The Real Spectrum of the Imaginary Cubic Oscillator: An Expository Proof

Giordanelli, Ilario; Graf, Gian Michele

doi:10.1007/s00023-014-0325-5

Cited by 11 publications

(10 citation statements)

References 18 publications

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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%

“…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%

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Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

Shapiro

Štampach

2017

Constr Approx

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We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to ∞. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in C\{0}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the principal submatrices have to be real. The latter claim is also proved for matrices given by a more general symbol. The special role of the Jordan curve is further demonstrated by a new formula for the limiting density of the asymptotic eigenvalue distribution for banded Toeplitz matrices from the studied class. Certain connections between the problem under investigation, Jacobi operators, and the Hamburger moment problem are also discussed. The main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically.

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Section: Example 2 (4-diagonal Case)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

Shapiro

Štampach

2017

Constr Approx

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“…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%

Pseudospectra in non-Hermitian quantum mechanics

Krejčiřı́k

Siegl

Tater

et al. 2015

Journal of Mathematical Physics

119

124

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We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.Comment: version accepted for publication in J. Math. Phys.: criterion excluding basis property (Proposition 6) added, unbounded time-evolution discussed, new reference

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“…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%

Approximations of spectra of Schrödinger operators with complex potentials on ℝ^d

Bögli

Siegl

Tretter

2017

Communications in Partial Differential Equations

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Abstract. We study spectral approximations of Schrödinger operators T = −∆ + Q with complex potentials on Ω = R d , or exterior domains Ω ⊂ R d , by domain truncation. Our weak assumptions cover wide classes of potentials Q for which T has discrete spectrum, of approximating domains Ωn, and of boundary conditions on ∂Ωn such as mixed Dirichlet/Robin type. In particular, Re Q need not be bounded from below and Q may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of T by those of the truncated operators Tn without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for d = 1, 2, 3, illustrate our results.

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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.

Cited by 11 publications

References 18 publications

Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

Pseudospectra in non-Hermitian quantum mechanics

Approximations of spectra of Schrödinger operators with complex potentials on ℝ^d

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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.

Cited by 11 publications

References 18 publications

Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

Non-Self-Adjoint Toeplitz Matrices Whose Principal Submatrices Have Real Spectrum

Pseudospectra in non-Hermitian quantum mechanics

Approximations of spectra of Schrödinger operators with complex potentials on ℝd

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Product

Resources

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Approximations of spectra of Schrödinger operators with complex potentials on ℝ^d