American Institute of Mathematical Sciences

October  2012, 5(5): 939-970. doi: 10.3934/dcdss.2012.5.939

The Evans function and the Weyl-Titchmarsh function

 1 Department of Mathematics, University of Missouri, Columbia, MO 65211, United States, United States

Received  March 2011 Revised  August 2011 Published  January 2012

We describe relations between the Evans function, a modern tool in the study of stability of traveling waves and other patterns for PDEs, and the classical Weyl-Titchmarsh function for singular Sturm-Liouville differential expressions and for matrix Hamiltonian systems. Also, for the scalar Schrödinger equation, we discuss a related issue of approximating eigenvalue problems on the whole line by that on finite segments.
Citation: Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
References:
 [1] M. Ablowitz, D. Kaup, A. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems,, Stud. Appl. Math., 53 (1974), 249. Google Scholar [2] M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform,", SIAM Studies in Applied Mathematics, 4 (1981). Google Scholar [3] Z. S. Agranovich and V. A. Marchenko, "The Inverse Problem of Scattering Theory,", Translated from the Russian by B. D. Seckler, (1963). Google Scholar [4] J. Alexander, R. Gardner and C. Jones, A topological invariant arising in the stability analysis of travelling waves,, J. Reine Angew. Math., 410 (1990), 167. Google Scholar [5] F. V. Atkinson, "Discrete and Continuous Boundary Problems,", Mathematics in Science and Engineering, (1964). Google Scholar [6] M. Beck, J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns,, SIAM J. Math. Anal., 41 (2009), 936. doi: 10.1137/080713306. Google Scholar [7] W.-J. Beyn and J. Lorenz, Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals,, Numer. Funct. Anal. Optim., 20 (1999), 201. Google Scholar [8] V. Borovyk and K. A. Makarov, On the weak and ergodic limit of the spectral shift function,, preprint, (). Google Scholar [9] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory,", 2nd edition, (1989). Google Scholar [10] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics for matrix-valued Schrödinger operators,, Proc. London Math. Soc. (3), 82 (2001), 701. doi: 10.1112/plms/82.3.701. Google Scholar [11] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators,, Trans. Amer. Math. Soc., 354 (2002), 3475. doi: 10.1090/S0002-9947-02-03025-8. Google Scholar [12] E. A. Coddington and N. Levinson, "The Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (1955). Google Scholar [13] N. Dunford and J. T. Schwartz, "Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space,", With the assistance of William G. Bade and Robert G. Bartle, (1963). Google Scholar [14] M. S. P. Eastham, "The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem,", London Mathematical Society Monographs, 4 (1989). Google Scholar [15] R. A. Gardner and C. K. R. T. Jones, A stability index for steady state solutions of boundary value problems for parabolic systems,, J. Diff. Eqns., 91 (1991), 181. doi: 10.1016/0022-0396(91)90138-Y. Google Scholar [16] R. A. Gardner and C. K. R. T. Jones, Travelling waves of a perturbed diffusion equation arising in a phase field model,, Indiana Univ. Math. J., 39 (1990), 1197. doi: 10.1512/iumj.1990.39.39054. Google Scholar [17] F. Gesztesy, Inverse spectral theory as influenced by Barry Simon, in "Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday," 741-820,, Proc. Sympos. Pure Math., 76 (2007). Google Scholar [18] F. Gesztesy, Y. Latushkin and K. A. Makarov, Evans functions, Jost functions, and Fredholm determinants,, Arch. Rat. Mech. Anal., 186 (2007), 361. doi: 10.1007/s00205-007-0071-7. Google Scholar [19] F. Gesztesy, Y. Latushkin, M. Mitrea and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications,, Russ. J. Math. Phys., 12 (2005), 