Solitons and the removal of eigenvalues for fourth-order differential operators
2006 (English)In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247Article in journal (Refereed) Published
A nonlinear functional Q[u, v] is given that governs the loss, respectively gain, of ( doubly degenerate) eigenvalues of fourth-order differential operators L = partial derivative(4) + partial derivative u partial derivative + v on the line. Apart from factorizing L as A*A + E-0, providing several explicit examples, and deriving various relations between u, v, and the eigenfunctions of L, we find u and v such that L is isospectral to the free operator L-0 = partial derivative(4) up to one (multiplicity 2) eigenvalue E-0 < 0. Not unexpectedly, this choice of u, v leads to exact solutions of the corresponding time-dependent PDE's. Removal of eigenvalues allows us to obtain a sharp Lieb-Thirring inequality for a class of operators L whose negative eigenvalues are of multiplicity two.
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IdentifiersURN: urn:nbn:se:kth:diva-15956ISI: 000240181600001ScopusID: 2-s2.0-33747029312OAI: oai:DiVA.org:kth-15956DiVA: diva2:333998
QC 201005252010-08-052010-08-05Bibliographically approved