# American Institute of Mathematical Sciences

March  2014, 13(2): 495-510. doi: 10.3934/cpaa.2014.13.495

## Schrödinger-like operators and the eikonal equation

 1 Departamento de Ciencias Básicas, UAM-A, Av. San Pablo 180, Col. Reynosa, Mèxico D. F. 02200, Mexico

Received  June 2012 Revised  August 2013 Published  October 2013

Let $V$ be a real-valued function of class $C^5$ on $\mathbb{R}^n$, $n \geq 2$, and suppose that $\partial^\alpha V(x)=O(|x|^{-|\alpha|})$, as $|x| \to \infty$, for $|\alpha| \leq 5$. For $\lambda > 0$ we set $W_\lambda(x) = 1-(V(x)/\lambda)$ and consider the Schrödinger-like operator $\mathcal{H}_\lambda=W_\lambda^{-{1/2}} H_0 W_\lambda^{-{1/2}}$ acting on $L^2(\mathbb{R}^n)$, where $H_0=-\Delta$ is the classical laplacian on $\mathbb{R}^n$. Using properties of the maximal solution to the eikonal equation $|\nabla S_\lambda|^2=W_\lambda$, for $\lambda$ sufficiently large we establish the behavior of $(\mathcal{H}_\lambda-z^2)^{-1}$ as Im $z\downarrow 0$ in the framework of Besov Spaces $B(\mathbb{R}^n)$. For $k\in \mathbb{R}\setminus\{0\}$ and $f\in B(\mathbb{R}^n)$ we find the unique solution to $-\Delta u-k^2 W_\lambda u = f$ on $\mathbb{R}^n$ that satisfies a certain radiation condition. These results can be applied to the study of the scattering theory of the Schrödinger operator $H=-\Delta+V$.
Citation: Jaime Cruz-Sampedro. Schrödinger-like operators and the eikonal equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495
##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, 1975. [2] S. Agmon, "Lectures on Elliptic Boundary Value Problems," D. Van Nostrand Co. In., 1965. [3] S. Agmon, "Unicité et convexité dans les problèmes différentiels," Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que., 1966. [4] S. Agmon, Lower bounds for solutions of Schrdinger equations, Journal D'analyse Mathématique, 23 (1970), 1-25, [5] S. Agmon, "Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations. Bounds on Eigenfunctions of N-Body Schrödinger Operators," Mathematical Notes 29, Princeton University Press, 1982. [6] S. Agmon, On the asymptotic behavior of solutions of Schröinger type equations in unbounded domains, Analyse mathématique et applications, 122, Gauthier-Villars, Montrouge, 1988. [7] S. Agmon, Representation theorems for solutions of the Helmholtz equation on $\mathbbR^n$, Differential Operators and Spectral Theory, 27-43, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999. [8] S. Agmon, J. Cruz-Sampedro and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero, Journal of Functional Analysis, 167 (1999), 345-369. [9] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, Journal D'Analyse Mathématique, 30 (1976). [10] S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math., 20 (1967), 207-229. [11] G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions, Commun. in Partial Differential Equations, 12 (1987), 263-283. [12] M. Ben-Artzi, Unitary equivalence and scattering theory for Stark-like Hamiltonians, J. Math. Phys., 25 (1984), 951-964. [13] P. Constantin, Scattering for Schröinger operators in a class of domains with noncompact boundaries, J. Funct. Anal., 44 (1981), 87-119. [14] J. Cruz-Sampedro, Exact asymptotic behavior at infinity of solutions to abstract second-order differential inequalities in Hilbert spaces, Math. Z., 237 (2001), 727-235. [15] J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076. [16] A. Hassell, R. Melrose and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math., 181 (2004), 1-87. [17] A. Hassell, R. Melrose and A. Vasy, Microlocal propagation near radial points and scattering for symbolic potentials of order zero, Anal. PDE, 1 (2008), 127-196. [18] L. Hörmander, "The Analysis of Linear Partial Differential Operators III," Springer-Verlag, Berlin, 1985. doi: 978-3-540-49938-1. [19] W. Jäger, Über das Dirichletsche Außenraumproblem für die Schwingungsgleichung, Math. Z., 95 (1967), 299-323. [20] W. Jäger, Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten, Math. Z., 102 (1967), 62-88. [21] W. Jäger, Das asymptotische Verhalten von Lsngen eines Typs von Differentialgleichungen, Math. Z., 112 (1969), 26-36. [22] W. Jäger and P. Rejto, Limiting absorption principle for some Schrödinger operators with exploding potentials. II, J. Math. Anal. Appl., 95 (1983), 169-194. [23] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. Math., 121 (1985), 463-494. [24] A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators, J. Operator Theory, 14 (1985), 181-188. [25] P. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," Pitman, London, 1982. [26] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, II Fourier Analysis Self-Adjontness," New York, Academic Press, 1978. [27] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory," New York, Academic Press, 1979. [28] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978. [29] Y. Saitō, "Spectral Representations for Schrödinger Operators with Long-range Potentials," Lecture Notes in Mathematics, 727. Springer, Berlin, 1979. doi: 978-3-540-35132-0. [30] Y. Saitō, Schrödinger operators with a nonspherical radiation condition, Pacific J. Math., 126 (1987), 331-359. [31] I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems," Lecture Notes in Mathematics 1011, Springer Verlag, 1983. doi: 978-3-540-38664-3.