443. Google Scholar [20] F. Gesztesy, Y. Latushkin and K. Zumbrun, Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves,, J. Math. Pures Appl. (9), 90 (2008), 160. doi: 10.1016/j.matpur.2008.04.001. Google Scholar [21] F. Gesztesy and K. A. Makarov, (Modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels revisited,, Integral Eq. Operator Theory, 47 (2003), 457. Google Scholar [22] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators,, Trans. Amer. Math. Soc., 348 (1996), 349. doi: 10.1090/S0002-9947-96-01525-5. Google Scholar [23] F. Gesztesy, B. Simon and G. Teschl, Spectral deformations of one-dimensional Schrödinger operators,, J. Anal. Math., 70 (1996), 267. doi: 10.1007/BF02820446. Google Scholar [24] F. Gesztesy, B. Simon and G. Teschl, Zeros of the Wronskian and renormalized oscillation theory,, Amer. J. Math., 118 (1996), 571. doi: 10.1353/ajm.1996.0024. Google Scholar [25] J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems,, Phys. D, 220 (2006), 116. doi: 10.1016/j.physd.2006.07.003. Google Scholar [26] D. B. Hinton and J. K. Shaw, On boundary value problems for Hamiltonian systems with two singular points,, SIAM J. Math. Anal., 15 (1984), 272. doi: 10.1137/0515022. Google Scholar [27] T. Kapitula and J. Rubin, Existence and stability of standing hole solutions to complex Ginzburg-Landau equations,, Nonlinearity, 13 (2000), 77. doi: 10.1088/0951-7715/13/1/305. Google Scholar [28] T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans function techniques,, SIAM J. Math. Anal., 33 (2002), 1117. doi: 10.1137/S0036141000372301. Google Scholar [29] T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans function,, Discrete Contin. Dyn. Syst., 10 (2004), 857. doi: 10.3934/dcds.2004.10.857. Google Scholar [30] K. Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions,, Amer. J. Math., 72 (1950), 502. doi: 10.2307/2372051. Google Scholar [31] A. Krall, "Hilbert Space, Boundary Value Problems and Orthogonal Polynomials,", Operator Theory: Advances and Applications, 133 (2002). Google Scholar [32] A. Krall, $M(\lambda)$ theory for singular Hamiltonian systems with two singular points,, SIAM J. Math. Anal., 20 (1989), 701. doi: 10.1137/0520048. Google Scholar [33] N. Kulagin, L. Lerman and T. Shmakova, Fronts and traveling fronts, and their stability in the generalized Swift-Hohenberg equation,, Comput. Math. Math. Phys., 48 (2008), 659. doi: 10.1134/S0965542508040131. Google Scholar [34] D. B. Pearson, "Quantum Scattering and Spectral Theory,", Techniques of Physics, 9 (1988). Google Scholar [35] R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves,, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47. doi: 10.1098/rsta.1992.0055. Google Scholar [36] J. Rademacher, B. Sandstede and A. Scheel, Computing absolute and essential spectra using continuation,, Phys. D, 229 (2007), 166. doi: 10.1016/j.physd.2007.03.016. Google Scholar [37] B. Sandstede, Stability of travelling waves,, in, (2002), 983. Google Scholar [38] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains,, Phys. D, 145 (2000), 233. doi: 10.1016/S0167-2789(00)00114-7. Google Scholar [39] B. Sandstede and A. Scheel, Evans function and blow-up methods in critical eigenvalue problems,, Discrete Contin. Dyn. Syst., 10 (2004), 941. doi: 10.3934/dcds.2004.10.941. Google Scholar [40] H. Sun and Y. Shi, Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints,, J. Funct. Anal., 259 (2010), 2003. doi: 10.1016/j.jfa.2010.06.008. Google Scholar [41] G. Teschl, On the approximation of isolated eigenvalues of ordinary differential operators,, Proc. Amer. Math. Soc., 136 (2008), 2473. doi: 10.1090/S0002-9939-08-09140-5. Google Scholar [42] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations,, in, III (2004), 311. Google Scholar [43] K. Zumbrun, Multidimensional stability of planar viscous shock waves,, in, 47 (2001), 307. Google Scholar [44] J. Weidmann, "Spectral Theory of Ordinary Differential Operators,", Lecture Notes in Mathematics, 1258 (1987). Google Scholar [45] J. Weidmann, Spectral theory of Sturm-Liouville operators. Approximation by regular problems,, in, (2005), 75. doi: 10.1007/3-7643-7359-8_4. Google Scholar

