Eigenvalues of the imaginary cubic oscillator $-y''+ix^3y$

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Numbers

$n$

$E_n$

1.1562670719881132937992191779999513764441864381057

4.1092287528096515358436684785613356944220484969282

7.5622738549788280413518091106314827208392311992988

11.314421820195804402233783948426989414322445163599

15.291553750392532388181630791751999233907426135766

Definition

This table gives the real eigenvalues $E_n$ of $-y''+ix^3y=Ey$ on $L^2(\mathbb{R})$, ordered increasingly with $n=0$ for the ground state [1] [2].

Parameters

$n$

— eigenvalue index ($n\geq0$)

Formulas

(1)

For $g>0$, the eigenvalues of $-d^2/dx^2+igx^3$ are $g^{2/5}E_n$.

Comments

(2)

The Hamiltonian is $-d^2/dx^2+ix^3$, with no factor of $1/2$ on the kinetic term.

(3)

The references usually write the potential as $-ix^3$ [3]. Replacing $x$ by $-x$ changes $ix^3$ to $-ix^3$, so the two signs have the same spectrum.

(4)

The operator is a standard example in PT-symmetric quantum mechanics [4].

Programs

(P1)

Sage

Run `generate.py` with SageMath. It forms the banded harmonic-basis Galerkin determinant and bisects its first five real roots.

References

[1]

Carl M. Bender and Stefan Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Physical Review Letters 80 (1998), 5243-5246. (arXiv) (doi)

[2]

Patrick Dorey, Clare Dunning and Roberto Tateo, Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics, Journal of Physics A: Mathematical and General 34 (2001), 5679-5704. (arXiv) (doi)

[3]

C. R. Handy, Generating converging eigenenergy bounds for the discrete states of the -ix^3 non-Hermitian potential, Journal of Physics A: Mathematical and General 34 (2001), L271-L277. (arXiv) (doi)

Links

[4]

Wikipedia: PT-symmetric quantum mechanics

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds the first five levels, the low-lying range for which [3] gives independent EMM bounds)

How they were obtained:

The generator uses the harmonic-oscillator basis for $-d^2/dx^2+\omega^2x^2$ and multiplies the $n$th basis vector by $i^n$, which turns the Galerkin matrix for $-d^2/dx^2+ix^3$ into a real banded matrix. It computes roots of $\det(H_N-E)$ by bisection with a banded Gaussian determinant at 560-bit precision, for $N=320,360$ and for $\omega=1.39,1.41$.

Each stored row is the real interval spanning those four computations; the widest interval retains 56.7 decimal digits. The first five intervals are checked to lie inside Handy's EMM bounds for $-ix^3$ [3]. A trial with $N=140,160$ and $\omega=1.3,1.5$ retained only 5.1 digits by $n=17$, so this draft stops at the first five levels rather than storing a longer weakly checked range.