Abstract
Using rearrangements of matrix-valued sequences, we prove that with certain boundary conditions the solution of the one-dimensional Schrödinger equation increases or decreases under monotone rearrangements of its potential.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baernstein, A. andTaylor, B. A., Spherical rearrangements, subharmonic functions, and*-functions inn-space,Duke Math. J. 43 (1976), 245–268.
Brock, F. andSolynin, A. Yu., An approach to symmetrization via polarization,Trans. Amer. Math. Soc. 352 (2000), 1759–1796.
Essén, M.,The cos π λ Theorem, Lecture Notes in Mathematics467, Springer-Verlag, Berlin-New York, 1975.
Essén, M., An estimate of harmonic measure inR d,d≥2,Proc. Roy. Soc. Edinburgh Sect. A 78 (1977), 129–138.
Essén, M., A theorem on convex sequences,Analysis 2 (1982), 231–252.
Essén, M., Optimization and α-disfocality for ordinary differential equations,Canad. J. Math. 37 (1985), 310–323.
Essén, M., Optimization and rearrangements of the coefficient in the operatord 2/dt 2−p(t) 2 on a finite interval.J. Math. Anal. Appl. 115 (1986), 278–304.
Essén, M. andHaliste, K., On Beurling's theorem for harmonic measure and the rings of Saturn,Complex Variables Theory Appl. 12 (1989), 137–152.
Luttinger, J. M., Generalized isoperimetric inequalities. II, III,J. Math. Phys. 14 (1973), 1444–1447.
Wolontis, V., Properties of conformal invariants,Amer. J. Math. 74 (1952), 587–606.
Author information
Authors and Affiliations
Additional information
Dedicated to the memory of Matts Essén
Rights and permissions
About this article
Cite this article
Kovalev, L.V. Comparison theorems for the one-dimensional Schrödinger equation. Ark. Mat. 43, 403–418 (2005). https://doi.org/10.1007/BF02384788
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02384788