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References:
 [1] M. Ablowitz, D. Kaup, A. Newell and H. Segur, The inverse scattering transform - Fourier analysis for nonlinear problems,, Stud. Appl. Math., 53 (1974), 249. Google Scholar [2] M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform,", SIAM Studies in Applied Mathematics, 4 (1981). Google Scholar [3] Z. S. Agranovich and V. A. Marchenko, "The Inverse Problem of Scattering Theory,", Translated from the Russian by B. D. Seckler, (1963). Google Scholar [4] J. Alexander, R. Gardner and C. Jones, A topological invariant arising in the stability analysis of travelling waves,, J. Reine Angew. Math., 410 (1990), 167. Google Scholar [5] F. V. Atkinson, "Discrete and Continuous Boundary Problems,", Mathematics in Science and Engineering, (1964). Google Scholar [6] M. Beck, J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns,, SIAM J. Math. Anal., 41 (2009), 936. doi: 10.1137/080713306. Google Scholar [7] W.-J. Beyn and J. Lorenz, Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals,, Numer. Funct. Anal. Optim., 20 (1999), 201. Google Scholar [8] V. Borovyk and K. A. Makarov, On the weak and ergodic limit of the spectral shift function,, preprint, (). Google Scholar [9] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory,", 2nd edition, (1989). Google Scholar [10] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics for matrix-valued Schrödinger operators,, Proc. London Math. Soc. (3), 82 (2001), 701. doi: 10.1112/plms/82.3.701. Google Scholar [11] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators,, Trans. Amer. Math. Soc., 354 (2002), 3475. doi: 10.1090/S0002-9947-02-03025-8. Google Scholar [12] E. A. Coddington and N. Levinson, "The Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (1955). Google Scholar [13] N. Dunford and J. T. Schwartz, "Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space,", With the assistance of William G. Bade and Robert G. Bartle, (1963). Google Scholar [14] M. S. P. Eastham, "The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem,", London Mathematical Society Monographs, 4 (1989). Google Scholar [15] R. A. Gardner and C. K. R. T. Jones, A stability index for steady state solutions of boundary value problems for parabolic systems,, J. Diff. Eqns., 91 (1991), 181. doi: 10.1016/0022-0396(91)90138-Y. Google Scholar [16] R. A. Gardner and C. K. R. T. Jones, Travelling waves of a perturbed diffusion equation arising in a phase field model,, Indiana Univ. Math. J., 39 (1990), 1197. doi: 10.1512/iumj.1990.39.39054. Google Scholar [17] F. Gesztesy, Inverse spectral theory as influenced by Barry Simon, in "Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday," 741-820,, Proc. Sympos. Pure Math., 76 (2007). Google Scholar [18] F. Gesztesy, Y. Latushkin and K. A. Makarov, Evans functions, Jost functions, and Fredholm determinants,, Arch. Rat. Mech. Anal., 186 (2007), 361. doi: 10.1007/s00205-007-0071-7. Google Scholar [19] F. Gesztesy, Y. Latushkin, M. Mitrea and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications,, Russ. J. Math. Phys., 12 (2005), 443. Google Scholar [20] F. Gesztesy, Y. Latushkin and K. Zumbrun, Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves,, J. Math. Pures Appl. (9), 90 (2008), 160. doi: 10.1016/j.matpur.2008.04.001. Google Scholar [21] F. Gesztesy and K. A. Makarov, (Modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels revisited,, Integral Eq. Operator Theory, 47 (2003), 457. Google Scholar [22] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators,, Trans. Amer. Math. Soc., 348 (1996), 349. doi: 10.1090/S0002-9947-96-01525-5. Google Scholar [23] F. Gesztesy, B. Simon and G. Teschl, Spectral deformations of one-dimensional Schrödinger operators,, J. Anal. Math., 70 (1996), 267. doi: 10.1007/BF02820446. Google Scholar [24] F. Gesztesy, B. Simon and G. Teschl, Zeros of the Wronskian and renormalized oscillation theory,, Amer. J. Math., 118 (1996), 571. doi: 10.1353/ajm.1996.0024. Google Scholar [25] J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems,, Phys. D, 220 (2006), 116. doi: 10.1016/j.physd.2006.07.003. Google Scholar [26] D. B. Hinton and J. K. Shaw, On boundary value problems for Hamiltonian systems with two singular points,, SIAM J. Math. Anal., 15 (1984), 272. doi: 10.1137/0515022. Google Scholar [27] T. Kapitula and J. Rubin, Existence and stability of standing hole solutions to complex Ginzburg-Landau equations,, Nonlinearity, 13 (2000), 77. doi: 10.1088/0951-7715/13/1/305. Google Scholar [28] T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans function techniques,, SIAM J. Math. Anal., 33 (2002), 1117. doi: 10.1137/S0036141000372301. Google Scholar [29] T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans function,, Discrete Contin. Dyn. Syst., 10 (2004), 857. doi: 10.3934/dcds.2004.10.857. Google Scholar [30] K. Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions,, Amer. J. Math., 72 (1950), 502. doi: 10.2307/2372051. Google Scholar [31] A. Krall, "Hilbert Space, Boundary Value Problems and Orthogonal Polynomials,", Operator Theory: Advances and Applications, 133 (2002). Google Scholar [32] A. Krall, $M(\lambda)$ theory for singular Hamiltonian systems with two singular points,, SIAM J. Math. Anal., 20 (1989), 701. doi: 10.1137/0520048. Google Scholar [33] N. Kulagin, L. Lerman and T. Shmakova, Fronts and traveling fronts, and their stability in the generalized Swift-Hohenberg equation,, Comput. Math. Math. Phys., 48 (2008), 659. doi: 10.1134/S0965542508040131. Google Scholar [34] D. B. Pearson, "Quantum Scattering and Spectral Theory,", Techniques of Physics, 9 (1988). Google Scholar [35] R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves,, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47. doi: 10.1098/rsta.1992.0055. Google Scholar [36] J. Rademacher, B. Sandstede and A. Scheel, Computing absolute and essential spectra using continuation,, Phys. D, 229 (2007), 166. doi: 10.1016/j.physd.2007.03.016. Google Scholar [37] B. Sandstede, Stability of travelling waves,, in, (2002), 983. Google Scholar [38] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains,, Phys. D, 145 (2000), 233. doi: 10.1016/S0167-2789(00)00114-7. Google Scholar [39] B. Sandstede and A. Scheel, Evans function and blow-up methods in critical eigenvalue problems,, Discrete Contin. Dyn. Syst., 10 (2004), 941. doi: 10.3934/dcds.2004.10.941. Google Scholar [40] H. Sun and Y. Shi, Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints,, J. Funct. Anal., 259 (2010), 2003. doi: 10.1016/j.jfa.2010.06.008. Google Scholar [41] G. Teschl, On the approximation of isolated eigenvalues of ordinary differential operators,, Proc. Amer. Math. Soc., 136 (2008), 2473. doi: 10.1090/S0002-9939-08-09140-5. Google Scholar [42] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations,, in, III (2004), 311. Google Scholar [43] K. Zumbrun, Multidimensional stability of planar viscous shock waves,, in, 47 (2001), 307. Google Scholar [44] J. Weidmann, "Spectral Theory of Ordinary Differential Operators,", Lecture Notes in Mathematics, 1258 (1987). Google Scholar [45] J. Weidmann, Spectral theory of Sturm-Liouville operators. Approximation by regular problems,, in, (2005), 75. doi: 10.1007/3-7643-7359-8_4. Google Scholar
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