Analytic description of the essential spectrum of a family of 3×3 operator matrices

Rasulov, Tulkin H.; Tosheva, Nargiza Ahmedovna

doi:10.17586/2220-8054-2019-10-5-511-519

Cited by 3 publications

(5 citation statements)

References 13 publications

(30 reference statements)

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“…The following theorem [4,5,15,27] describes the location of the essential spectrum of the operator H(K) by the spectrum of the family h(k) of Friedrichs models.…”

Section: Family Of 3 × 3 Operator Matrices and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

Muminov¹,

Rasulov²,

Tosheva³

2020

Preprint

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We consider the family of 3 × 3 operator matrices H(K), K ∈ T 3 := (−π; π] 3 associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set Λ ⊂ T 3 to prove the existence of infinitely many eigenvalues of H(K) for all K ∈ Λ when the associated Friedrichs model has a zero energy resonance. It is found that for every K ∈ Λ, the number N (K, z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim z→−0 N (K, z)| log |z|| −1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ. Moreover, we prove that for any K ∈ Λ the operator H(K) has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.

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“…The following theorem [4,5,15,27] describes the location of the essential spectrum of the operator H(K) by the spectrum of the family h(k) of Friedrichs models.…”

Section: Family Of 3 × 3 Operator Matrices and Main Resultsmentioning

confidence: 99%

“…This paper is devoted to the detailed proof of these results with respect to the number of eigenvalues. The result related with the essential spectrum of H(K) was discussed in [27].…”

Section: Introductionmentioning

confidence: 99%

Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

Muminov¹,

Rasulov²,

Tosheva³

2020

Preprint

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“…The inclusion Σ µ ⊂ σ ess (A µ ) in the proof of Theorem 2.2 is established with the use of a well-known Weyl creterion, see for example [11]. An application of Theorem 2.1 and analytic Fredholm theorem (see, e.g., Theorem VI.14 in [18]) proves inclusion σ ess (A µ ) ⊂ Σ µ .…”

Section: Faddeev's Equation and Essential Spectrum Of A µmentioning

confidence: 99%

“…For the case when the underlying domain is a torus, the spectral properties of some versions of H were investigated in [8][9][10][11]. An important problem of the spectral theory of such matrix operators is the infiniteness of the number of eigenvalues located outside the essential spectrum.…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Analysis of the spectrum of a 2x2 operator matrix. Discrete spectrum asymptotics

Rasulov¹,

Dilmurodov²

2020

Nanosystems: Phys. Chem. Math.

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We consider a 2 × 2 operator matrix Aµ, µ > 0 related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analog of the Faddeev equation and its symmetric version for the eigenfunctions of Aµ. We describe the new branches of the essential spectrum of Aµ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of Aµ consists the union of at most three bounded closed intervals and their location is studied. For the critical value µ 0 of the coupling constant µ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of Aµ. In this case, an asymptotic formula for the discrete spectrum of Aµ is found.

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“…The threshold eigenvalues and virtual levels of a slightly simpler version of A µ (k) were investigated in [15], and the structure of the numerical range are studied using similar results. In [16], the essential spectrum of the family of 3 × 3 operator matrices H(K) is described by the spectrum of the family of 2 × 2 operator matrices. The results of the present paper are play important role in the investigations of the operator H(K), see [10].…”

Section: Introductionmentioning

confidence: 99%

Threshold analysis for a family of 2×2 operator matrices

Rasulov¹,

Dilmurodov²

2019

Nanosystems: Phys. Chem. Math.

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We consider a family of 2 × 2 operator matrices Aµ(k), k ∈ T 3 := (−π, π] 3 , µ > 0, acting in the direct sum of zero-and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z 3 , interacting via annihilation and creation operators. We find a set Λ := {k (1) , ..., k (8) } ⊂ T 3 and a critical value of the coupling constant µ to establish necessary and sufficient conditions for either z = 0 = min k∈T 3 σess(Aµ(k)) ( or z = 27/2 = max k∈T 3 σess(Aµ(k)) is a threshold eigenvalue or a virtual level of Aµ(k (i) ) for some k (i) ∈ Λ.

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Analytic description of the essential spectrum of a family of 3×3 operator matrices

Cited by 3 publications

References 13 publications

Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

Analysis of the spectrum of a 2x2 operator matrix. Discrete spectrum asymptotics

Threshold analysis for a family of 2×2 operator matrices

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