On characteristic determinants of boundary value problems for Dirac type systems

A. Lunev   ; M. Malamud

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 69-120

Résumé

The paper is concerned with the asymptotic behavior of the eigenvalues of the following $n \times n$ Dirac type equation $$ y' + Q(x) y = i \lambda B(x) y, y = \mathrm{col}\,(y_1, \ldots, y_n), x \in [0,\ell], $$ on a finite interval $[0,\ell]$ subject to general regular boundary conditions $C y(0) + D y(\ell) = 1$ with $C, D \in \mathbb{C}^{n \times n}$. Here $Q = (Q_{jk})_{j,k=1}^n$ is an integrable potential matrix and $ B = \mathrm{diag}\,(\beta_1, \ldots, \beta_n) = B^* \in L^1([0,\ell];\mathbb{R}^{n \times n}) $ is a diagonal matrix “weight”. If $n=2m$ and $ B(x) = \mathrm{diag}\,(-I_m, I_m) $ this equation is equivalent to $n\times n$ Dirac equation. Under the assumption $\mathrm{supp}\,(Q_{jk}) \subset \mathrm{supp}\,(\beta_k - \beta_j)$, we show that the deviation of the characteristic determinants $\Delta_Q(\cdot)$ and $\Delta_0(\cdot)$ of this boundary value problem (BVP) and the unperturbed BVP (with $Q \equiv 0$) is a Fourier transform of some integrable function, $$ \Delta_Q(\lambda) = \Delta_0(\lambda) + \int\limits_{\widetilde{b}_-}^{\widetilde{b}_+} g(u) e^{i \lambda u} du, g \in L^1[\widetilde{b}_-, \widetilde{b}_+]. $$ We apply this representation to study of zeros distribution of the characteristic determinant $\Delta_Q(\cdot)$ (eigenvalues of the above BPV) and show that $\Delta_Q(\cdot)$ is always an entire class $A$ function of exponential type, which is bounded on the real axis. We also find conditions guaranteeing that $\Delta_Q(\cdot)$ is a sine-type function and provide sharp asymptotic formula for its zeros. Finally, we show that if the entries of matrix $B(\cdot)$ can change sign within the segment $[0,\ell]$, then in general even in the case of regular boundary conditions eigenvalues split into two branches: the “good” branch lies in the horizontal strip and is close to the eigenvalues of the unperturbed BVP, while the “bad” branch has non-zero density and imaginary parts that tend to infinity. We illustrate this effect on a concrete $2 \times 2$ example.

Bibliographie

[1] G. D. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations”, Trans. Amer. Math. Soc., 9 (1908), 373–395 | DOI | MR

[2] G. D. Birkhoff, R. E. Langer, “The boundary problems and developments associated with a system of ordinary differential equations of the first order”, Proc. Amer. Acad. Arts Sci., 58 (1923), 49–128 | DOI | MR

[3] J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, X, Academic Press, New York–London, 1960 | MR

[4] P. Djakov, B. Mityagin, “Bari–Markus property for Riesz projections of 1D periodic Dirac operators”, Math. Nachr., 283:3 (2010), 443–462 | DOI | MR

[5] P. Djakov and B. Mityagin, “Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions”, Indiana Univ. Math. J., 61:1 (2012), 359–398 | DOI | MR

[6] F. Gantmacher, Theory of Matrices, AMS Chelsea publishing, 1959 | MR

[7] A. M. Gomilko, L. Rzepnicki, “On asymptotic behaviour of solutions of the Dirac system and applications to the Sturm-Liouville problem with a singular potential”, J. Spect. Theory, 10:3 (2020), 747–786 | DOI | MR

[8] V.É. Katsnel'son, “Exponential bases in $L^2$”, Funct. Anal. Appl., 5:1 (1971), 31–38 | DOI

[9] B. Ya. Levin, “Exponential bases in $L^2$”, Zapiski Matem. Otd. Fiz.-Matem. F-ta Khar'kovskogo Un-ta i Khar'kovskogo Matem. Ob-va, 27:4 (1961), 39–48

[10] B. Ya. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monographs, Amer. Math. Soc., Providence, 1964 | DOI | MR

[11] B. Ya. Levin, Lectures on Entire Functions, Transl. Math. Monographs, 150, Amer. Math. Soc., Providence, RI, 1996 (with Yu. Lyubarskii, M. Sodin, V. Tkachenko) | DOI | MR

