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 and 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), 249-315. [2] M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform," SIAM Studies in Applied Mathematics, 4, SIAM, Philadelphia, Pa., 1981. [3] Z. S. Agranovich and V. A. Marchenko, "The Inverse Problem of Scattering Theory," Translated from the Russian by B. D. Seckler, Gordon and Breach Science Publishers, New York-London, 1963. [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), 167-212. [5] F. V. Atkinson, "Discrete and Continuous Boundary Problems," Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. [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), 936-972. doi: 10.1137/080713306. [7] W.-J. Beyn and J. Lorenz, Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals, Numer. Funct. Anal. Optim., 20 (1999), 201-244. [8] V. Borovyk and K. A. Makarov, On the weak and ergodic limit of the spectral shift function, preprint, arXiv:0911.3880v1. [9] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory," 2nd edition, With a foreword by R. G. Newton, Texts and Monographs in Physics, Springer-Verlag, New York, 1989. [10] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics for matrix-valued Schrödinger operators, Proc. London Math. Soc. (3), 82 (2001), 701-724. doi: 10.1112/plms/82.3.701. [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), 3475-3534. doi: 10.1090/S0002-9947-02-03025-8. [12] E. A. Coddington and N. Levinson, "The Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [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, Interscience Publishers, John Wiley & Sons, New York-London, 1963. [14] M. S. P. Eastham, "The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem," London Mathematical Society Monographs, New Series, 4, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1989. [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), 181-203. doi: 10.1016/0022-0396(91)90138-Y. [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), 1197-1222. doi: 10.1512/iumj.1990.39.39054. [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, Part 2, AMS, Providence, RI, 2007. [18] F. Gesztesy, Y. Latushkin and K. A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal., 186 (2007), 361-421. doi: 10.1007/s00205-007-0071-7. [19] F. Gesztesy, Y. Latushkin, M. Mitrea and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys., 12 (2005), 443-471. [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), 160-200. doi: 10.1016/j.matpur.2008.04.001. [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), 457-497; See also Erratum, 48 (2004), 425-426 and the corrected electronic only version in, 48 (2004), 561-602. [22] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc., 348 (1996), 349-373. doi: 10.1090/S0002-9947-96-01525-5. [23] F. Gesztesy, B. Simon and G. Teschl, Spectral deformations of one-dimensional Schrödinger operators, J. Anal. Math., 70 (1996), 267-324. doi: 10.1007/BF02820446. [24] F. Gesztesy, B. Simon and G. Teschl, Zeros of the Wronskian and renormalized oscillation theory, Amer. J. Math., 118 (1996), 571-594. doi: 10.1353/ajm.1996.0024. [25] J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D, 220 (2006), 116-126. doi: 10.1016/j.physd.2006.07.003. [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), 272-286. doi: 10.1137/0515022. [27] T. Kapitula and J. Rubin, Existence and stability of standing hole solutions to complex Ginzburg-Landau equations, Nonlinearity, 13 (2000), 77-112. doi: 10.1088/0951-7715/13/1/305. [28] T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans function techniques, SIAM J. Math. Anal., 33 (2002), 1117-1143. doi: 10.1137/S0036141000372301. [29] T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans function, Discrete Contin. Dyn. Syst., 10 (2004), 857-869. doi: 10.3934/dcds.2004.10.857. [30] K. Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Amer. J. Math., 72 (1950), 502-544. doi: 10.2307/2372051. [31] A. Krall, "Hilbert Space, Boundary Value Problems and Orthogonal Polynomials," Operator Theory: Advances and Applications, 133, Birkhauser Verlag, Basel, 2002. [32] A. Krall, $M(\lambda)$ theory for singular Hamiltonian systems with two singular points, SIAM J. Math. Anal., 20 (1989), 701-715. doi: 10.1137/0520048. [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), 659-676. doi: 10.1134/S0965542508040131. [34] D. B. Pearson, "Quantum Scattering and Spectral Theory," Techniques of Physics, 9, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1988. [35] R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94. doi: 10.1098/rsta.1992.0055. [36] J. Rademacher, B. Sandstede and A. Scheel, Computing absolute and essential spectra using continuation, Phys. D, 229 (2007), 166-183. doi: 10.1016/j.physd.2007.03.016. [37] B. Sandstede, Stability of travelling waves, in "Handbook of Dynamical Systems," Vol. 2 (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, (2002), 983-1055. [38] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7. [39] B. Sandstede and A. Scheel, Evans function and blow-up methods in critical eigenvalue problems, Discrete Contin. Dyn. Syst., 10 (2004), 941-964. doi: 10.3934/dcds.2004.10.941. [40] H. Sun and Y. Shi, Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints, J. Funct. Anal., 259 (2010), 2003-2027. doi: 10.1016/j.jfa.2010.06.008. [41] G. Teschl, On the approximation of isolated eigenvalues of ordinary differential operators, Proc. Amer. Math. Soc., 136 (2008), 2473-2476. doi: 10.1090/S0002-9939-08-09140-5. [42] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, in "Handbook of Mathematical Fluid Dynamics," III, With an appendix by Helge Kristian Jenssen and Gregory Lyng, North-Holland, Amsterdam, (2004), 311-533. [43] K. Zumbrun, Multidimensional stability of planar viscous shock waves, in "Advances in the Theory of Shock Waves" (eds. T.-P. Liu, H. Freistühler and A. Szepessy), Progress Nonlin. Diff. Eqs. Appls., 47, Birkhäuser Boston, Boston, (2001), 307-516. [44] J. Weidmann, "Spectral Theory of Ordinary Differential Operators," Lecture Notes in Mathematics, 1258, Springer-Verlag, Berlin, 1987. [45] J. Weidmann, Spectral theory of Sturm-Liouville operators. Approximation by regular problems, in "Sturm-Liouville Theory" (eds. W. O. Amrein, A. M. Hinz and D. B. Pearson), Birkhäuser, Basel, (2005), 75-98. doi: 10.1007/3-7643-7359-8_4.

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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), 249-315. [2] M. J. Ablowitz and H. Segur, "Solitons and the Inverse Scattering Transform," SIAM Studies in Applied Mathematics, 4, SIAM, Philadelphia, Pa., 1981. [3] Z. S. Agranovich and V. A. Marchenko, "The Inverse Problem of Scattering Theory," Translated from the Russian by B. D. Seckler, Gordon and Breach Science Publishers, New York-London, 1963. [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), 167-212. [5] F. V. Atkinson, "Discrete and Continuous Boundary Problems," Mathematics in Science and Engineering, Vol. 8, Academic Press, New York-London, 1964. [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), 936-972. doi: 10.1137/080713306. [7] W.-J. Beyn and J. Lorenz, Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals, Numer. Funct. Anal. Optim., 20 (1999), 201-244. [8] V. Borovyk and K. A. Makarov, On the weak and ergodic limit of the spectral shift function, preprint, arXiv:0911.3880v1. [9] K. Chadan and P. C. Sabatier, "Inverse Problems in Quantum Scattering Theory," 2nd edition, With a foreword by R. G. Newton, Texts and Monographs in Physics, Springer-Verlag, New York, 1989. [10] S. Clark and F. Gesztesy, Weyl-Titchmarsh $M$-function asymptotics for matrix-valued Schrödinger operators, Proc. London Math. Soc. (3), 82 (2001), 701-724. doi: 10.1112/plms/82.3.701. [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), 3475-3534. doi: 10.1090/S0002-9947-02-03025-8. [12] E. A. Coddington and N. Levinson, "The Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [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, Interscience Publishers, John Wiley & Sons, New York-London, 1963. [14] M. S. P. Eastham, "The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem," London Mathematical Society Monographs, New Series, 4, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1989. [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), 181-203. doi: 10.1016/0022-0396(91)90138-Y. [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), 1197-1222. doi: 10.1512/iumj.1990.39.39054. [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, Part 2, AMS, Providence, RI, 2007. [18] F. Gesztesy, Y. Latushkin and K. A. Makarov, Evans functions, Jost functions, and Fredholm determinants, Arch. Rat. Mech. Anal., 186 (2007), 361-421. doi: 