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Generalized Euclidean Bosonic String Equations

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Spectral Analysis of Quantum Hamiltonians

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 224))

Abstract

We consider nonlinear equations of the form p(Δ)u = U(x, u(x)), in which p is a real-valued function satisfying some suitable technical conditions, and Δ stands for the Laplacian operator. We formulate a functional calculus appropriate for the study of such equations, and we establish results on the existence and regularity of solutions to the Euclidean bosonic string equation Δexp(-c Δ) u = U(x, u(x)), and we introduce a functional calculus appropriate for the study of very general nonlinear equations depending on functions of the Laplace operator. We also prove that under some further technical conditions, these “nonlocal” equations admit smooth, and even realanalytic, solutions. Our motivation comes from recent developments in string theory and nonlocal cosmology.

Mathematics Subject Classification (2010). 35Sxx, 44Axx, 81T30, 83F05.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechniki 1, PL-00-661, Warsaw, Poland

    P. Górka

  2. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile

    H. Prado & E. G. Reyes

Authors
  1. P. Górka
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  2. H. Prado
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  3. E. G. Reyes
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Corresponding author

Correspondence to P. Górka .

Editor information

Editors and Affiliations

  1. , Departamento de Física, P. Universidad Católica de Chile, Santiago, 22, Chile

    Rafael Benguria

  2. , Departamento de Matemáticas, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Eduardo Friedman

  3. , Departamento de Matemática, Universidad de Chile, Las Palmeras, Nuñoa 3425, Santiago, Chile

    Marius Mantoiu

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Górka, P., Prado, H., Reyes, E.G. (2012). Generalized Euclidean Bosonic String Equations. In: Benguria, R., Friedman, E., Mantoiu, M. (eds) Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol 224. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0414-1_8

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