[12] B. M. Levitan, I. S. Sargsyan, Sturm–Liouville And Dirac Operators, Kluwer, Dordrecht, 1991 | MR

[13] A. A. Lunyov, M. M. Malamud, “On spectral synthesis for dissipative Dirac type operators”, Integr. Equ. Oper. Theory, 90 (2014), 79–106 | DOI | MR

[14] A. A. Lunyov, M. M. Malamud, “On the Riesz dasis property of the root vector system for Dirac type $2 \times 2$ systems”, Dokl. Math., 90:2 (2014), 556–561 | DOI | MR

[15] A. A. Lunyov, M. M. Malamud, “On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications”, J. Spectral Theory, 5:1 (2015), 17–70 | DOI | MR

[16] A. A. Lunyov, M. M. Malamud, “On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators”, J. Math. Anal. Appl., 441 (2016), 57–103 | DOI | MR

[17] A. A. Lunyov, M. M. Malamud, On transformation operators and Riesz basis property of root vectors system for $n \times n$ Dirac type operators. Application to the Timoshenko beam model, 2021, arXiv: 2112.07248

[18] A. A. Lunyov, M. M. Malamud, “Stability of spectral characteristics of boundary value problems for $2 \times 2$ Dirac type systems. Applications to the damped string”, J. Diff. Eq., 313 (2022), 633–742 | DOI | MR

[19] A. S. Makin, “Regular boundary value problems for the Dirac operator”, Doklady Mathematics, 101:3 (2020), 214–217 | DOI | MR

[20] A. S. Makin, “On the spectrum of two-point boundary value problems for the Dirac operator”, Diff. Eq., 57:8 (2021), 993–1002 | DOI | MR

[21] A. S. Makin, “On convergence of spectral expansions of Dirac operators with regular boundary conditions”, Math. Nachr., 295:1 (2022), 189–210 | DOI | MR

[22] M. M. Malamud, “Similarity of Volterra operators and related questions of the theory of differential equations of fractional order”, Trans. Moscow Math. Soc., 55 (1994), 57–122 | MR

[23] M. M. Malamud, L. L. Oridoroga, “On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations”, J. Funct. Anal., 263 (2012), 1939–1980 | DOI | MR

[24] V. A. Marchenko, Sturm–Liouville operators and applications, Operator Theory: Advances and Appl., 22, Birkhäuser Verlag, Basel, 1986 | MR

[25] M. Marcus, “Determinants of Sums”, College Math. J., 1990, March

[26] J. S. Muldowney, “Compound matrices and ordinary differential equations”, Rocky Mountain J. Math., 20:4 (1990), 857–872 | DOI | MR

[27] M. A. Naimark, Linear Differential Operators, v. I, Frederick Ungar Publishing Co., New York, 1967 | MR

[28] S. P. Novikov, S. V. Manakov, L. P. Pitaevskij, V. E. Zakharov, Theory of solitons. The inverse scattering method, Springer-Verlag, 1984 | MR

[29] L. Rzepnicki, “Asymptotic behavior of solutions of the Dirac system with an integrable potential”, Integral Equations Operator Theory, 93 (2021), 55, 24 pp., arXiv: 2011.06510 | DOI | MR

[30] A. M. Savchuk, I. V. Sadovnichaya, “Spectral analysis of a one-dimensional Dirac system with summable potential and a Sturm–Liouville operator with distribution coefficients”, Sovrem. Mat. Fundam. Napravl., 66:3 (2020), 373–530 | MR

[31] A. M. Savchuk, A. A. Shkalikov, “The Dirac operator with complex-valued summable potential”, Math. Notes, 96:5–6 (2014), 777–810 | DOI | MR

[32] B. Schwarz, “Totally positive differential systems”, Pacific J. Math., 32 (1970), 203–229 | DOI | MR

[33] A. A. Shkalikov, “Regular spectral problems of hyperbolic type for systems of ordinary differential equations of the first order”, Math. Notes, 110:5 (2021), 806–810 | DOI | MR

[34] J. D. Tamarkin, “Sur quelques points de la theorie des equations differentielles lineaires ordinaires et sur la generalisation de la serie de Fourier”, Rend. Circ. Mat. Palermo, 34:2 (1912), 345–382 | DOI

[35] J. D. Tamarkin, On some general problems of the theory of ordinary linear differential operators and the expansion of arbitrary functions into series, Petrograd, 1917 | MR