10.1007/s00205-007-0071-7. [19] F. Gesztesy, Y. Latushkin, M. Mitrea and M. Zinchenko, Nonselfadjoint operators, infinite determinants, and some applications, Russ. J. Math. Phys., 12 (2005), 443-471. [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), 160-200. doi: 10.1016/j.matpur.2008.04.001. [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), 457-497; See also Erratum, 48 (2004), 425-426 and the corrected electronic only version in, 48 (2004), 561-602. [22] F. Gesztesy and B. Simon, Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators, Trans. Amer. Math. Soc., 348 (1996), 349-373. doi: 10.1090/S0002-9947-96-01525-5. [23] F. Gesztesy, B. Simon and G. Teschl, Spectral deformations of one-dimensional Schrödinger operators, J. Anal. Math., 70 (1996), 267-324. doi: 10.1007/BF02820446. [24] F. Gesztesy, B. Simon and G. Teschl, Zeros of the Wronskian and renormalized oscillation theory, Amer. J. Math., 118 (1996), 571-594. doi: 10.1353/ajm.1996.0024. [25] J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D, 220 (2006), 116-126. doi: 10.1016/j.physd.2006.07.003. [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), 272-286. doi: 10.1137/0515022. [27] T. Kapitula and J. Rubin, Existence and stability of standing hole solutions to complex Ginzburg-Landau equations, Nonlinearity, 13 (2000), 77-112. doi: 10.1088/0951-7715/13/1/305. [28] T. Kapitula and B. Sandstede, Edge bifurcations for near integrable systems via Evans function techniques, SIAM J. Math. Anal., 33 (2002), 1117-1143. doi: 10.1137/S0036141000372301. [29] T. Kapitula and B. Sandstede, Eigenvalues and resonances using the Evans function, Discrete Contin. Dyn. Syst., 10 (2004), 857-869. doi: 10.3934/dcds.2004.10.857. [30] K. Kodaira, On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Amer. J. Math., 72 (1950), 502-544. doi: 10.2307/2372051. [31] A. Krall, "Hilbert Space, Boundary Value Problems and Orthogonal Polynomials," Operator Theory: Advances and Applications, 133, Birkhauser Verlag, Basel, 2002. [32] A. Krall, $M(\lambda)$ theory for singular Hamiltonian systems with two singular points, SIAM J. Math. Anal., 20 (1989), 701-715. doi: 10.1137/0520048. [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), 659-676. doi: 10.1134/S0965542508040131. [34] D. B. Pearson, "Quantum Scattering and Spectral Theory," Techniques of Physics, 9, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1988. [35] R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A, 340 (1992), 47-94. doi: 10.1098/rsta.1992.0055. [36] J. Rademacher, B. Sandstede and A. Scheel, Computing absolute and essential spectra using continuation, Phys. D, 229 (2007), 166-183. doi: 10.1016/j.physd.2007.03.016. [37] B. Sandstede, Stability of travelling waves, in "Handbook of Dynamical Systems," Vol. 2 (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, (2002), 983-1055. [38] B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D, 145 (2000), 233-277. doi: 10.1016/S0167-2789(00)00114-7. [39] B. Sandstede and A. Scheel, Evans function and blow-up methods in critical eigenvalue problems, Discrete Contin. Dyn. Syst., 10 (2004), 941-964. doi: 10.3934/dcds.2004.10.941. [40] H. Sun and Y. Shi, Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints, J. Funct. Anal., 259 (2010), 2003-2027. doi: 10.1016/j.jfa.2010.06.008. [41] G. Teschl, On the approximation of isolated eigenvalues of ordinary differential operators, Proc. Amer. Math. Soc., 136 (2008), 2473-2476. doi: 10.1090/S0002-9939-08-09140-5. [42] K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, in "Handbook of Mathematical Fluid Dynamics," III, With an appendix by Helge Kristian Jenssen and Gregory Lyng, North-Holland, Amsterdam, (2004), 311-533. [43] K. Zumbrun, Multidimensional stability of planar viscous shock waves, in "Advances in the Theory of Shock Waves" (eds. T.-P. Liu, H. Freistühler and A. Szepessy), Progress Nonlin. Diff. Eqs. Appls., 47, Birkhäuser Boston, Boston, (2001), 307-516. [44] J. Weidmann, "Spectral Theory of Ordinary Differential Operators," Lecture Notes in Mathematics, 1258, Springer-Verlag, Berlin, 1987. [45] J. Weidmann, Spectral theory of Sturm-Liouville operators. Approximation by regular problems, in "Sturm-Liouville Theory" (eds. W. O. Amrein, A. M. Hinz and D. B. Pearson), Birkhäuser, Basel, (2005), 75-98. doi: 10.1007/3-7643-7359-8_4.
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