 . Thus we arrive at the familiar expression g 1 x = 1 λ − V x 4 exp − i ∫ 0 x λ − V t   d t . Then H V − λ g 1 = r 1   g 1 with r 1 ≔ − 5 16 V ′ 2 λ − V 2 − 1 4 V ′ ′ λ − V , where the new error term r 1 can be estimated as follows: ‖ r 1 ‖ ∞ ≤ 1 R λ 2 1 − δ 5   | V ′ | 2 16   | V | 1 + δ + | V ′ ′ | 4   | V | δ ∞ . This result is an improvement upon (Equation 4) with (Equation 5) in two respects. First, if the supremum norm is bounded for δ = 0 , we get a pseudomode with an improved decay ‖ ( H V − λ ) g 1 ‖ = O ( ( R λ ) − 1 )   ‖ g 1 ‖ as R λ → ∞ . This is the case of V 1 and V 2 with m = 1 from examples (Equation 3). Second, keeping the decay O ( ( R λ ) − 1 / 2 ) by the choice δ = 1 / 2 , we can now cover V 2 with m = 3 from examples (Equation 3).

The above scheme can be continued by employing the general ansatz in square-root powers of λ : g k = exp − λ 1 / 2   ψ − 1 + λ − 0 / 2   ψ 0 + λ − 1 / 2   ψ 1 + ⋯ + λ − k − 1 / 2   ψ k − 1 , (6) where ψ − 1 ( x ) ≔ i λ − 1 / 2 ∫ 0 x λ − V ( t )   d t and ψ k − 1 with k ∈ N is iteratively chosen in such a way to annihilate the previous error term r k − 1 . By enlarging k , more derivatives of V are required. On the other hand, a better decay (in negative powers of R λ → ∞ ) of the new error term is achieved and a larger class of potentials can be covered. For instance, all the examples (Equation 3) are already covered by the choice k = 2 , namely, ‖ ( H V − λ ) g 2 ‖ = O ( ( R λ ) − 1 / 2 )   ‖ g 2 ‖ as R λ → ∞ .

To make the above procedure rigorous, it is important to ensure that g 0 in the expansion (Equation 6) is dominant, in order to guarantee that g k ( x ) have appropriate decay properties at x = ± ∞ . One of the main achievements of [4] is the formulation of the robust sufficient condition | V n x | | V x | = O | x | n ν and | x | 4 1 + ν = O | V x | (7) to hold as | x | → ∞ with some real number ν ≤ 0 for every n = 1 , … , k + 1 . Note that ν = − 2 , − 1 and 0 for the potentials V 1 , V 2 and V 3 of Equation 3, respectively. In fact, it is possible to allow for ν > 0 (corresponding to superexponentially growing potentials). Moreover, different behaviours at x → ± ∞ may be allowed. However, let us stick to Equation 7 to make the presentation here as simple as possible.

To get a compactly supported pseudomode, it turns out that the adequate λ -dependent cut-off function should be supported in the interval [ − l − , l + ] , where (denoting ⟨ l ⟩ ≔ ( 1 + l 2 ) 1 / 2 ) l ± ≔ inf l ≥ 0   :   | V ± l | 2 ⟨ l ⟩ 4 1 + ν = λ if  V  is unbounded at  ± ∞ , λ 1 − ν 4 if  V  is bounded at  ± ∞ . Recall that we assume I λ = 0 and note that l ± → ∞ as λ → ∞ . In particular, l ± = λ 3 / 2 , λ 1 / ( 2 m ) and log ⁡ λ as λ → ∞ for the potentials V 1 , V 2 and V 3 of Equation 3, respectively.

Under the present simplifying hypotheses (in particular, I λ = 0 , R V = 0 and ν ≤ 0 ), the general result of Krejčiřík and Siegl [4] (Thm. 3.7) can be formulated as follows.

Theorem 1

Let V : R → i R be smooth satisfying Equations 2, 7 with given k ∈ N . If λ − k + 1 / 2 sup x ∈ − l − , l + | V x |   ⟨ x ⟩ k + 1 ν → λ → + ∞ 0 , (8) then there exists { ψ λ } λ ⊂ C 0 ∞ ( R ) such that ‖ ψ λ ‖ = 1 and lim λ → + ∞ ‖ H V − λ ψ λ ‖ = 0 . (9)

The extra condition (Equation 8) with the choice k = 0 is clearly satisfied for the potential V 1 of Equation 3 (in fact, for any bounded potential satisfying Equations 2, 7). To satisfy Equation 8 for all the polynomial potentials V 2 of Equation 3, it is sufficient to take k = 1 . Finally, Equation 8 is verified for the exponential potential V 3 of Equation 3 with k = 2 .

In Krejčiřík and Siegl [4], the decay rate in Equation 9 is carefully quantified in terms of the left-hand side of Equation 8 and other quantities related to the behaviour of a general potential V at infinity.