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

show all references

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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