Volume 1

Preface

xiii

(2)

John A. Nohel

David H. Sattinger

Foreword

(1)

Henry P. McKean, Jr.

Biographical Sketch of Norman Levinson

xvi

(2)

Publications of Norman Levinson

xviii

(9)

Ph.D. Students of Norman Levinson

xxvii

(2)

Remarks

xxix

(3)

Zipporah (Fagi) Levinson

Remarks

xxxii

(4)

William T. Martin

Norman Levinson: Colleague and Friend

xxxvi

(1)

Bertram Kostant

Norman Levinson

xxxvii

(2)

Gian-Carlo Rota

Acknowledgments

xxxix

Throughout these two volumes, numbers in brackets preceded by the letter "L" refer to entries in the complete list of Levinson's publications in order to distinguish his papers from those by other authors referenced in the commentaries.

I. Stability and Asymptotic Behavior of Solutions of Ordinary Differential Equations

(63)

Commentary on [L 31] and [L 36]

(9)

John Nohel

[L 20] The Growth of the Solutions of a Differential Equation (1941)

(10)

[L 24] (with Mary L. Boas and R. P. Boas, Jr.), The Growth of the Solutions of a Differential Equation (1942)

(7)

[L 31] The Asymptotic Behavior of a System of Linear Differential Equations (1946)

(6)

[L 36] The Asymptotic Nature of Solutions of Linear Systems of Differential Equations (1948)

(17)

[L 40] On Stability of Non-Linear Systems of Differential Equations (1949)

(7)

[L 68] (with R. R. D. Kemp), On u" + (1 + Lambda(g)(x))u = 0 for Integral^Infinity(0) |g(x)|dx is Less Than Infinity (1959)

(7)

II. Nonlinear Oscillations and Dynamical Systems

(94)

Commentary on [L 29], [L 38] and [L 47]

(4)

Jurgen Moser

[L 23] (with O. K. Smith), A General Equation for Relaxation Oscillations (1942)

(22)

[L 25] On the Existence of Periodic Solutions for Second Order Differential Equations with a Forcing Term (1943)

(8)

[L 29] Transformation Theory of Non-Linear Differential Equations of the Second Order (1944) and Correction (1948)

(15)

[L 38] A Second Order Differential Equation with Singular Solutions (1949)

114

(27)

[L 47] Small Periodic Perturbations of an Autonomous System with a Stable Orbit (1950)

141

(11)

[L 52] (with A. B. Farnell and C. E. Langenhop), Forced Periodic Solutions of a Stable Non-Linear System of Differential Equations (1951)

152

(3)

[L 57] (with G. F. D. Duff), On the Non-Uniqueness of Periodic Solutions for an Asymmetric Lienard Equation (1952)

155

(4)

III. Inverse Problems for Sturm-Liouville and Schrodinger Operators

159

(44)

Commentary on [L 41], [L 43] and [L 58]

159

(5)

D. H. Sattinger

[L 41] On the Uniqueness of the Potential in a Schrodinger Equation for a Given Asymptotic Phase (1949)

164

(28)

[L 42] Determination of the Potential from the Asymptotic Phase (1949)

192

(1)

[L 43] The Inverse Sturm-Liouville Problem (1949)

193

(6)

[L 58] Certain Explicit Relationships between Phase Shift and Scattering Potential (1953)

199

(4)

IV. Eigenfunction Expansions and Spectral Theory for Ordinary Differential Equations

203

(64)

Commentary on [L 49], [L 51], and [L 59]

203

(3)

D. H. Sattinger

Boris Levitan

[L 39] Criteria for the Limit-Point Case for Second Order Linear Differential Operators (1949)

206

(5)

[L 49] A Simplified Proof of the Expansions Theorem for Singular Second Order Linear Differential Equations (1951)

211

(15)

[L 50] Addendum to "A Simplified Proof of the Expansions Theorem for Singular Second Order Linear Differential Equations" (1951)

226

(4)

[L 51] (with E. A. Coddington), On the Nature of the Spectrum of Singular Second Order Linear Differential Equations (1951)

230

(4)

[L 53] The XXX-Closure of Eigenfunctions Associated with Selfadjoint Boundary Value Problems (1952)

234

(4)

[L 59] The Expansion Theorem for Singular Self-Adjoint Linear Differential Operators (1954)

238

(16)

[L 65] Transform and Inverse Transform Expansions for Singular Self-Adjoint Differential Operators (1958)

254

(13)

V. Singular Perturbations of Ordinary and Partial Differential Equations

267

(148)

Commentary on [L 45], [L 48], [L 60], [L 62], [L 63], [L 67], [L 56] and [L 46]

267

(21)

John Nohel

[L 45] Perturbations of Discontinuous Solutions of Non-Linear Systems of Differential Equations (1950)

288

(37)

[L 48] An Ordinary Differential Equation with an Interval of Stability, a Separation Point, and an Interval of Instability (1950)

325

(8)

[L 60] (with J. J. Levin), Singular Perturbations of Non-Linear Systems of Differential Equations and an Associated Boundary Layer Equation (1954)

333

(24)

[L 62] (with L. Flatto), Periodic Solutions of Singularly Perturbed Systems (1955)

357

(9)

[L 56] (with E. A. Coddington), A Boundary Value Problem for a Nonlinear Differential Equation with a Small Parameter (1952)

366

(10)

[L 63] (with S. Haber), A Boundary Value Problem for a Singularly Perturbed Differential Equation (1955)

376

(8)

[L 67] A Boundary Value Problem for a Singularly Perturbed Differential Equation (1958)

384

(12)

[L 46] The First Boundary Value Problem for XXX(Triangle)u + A(x, y)u(x) + B(x, y)u(y) + C(x, y)u = D(x, y) for small XXX (1950)

396

(19)

VI. Elliptic Partial Differential Equations

415

(47)

Commentary on [L 75], [L 78], [L 87]

415

(8)

D. H. Sattinger

Eberhard Zeidler

[L 75] Positive Eigenfunctions for (Triangle)u + (Lambda)f(u) = 0 (1962)

423

(16)

[L 78] Dirichlet Problem for (Triangle)u = f(P, u) (1963)

439

(10)

[L 87] One-Sided Inequalities for Elliptic Differential Operators (1965)

449

(13)

VII. Integral Equations

462

Commentary on [L 73]

462

(5)

John Nohel

[L 32] On the Asymptotic Shape of the Cavity Behind an Axially Symmetric Nose Moving Through an Ideal Fluid (1946)

467

(27)

[L 73] A Nonlinear Volterra Equation Arising in the Theory of Superfluidity (1960)

494

(11)

[L 89] Simplified Treatment of Integrals of Cauchy Type, the Hilbert Problem and Singular Integral Equations.

Appendix: Poincare-Bertrand Formula (1965)

505

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Selected Papers of Norman Levinson

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