Isospectral Mathieu–Hill Operators

Veliev, O. A.

doi:10.1007/s11005-013-0627-4

Cited by 13 publications

(14 citation statements)

References 11 publications

(44 reference statements)

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“…Suppose that λ n (t) is nonreal. Since F (λ n (t), t) is real number (see (27)), from (20)- (22) one can readily see that F (λ n (t), t) = F λ n (t), t from which by using (24) we obtain the following contradiction…”

Section: Remark 2 Formulamentioning

confidence: 98%

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On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

Veliev

2017

Int. J. Geom. Methods Mod. Phys.

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In this paper we investigate the spectrum and spectrality of the one-dimensional Schrodinger operator with a periodic PT-symmetric complex-valued potential.Key Words: Schrodinger operator, PT-symmetric periodic potential, Real spectrum.AMS Mathematics Subject Classification: 34L05, 34L20. Introduction and Preliminary FactsIn this paper we investigate the one dimensional Schrödinger operator L(q) generated inwhere q is complex-valued, locally integrable, periodic and PT-symmetric. Without loss of generality, we assume that the period of q is 1 and the integral of q over [0, 1] is zero. ThusA basic mathematical question of PT-symmetric quantum mechanics concerns the reality of the spectrum of the considered Hamiltonian (see [2, 15 and references of them]). In the first papers [1,3,5,6,10] about the PT-symmetric periodic potential, the appearance and disappearance of real energy bands for some complex-valued PT-symmetric periodic potentials under perturbations have been reported. Shin [17] showed that the appearance and disappearance of such real energy bands imply the existence of nonreal band spectra. He involved some condition on the Hill discriminant to show the existence of nonreal curves in the spectrum. Caliceti and Graffi [4] found explicit condition on the Fourier coefficient of the potential providing the nonreal spectra for small potentials. Besides, they proved that if all gaps of the spectrum of the Hill operator L(q) with distributional potential q are open and the width of the n-th gap does not vanish as n → ∞, then the spectrum of L(q) + gW, where W is a bounded periodic function and g is a small number, is real. This result can not be used for the locally integrable periodic potentials (2), since the width of the n-th gap vanishes as n → ∞.In this paper, we first consider the general spectral property of the spectrum of L(q) under conditions (2) and prove that the main part of its spectrum is real and contains the large part of [0, ∞). Using this we find necessary and sufficient condition on the potential 1 2 for finiteness of the number of the nonreal arcs in the spectrum of L(q). Besides we find necessary and sufficient conditions for the equality of the spectrum of L(q) to the half line. Moreover, we consider the connections between spectrality of L(q) and the reality of its spectrum for some class of PT-symmetric periodic potentials. Finally, we find explicit conditions on the potential q for which the number of gaps in the real part of the spectrum of L(q) is finite. Now let us list the well-known results, as Summary 1-Sammary 7, about L(q) which will be essentially used in this paper. Note that we formulate the well-known results from the books [7,19] and the papers [11,13,14,16,17] in the suitable form for this paper and by using the unique notation, since the different notations were used in those references. Summary 1The spectrum σ(L) of the operator L is the union of the spectra σ(L t ) of the operators L t for t ∈ (−π, π] generated in L 2 [0, 1] by (1) and the boundary conditions...

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Section: Remark 2 Formulamentioning

confidence: 98%

“…To consider the spectrum of these operators we use the following results of [24] formulated as Summary 10 (see Theorem 1 and (26) of [24]).…”

Section: Remark 2 Formulamentioning

confidence: 99%

On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

Veliev

2017

Int. J. Geom. Methods Mod. Phys.

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“…But as Veliev [30] observed, the operators (1.1) generated by potentials (1.12) with cd = const are isospectral if considered with periodic or antiperiodic boundary conditions. Therefore, from (1.11) with a = √ cd it follows that…”

Section: Introductionmentioning

confidence: 99%

Refined asymptotics of the spectral gap for the Mathieu operator

Anahtarcı

Djakov

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tThe Mathieu operator L(y) = −y ′′ + 2a cos(2x)y, a ∈ C, a ̸ = 0, considered with periodic or anti-periodic boundary conditions has, close to n 2 for large enough n, two periodic (if n is even) or anti-periodic (if n is odd) eigenvalues λ − n , λ + n . For fixed a, we show thatThis result extends the asymptotic formula of Harrell-Avron-Simon by providing more asymptotic terms.

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“…As we noted above in [11] it was proved that the investigation of the operator L(q) with potential (2) can be reduced to the investigation of the Mathieu operator. Besides in [15] (see Theorem 1 and (26) of [15]) we proved that if ab = cd, where a, b, c, and d are arbitrary complex numbers, then the operators L(q) and L(p) with potentials q(x) = ae −i2x + be i2x and p(x) = ce −i2x + de i2x have the same Hill discriminant F (λ) and hence the same Bloch eigenvalues and spectrum. Therefore we have σ(L(V )) = σ(H(a)), σ(L t (V )) = σ(H t (a)), ∀t ∈ [0, π], a = 1 − 4V 2 ,…”

Section: Introduction and Preliminary Factsmentioning

confidence: 99%

“…Therefore using Theorem 10 and taking into account that if λ lies inside γ 1 then λ does not lie inside γ 2 , we obtain that λ 0 (a) and λ 2 (a) are the real eigenvalues lying respectively inside γ 1 and γ 2 . (b) The roots of P (a, λ) for the cases a 2 = −2.157281295 are 2.088698925 ± 0.000232839i, 15.98321016 ± 0.11878599i,15.85581654, 36.00018270 ± 0.00333046i, 63.99999991.…”

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confidence: 98%

The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential

Veliev

2017

Int. J. Geom. Methods Mod. Phys.

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We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with potential 4 cos 2 x+4iV sin 2x, where V ∈ (0, ∞). We prove that the second critical point V2, after which the real parts of the first and second band disappear, is a number between 0.8884370025 and 0.8884370117. Moreover we prove that V2 is the degeneration point for the first periodic eigenvalue. Besides, we give a scheme by which one can find arbitrary precise value of the second critical point as well as the k-th critical points after which the real parts of the (2k − 3)-th and (2k − 2)-th bands disappear, where k = 3, 4, ...

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Isospectral Mathieu–Hill Operators

Cited by 13 publications

References 11 publications

On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

Refined asymptotics of the spectral gap for the Mathieu operator

The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential

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