Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

One dimensional Dirac operators L"b"c(v)y=i(100-1)dydx+v(x)y,y=(y"1y"2),[email protected]?[0,@p], considered with L^2-potentials v(x)=(0P(x)Q(x)0) and subject to regular boundary conditions (bc), have discrete spectrum. For strictly regular bc, the spectrum of the free operator L"b"c(0) is simple while the spectrum of L"b"c(v) is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval [0,@p]. Analogous results are obtained for regular but not strictly regular bc.

[1]  George D. Birkhoff,et al.  Boundary value and expansion problems of ordinary linear differential equations , 1908 .

[2]  H. R. Dowson,et al.  LINEAR OPERATORS PART III: SPECTRAL OPERATORS , 1974 .

[3]  On the completeness of the system of root vectors for first-order systems , 2011, 1109.3683.

[4]  Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions , 2010, 1008.4095.

[5]  B. Mityagin,et al.  SPECTRA OF 1D PERIODIC DIRAC OPERATORS AND SMOOTHNESS OF POTENTIALS , 2003 .

[6]  P. Goldbart,et al.  Linear differential operators , 1967 .

[7]  Bari–Markus property for Riesz projections of 1D periodic Dirac operators , 2009, 0901.0856.

[8]  A. Shkalikov.  ON THE ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS OF A POLYHARMONIC EQUATION IN A NEIGHBOURHOOD OF NON-REGULAR BOUNDARY POINTS AND AT INFINITY , 1979 .

[9]  L. Carleson.  Selected Problems on Exceptional Sets , 1998 .

[10]  On the completeness of the root vectors of first-order systems , 2010 .

[11]  B. Mityagin,et al.  Instability Zones of a Periodic 1D Dirac Operator and Smoothness of its Potential , 2005 .

[12]  P. Heywood.  Trigonometric Series , 1968, Nature.

[13]  Radosław Szmytkowski.  Discontinuities in Dirac eigenfunction expansions , 2001 .

[14]  N. Dunford.  A survey of the theory of spectral operators , 1958 .

[15]  J. Tamarkine.  Sur quelques points de la théorie des équations différentielles linéaires ordinaires et sur la généralisation de la série de fourier , 1912 .

[16]  J. Tamarkin,et al.  Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions , 1928 .

[17]  M. Yamamoto,et al.  Spectral properties and an inverse eigenvalue problem for non-symmetric systems of ordinary differential operators , 2002 .

[18]  Y. Masahiro,et al.  Riesz basis of root vectors of a non-symmetric system of first-order ordinary differential operators and application to inverse eigenvalue problems , 2001 .

[19]  B. Mityagin.  Spectral expansions of one-dimensional periodic Dirac operators , 2004 .

[20]  Equiconvergence theorems for differential operators , 1999, math/0602406.

[21]  F. Smithies.  Linear Operators , 2019, Nature.

[22]  Completeness Theorems for Systems of Differential Equations , 2000 .

[23]  A. Shkalikov,et al.  Boundary-value problems for ordinary differential equations with a parameter in the boundary conditions , 1982 .

[24]  George D. Birkhoff,et al.  On the asymptotic character of the solutions of certain linear differential equations containing a parameter , 1908 .

[25]  Boris Mityagin,et al.  Smoothness of Schrödinger Operator Potential in the Case of Gevrey Type Asymptotics of the Gaps , 2002 .

[26]  B. Mityagin,et al.  Instability zones of periodic 1-dimensional Schrödinger and Dirac operators , 2006 .

[27]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[28]  R. A. Hunt,et al.  Weighted norm inequalities for the conjugate function and Hilbert transform , 1973 .

[29]  Rudolph Ernest Langer,et al.  The Boundary Problems and Developments Associated with a System of Ordinary Linear Differential Equations of the First Order , 1923 .

[30]  B. Mityagin,et al.  Eigensystem of an L2-perturbed harmonic oscillator is an unconditional basis , 2009, 0912.2722.

[31]  Boris Mityagin,et al.  Spectral triangles of Schrödinger operators with complex potentials , 2003 .

[32]  Theorem of Completeness for a Dirac-Type Operator with Generalized λ-Depending Boundary Conditions , 2009 .