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 & Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495
References:
[1]

R. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (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, (1965).

[4]

S. Agmon, Lower bounds for solutions of Schrdinger equations,, Journal D'analyse Mathématique, 23 (1970), 1.

[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, (1982).

[6]

S. Agmon, On the asymptotic behavior of solutions of Schröinger type equations in unbounded domains,, Analyse mathématique et applications, (1988).

[7]

S. Agmon, Representation theorems for solutions of the Helmholtz equation on $\mathbbR^n$,, Differential Operators and Spectral Theory, (1999), 27.

[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.

[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.

[11]

G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions,, Commun. in Partial Differential Equations, 12 (1987), 263.

[12]

M. Ben-Artzi, Unitary equivalence and scattering theory for Stark-like Hamiltonians,, J. Math. Phys., 25 (1984), 951.

[13]

P. Constantin, Scattering for Schröinger operators in a class of domains with noncompact boundaries,, J. Funct. Anal., 44 (1981), 87.

[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.

[15]

J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero,, Discrete Contin. Dyn. Syst., 33 (2013), 1061.

[16]

A. Hassell, R. Melrose and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero,, Adv. Math., 181 (2004), 1.

[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.

[18]

L. Hörmander, "The Analysis of Linear Partial Differential Operators III,", Springer-Verlag, (1985). doi: 978-3-540-49938-1.

[19]

W. Jäger, Über das Dirichletsche Außenraumproblem für die Schwingungsgleichung,, Math. Z., 95 (1967), 299.

[20]

W. Jäger, Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten,, Math. Z., 102 (1967), 62.

[21]

W. Jäger, Das asymptotische Verhalten von Lsngen eines Typs von Differentialgleichungen,, Math. Z., 112 (1969), 26.

[22]

W. Jäger and P. Rejto, Limiting absorption principle for some Schrödinger operators with exploding potentials. II,, J. Math. Anal. Appl., 95 (1983), 169.

[23]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators,, Ann. Math., 121 (1985), 463.

[24]

A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators,, J. Operator Theory, 14 (1985), 181.

[25]

P. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Pitman, (1982).

[26]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, II Fourier Analysis Self-Adjontness,", New York, (1978).

[27]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory,", New York, (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, (1979). doi: 978-3-540-35132-0.

[30]

Y. Saitō, Schrödinger operators with a nonspherical radiation condition,, Pacific J. Math., 126 (1987), 331.

[31]

I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems,", Lecture Notes in Mathematics 1011, (1011). doi: 978-3-540-38664-3.

show all references

References:
[1]

R. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (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, (1965).

[4]

S. Agmon, Lower bounds for solutions of Schrdinger equations,, Journal D'analyse Mathématique, 23 (1970), 1.

[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, (1982).

[6]

S. Agmon, On the asymptotic behavior of solutions of Schröinger type equations in unbounded domains,, Analyse mathématique et applications, (1988).

[7]

S. Agmon, Representation theorems for solutions of the Helmholtz equation on $\mathbbR^n$,, Differential Operators and Spectral Theory, (1999), 27.

[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.

[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.

[11]

G. Barles, On eikonal equations associated with Schrödinger operators with nonspherical radiation conditions,, Commun. in Partial Differential Equations, 12 (1987), 263.

[12]

M. Ben-Artzi, Unitary equivalence and scattering theory for Stark-like Hamiltonians,, J. Math. Phys., 25 (1984), 951.

[13]

P. Constantin, Scattering for Schröinger operators in a class of domains with noncompact boundaries,, J. Funct. Anal., 44 (1981), 87.

[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.

[15]

J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero,, Discrete Contin. Dyn. Syst., 33 (2013), 1061.

[16]

A. Hassell, R. Melrose and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero,, Adv. Math., 181 (2004), 1.

[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.

[18]

L. Hörmander, "The Analysis of Linear Partial Differential Operators III,", Springer-Verlag, (1985). doi: 978-3-540-49938-1.

[19]

W. Jäger, Über das Dirichletsche Außenraumproblem für die Schwingungsgleichung,, Math. Z., 95 (1967), 299.

[20]

W. Jäger, Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Außengebieten,, Math. Z., 102 (1967), 62.

[21]

W. Jäger, Das asymptotische Verhalten von Lsngen eines Typs von Differentialgleichungen,, Math. Z., 112 (1969), 26.

[22]

W. Jäger and P. Rejto, Limiting absorption principle for some Schrödinger operators with exploding potentials. II,, J. Math. Anal. Appl., 95 (1983), 169.

[23]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators,, Ann. Math., 121 (1985), 463.

[24]

A. Jensen and P. Perry, Commutator methods and Besov space estimates for Schrödinger operators,, J. Operator Theory, 14 (1985), 181.

[25]

P. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Pitman, (1982).

[26]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, II Fourier Analysis Self-Adjontness,", New York, (1978).

[27]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III Sacattering Theory,", New York, (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, (1979). doi: 978-3-540-35132-0.

[30]

Y. Saitō, Schrödinger operators with a nonspherical radiation condition,, Pacific J. Math., 126 (1987), 331.

[31]

I. Sigal, "Scattering Theory for Many-Body Quantum Mechanical Systems,", Lecture Notes in Mathematics 1011, (1011). doi: 978-3-540-38664-3.

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