Exact Wavefunctions for a Delta Function Bose Gas with Higher Derivatives

A quantum mechanical system describing bosons in one space dimension with a kinetic energy of arbitrary order in derivatives and a delta function interaction is studied. Exact wavefunctions for an arbitrary number of particles and order of derivative are constructed. Also, equations determining the spectrum of eigenvalues are found. PACS numbers: 02.30Jr, 05.30Jp Physical systems whose equations contains derivatives higher than two arise in a number of contexts. In classical mechanics, for example, the Korteweq-deVries equation, which describes shallow water waves, contains a third order derivative term [1]. Higher derivative theories of classical gravity, first introduced by Weyl [2], have some attractive cosmological properties, such as gravity driven expansion [3]. A model of quantum gravity with higher derivative terms has been shown to be renormalizable [4] and asympotically free [5]. In particle physics, the NambuJona-Lasinio [6], Schwinger [7], and Skyrme [8] models with higher derivative terms added have been considered. Also, in the superfield formulation of supersymmetric field theories, higher derivative terms arise naturally [9]. In this work, a family of higher derivative generalizations of Schrodinger’s equation is studied. The system describes bosons where the “kinetic energy” is a polynomial of order n in derivatives 1 and the interaction is a delta function potential. A particular choice of polynomial corresponds to the much studied delta function bose gas (or quantum nonlinear Schrodinger equation)[10, 11]. The main result here is that for an arbitrary polynomial in derivatives and arbitrary number of particles, exact eigenfunctions for the system may be constructed. Formulating the system under study, consider the eigenvalue problem HΨ = EΨ, (1) where the hamiltonian H is the partial differential operator H = N ∑