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Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
High Energy Physics - Theory · Submitted 7 Mar 2001
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Finally we generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT-symmetric quantum-mechanical systems.
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The abstract, sentence by sentence
The one-dimensional Schrodinger equation for the potential $x^6+\alpha x^2 +l(l+1)/x^2$ has many interesting properties. For certain values of the parameters l and alpha the equation is in turn supersymmetric (Witten), quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to high energy QCD. In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. These relationships are obtained via a recently-observed connection between the theories of ordinary differential equations and integrable models. Generalised supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. Finally we generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT-symmetric quantum-mechanical systems.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The one-dimensional Schrodinger equation for the potential $x^6+\alpha x^2 +l(l+1)/x^2$ has many interesting properties. | background 1.00 | 0.09 | 0.02 | asserted 1.00 |
| s1 | For certain values of the parameters l and alpha the equation is in turn supersymmetric (Witten), quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to high energy QCD. | background 1.00 | 0.06 | 0.03 | asserted 0.98 |
| s2 | In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. | contribution 0.69 | 0.87 | 0.23 | asserted 0.72 |
| s3 | These relationships are obtained via a recently-observed connection between the theories of ordinary differential equations and integrable models. | method 0.93 | 0.43 | 0.05 | asserted 1.00 |
| s4 | Generalised supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. | contribution 0.72 | 0.92 | 0.07 | asserted 1.00 |
| s5 | Finally we generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT-symmetric quantum-mechanical systems. | result 0.73 | 0.92 | 0.65 | asserted 1.00 |
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The paper
Original abstract
The one-dimensional Schrodinger equation for the potential $x^6+\alpha x^2 +l(l+1)/x^2$ has many interesting properties. For certain values of the parameters l and alpha the equation is in turn supersymmetric (Witten), quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to high energy QCD. In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. These relationships are obtained via a recently-observed connection between the theories of ordinary differential equations and integrable models. Generalised supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. Finally we generalise slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain PT-symmetric quantum-mechanical systems.
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References
How the literature uses it
38 arXiv papers cite this in 54 sentences
Papers citing it · 33 sentences without a confident label
Cites it as support 2 papers
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It is clear that the pseudo-Hermiticity itself is much less relevant because once we get through the difficult proof of the necessary reality of the spectrum [ref] , the pseudo-Hermiticity of a given $H$ becomes in fact fully equivalent to its quasi-Hermiticity [ref] .
On the Role of the Normalization Factors $\kappa_n$ and of the Pseudo-Metric ${\cal P} \neq {\cal P}^\dagger$ in Crypto-Hermitian Quantum Models · Pseudo-Hermitian Hamiltonians · 0.97
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Again, as proved in Appendix B of [ref] , the spectrum of ([ref]) is real and positive for $M≥1$ and $|2l+1|< M+1$.
ODE/IM Correspondence in Toda Field Theories and Fermionic Basis in sin(h)-Gordon Model · $𝒫𝒯$ symmetric Hamiltonians · 0.97
Cites it as background 17 papers
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In the past few years it has been demonstrated that some very simple non-Hermitian Hamiltonians provide well-defined quantum theories [ref] .
Electric-Magnetic Duality in Massless QED? · Introduction · 0.86
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In the context of field theory, typically, $P$ is chosen as parity while the antilinear Hermitian-conjugation factor $T$ mimics time reversal [ref] .
On the Role of the Normalization Factors $\kappa_n$ and of the Pseudo-Metric ${\cal P} \neq {\cal P}^\dagger$ in Crypto-Hermitian Quantum Models · Introduction · 0.99
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The next important development happened in the context of ODE/IM correspondence [ref] .
On Bethe equations of 2d conformal field theory · Introduction · 0.97
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