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##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, 1975. [2] S. Agmon, "Lectures on Elliptic Boundary Value Problems," D. Van Nostrand Co. In., 1965. [3] S. Agmon, "Unicité et convexité dans les problèmes différentiels," Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que., 1966. [4] S. Agmon, Lower bounds for solutions of Schrdinger equations, Journal D'analyse Mathématique, 23 (1970), 1-25, [5] S. Agmon, "Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations. Bounds on Eigenfunctions of N-Body Schrödinger Operators," Mathematical Notes 29, Princeton University Press, 1982. [6] S. Agmon, On the asymptotic behavior of solutions of Schröinger type equations in unbounded domains, Analyse mathématique et applications, 122, Gauthier-Villars, Montrouge, 1988. [7] S. Agmon, Representation theorems for solutions of the Helmholtz equation on $\mathbbR^n$, Differential Operators and Spectral Theory, 27-43, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999. [8] S. Agmon, J. Cruz-Sampedro and I. Herbst, Generalized Fourier transform for Schrödinger operators with potentials of order zero, Journal of Functional Analysis, 167 (1999), 345-369. [9] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, Journal D'Analyse Mathématique, 30 (1976). [10] S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math., 20 (1967), 207-229. [11] G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions, Commun. in Partial Differential Equations, 12 (1987), 263-283. [12] M. Ben-Artzi, Unitary equivalence and scattering theory for Stark-like Hamiltonians, J. Math. Phys., 25 (1984), 951-964. [13] P. Constantin, Scattering for Schröinger operators in a class of domains with noncompact boundaries, J. Funct. Anal., 44 (1981), 87-119. [14] J. Cruz-Sampedro, Exact asymptotic behavior at infinity of solutions to abstract second-order differential inequalities in Hilbert spaces, Math. Z., 237 (2001), 727-235. [15] J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076. [16] A. Hassell, R. Melrose and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math., 181 (2004), 1-87. [17] A. Hassell, R. Melrose and A. Vasy, Microlocal propagation near radial points and scattering for symbolic potentials of order zero, Anal. PDE, 1 (2008), 127-196. [18] L. Hörmander, "The Analysis of Linear Partial Differential Operators III," Springer-Verlag, Berlin, 1985. doi: 978-3-540-49938-1. [19] W. Jäger, Über das Dirichletsche Außenraumproblem für die Schwingungsgleichung, Math. Z., 95 (1967), 299-323. [20] W. Jäger, Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten, Math. Z., 102 (1967), 62-88. [21] W. Jäger, Das asymptotische Verhalten von Lsngen eines Typs von Differentialgleichungen, Math. Z., 112 (1969), 26-36. [22] W. Jäger and P. Rejto, Limiting absorption principle for some Schrödinger operators with exploding potentials. II, J. Math. Anal. Appl., 95 (1983), 169-194. [23] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. Math., 121 (1985), 463-494. [24] A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators, J. Operator Theory, 14 (1985), 181-188. [25] P. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," Pitman, London, 1982. [26] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, II Fourier Analysis Self-Adjontness," New York, Academic Press, 1978. [27] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory," New York, Academic Press, 1979. [28] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978. [29] Y. Saitō, "Spectral Representations for Schrödinger Operators with Long-range Potentials," Lecture Notes in Mathematics, 727. Springer, Berlin, 1979. doi: 978-3-540-35132-0. [30] Y. Saitō, Schrödinger operators with a nonspherical radiation condition, Pacific J. Math., 126 (1987), 331-359. [31] I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems," Lecture Notes in Mathematics 1011, Springer Verlag, 1983. doi: 978-3-540-38664-3.
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