Some remarks on the \(\delta\) ’-interaction in one dimension. (English) Zbl 0638.70016

Summary: We discuss the existence and the physical properties of the one- dimensional \(\delta\) ’-interaction Hamiltonian. We show that the so called \(\delta\) ’-interaction Hamiltonian which appears in the literature represents a self-adjoint realization of the heuristic operator \(-d^ 2/dx^ 2+\beta | \delta'(x)><\delta'(x)|\) with a renormalized coupling constant \(\beta\). We investigate also a possible self-adjoint realization of the formal Hamiltonian \(-d^ 2/dx^ 2+\beta \delta'(x).\) We show that this realization coincides with the ordinary one-dimensional point interaction Hamiltonian \(H_ A=-d^ 2/dx^ 2+A\delta(x)\).


70H99 Hamiltonian and Lagrangian mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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