On the basis property of the system of eigenfunctions and associated functions of a one-dimensional Dirac operator

2018 ◽  
Vol 82 (2) ◽  
pp. 351-376 ◽  
Author(s):  
A M Savchuk
Keyword(s):  
Dirac Operator ◽  
Basis Property ◽  
One Dimensional ◽  
Associated Functions ◽  
Eigenfunctions And Associated Functions

scholarly journals On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
N. S. Imanbaev
Keyword(s):  
Boundary Conditions ◽  
Liouville Operator ◽  
Basis Property ◽  
Differentiation Operator ◽  
Stability And Instability ◽  
Associated Functions ◽  
Sturm Liouville ◽  
Type Boundary ◽  
Eigenfunctions And Associated Functions ◽  
Integral Perturbation

We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of Samarskii-Ionkin type boundary conditions.

scholarly journals Spectral Analysis of One-Dimensional Dirac System with Summable Potential and Sturm- Liouville Operator with Distribution Coefficients

2020 ◽  
Vol 66 (3) ◽  
pp. 373-530
Author(s):  
A. M. Savchuk ◽  
I. V Sadovnichaya
Keyword(s):  
Asymptotic Formulas ◽  
Regular Boundary ◽  
Regular Case ◽  
One Dimensional ◽  
Spectral Decompositions ◽  
Associated Functions ◽  
Strongly Regular ◽  
Eigenfunctions And Associated Functions ◽  
Regular Boundary Conditions ◽  
Summable Potential

We consider one-dimensional Dirac operatorLP,U with Birkhoff regular boundary conditions and summable potential P(x) on[0, ]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,] with p [1,2] and the Besov space Bp,p'[0,] with p [1,2] and (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,R Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x [0,]. In the weakly regular case, the eigenvalues 2n and 2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,])2 for a system of eigenfunctions and associated functions of an arbitrary strongly regular operatorLP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved. In parallel with theLP,U operator, we consider the SturmLiouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, ]) and Birkhoff-regular boundary conditions. We reduce to this case -(1y')'+i(y)'+iy'+0y, operators of more general form where '1,,0(-1)L2and 10. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U . Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,R for p1 or 0. The problem of conditional basicity is naturally generalized to the problem of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: fL[0,] (decomposable function), PL[0,] (potential), and Sm-Sm00,m in L[0,] (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces L[0,],2, and in various Besov spaces Bp,q[0,].

scholarly journals Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Kun Li ◽  
Maozhu Zhang ◽  
Jinming Cai ◽  
Zhaowen Zheng
Keyword(s):  
Boundary Condition ◽  
Dirac Operator ◽  
Completeness Theorem ◽  
Scattering Matrix ◽  
Dissipative Operator ◽  
Limit Circle ◽  
Associated Functions ◽  
Limit Circle Case ◽  
Eigenfunctions And Associated Functions ◽  
Circle Case

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

scholarly journals Uniform Basis Property of the System of Root Vectors of the Dirac Operator

2018 ◽  
Vol 64 (1) ◽  
pp. 180-193
Author(s):  
A M Savchuk ◽  
I V Sadovnichaya
Keyword(s):  
Dirac Operator ◽  
Finite Number ◽  
Basis Property ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Root Functions ◽  
Strongly Regular ◽  
System Of Root Functions ◽  
Complex Valued ◽  
Summable Potential

We study one-dimensional Dirac operator L on the segment [0,π] with regular in the sense of Birkhoff boundary conditions U and complex-valued summable potential P=(pij(x)), i,j=1,2. We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator L assuming that boundary-value conditions U and the number ∫(p1(x)-p4(x))dx are fixed and the potential P takes values from the ball B(0,R) of radius R in the space Lϰ for ϰ>1. Moreover, we can choose the system of root functions so that it consists of eigenfunctions of the operator L except for a finite number of root vectors that can be uniformly estimated over the ball ∥P∥ϰ≤R.

On the Riesz Inequality and the Basis Property of Systems of Root Vector Functions of a Discontinuous Dirac Operator

2019 ◽  
Vol 55 (8) ◽  
pp. 1045-1055
Author(s):  
V. M. Kurbanov ◽  
L. Z. Buksaeva
Keyword(s):  
Dirac Operator ◽  
Basis Property ◽  
Root Vector ◽  
Riesz Inequality

scholarly journals Dynamical Generation of Solitons in a 1 + 1 Dimensional Chiral Field Theory: Non-Perturbative Dirac Operator Resolvent Analysis

1997 ◽  
Vol 12 (06) ◽  
pp. 1133-1142 ◽  
Author(s):  
Joshua Feinberg ◽  
A. Zee
Keyword(s):  
Field Theory ◽  
Dirac Operator ◽  
Inverse Scattering ◽  
Saddle Points ◽  
Inverse Scattering Method ◽  
Chiral Field ◽  
Scattering Method ◽  
One Dimensional ◽  
Dynamical Generation ◽  
Static Space

We analyze the 1 + 1 dimensional Nambu–Jona–Lasinio model non-perturbatively. We study non-trivial saddle points of the effective action in which the composite fields [Formula: see text] and [Formula: see text] form static space dependent configurations. These configurations may be viewed as one dimensional chiral bags that trap the original fermions ("quarks") into stable extended entities ("hadrons"). We provide explicit expressions for the profiles of some of these objects and calculate their masses. Our analysis of these saddle points, and in particular, the proof that the σ(x), π(x) condensations must give rise to a reflectionless Dirac operator, appear to us simpler and more direct than the calculations previously done by Shei, using the inverse scattering method following Dashen, Hasslacher, and Neveu.

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