2014
DOI: 10.1007/s00023-014-0325-5
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The Real Spectrum of the Imaginary Cubic Oscillator: An Expository Proof

Abstract: We give a partially alternate proof of the reality of the spectrum of the imaginary cubic oscillator in quantum mechanics.

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Cited by 11 publications
(10 citation statements)
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“…Taking into account that b(γ(0)) = (r + s) r+s r r s s and b(γ(π)) = 0, we see that b • γ is strictly decreasing on (0, π). Consequently, by Theorem 15, the limiting mesure is supported on the interval supp µ = 0, r −r s −s (r + s) r+s (43) and, according to (15), the distribution function of µ satisfies…”
Section: Example 2 (4-diagonal Case)mentioning
confidence: 99%
See 1 more Smart Citation
“…Taking into account that b(γ(0)) = (r + s) r+s r r s s and b(γ(π)) = 0, we see that b • γ is strictly decreasing on (0, π). Consequently, by Theorem 15, the limiting mesure is supported on the interval supp µ = 0, r −r s −s (r + s) r+s (43) and, according to (15), the distribution function of µ satisfies…”
Section: Example 2 (4-diagonal Case)mentioning
confidence: 99%
“…This is usually a mathematically challenging problem (see, for example, the proof of reality of the spectrum of the imaginary cubic oscillator in [27]) which, in addition, may be of physical relevance. At the moment, the non-self-adjoint operators whose spectrum is known to be real mainly comprise either very specific operators [27,15,34,29] or operators which are in a certain sense close to being self-adjoint [21,6,22]. In particular, there are almost no criteria for famous non-self-adjoint families (such as Toeplitz, Jacobi, Hankel, Schrödinger, etc.)…”
Section: Introductionmentioning
confidence: 99%
“…for λ 1 , λ 2 ∈ R. Applying a scaling like (62) and scaling µ allows us to assume that the coefficients of x 2 and ξ 2 have the same modulus. It is then evident that the resulting symbol is a multiple of that of a rotated harmonic oscillator (35). If one wishes only to identify the parameters of the rotated harmonic oscillator involved, one may appeal to the spectral decomposition of the fundamental matrix and the growth of the spectral projections.…”
Section: Elliptic Quadratic Operatorsmentioning
confidence: 99%
“…The imaginary cubic oscillator and related operators have been studied extensively, cf. [17,9,42,44,27,31,26]. It is known that all eigenvalues λ (k) of T are real and simple, behave asymptotically as k 6/5 , and the system of eigenfunctions of T is complete in L 2 (R, C) but does not form a basis, cf.…”
Section: Examplesmentioning
confidence: 99%