We construct Hamiltonians for systems of nonrelativistic particles linearly coupled to massive scalar bosons using abstract boundary conditions. The construction yields an explicit characterisation of the domain of self-adjointness in terms of boundary conditions that relate sectors with different numbers of bosons. We treat both models in which the Hamiltonian may be defined as a form perturbation of the free operator, such as Fröhlich's polaron, and renormalisable models, such as the massive Nelson model.
We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cutoff (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC. MSC: 81T10, 81Q10, 47F05. Key words: Ultraviolet divergence problem; renormalization in quantum field theory; self-adjoint Hamiltonian; self-adjoint extensions of the Laplace operator; particle-position representation; ultraviolet cut-off.
We construct a Hamiltonian for a quantum-mechanical model of nonrelativistic particles in three dimensions interacting via the creation and annihilation of a second type of nonrelativistic particles, which are bosons. The interaction between the two types of particles is a point interaction concentrated on the points in configuration space where the positions of two different particles coincide. We define the operator, and its domain of self-adjointness, in terms of co-dimension-three boundary conditions on the set of collision configurations relating sectors with different numbers of particles.
We study general quantum waveguides and establish explicit effective Hamiltonians for the Laplacian on these spaces. A conventional quantum waveguide is an ε-tubular neighbourhood of a curve in R 3 and the object of interest is the Dirichlet Laplacian on this tube in the asymptotic limit ε → 0. We generalise this by considering fibre bundles M over a d-dimensional submanifold B ⊂ R d+k with fibres diffeomorphic to F ⊂ R k , whose total space is embedded into an ε-neighbourhood of B. From this point of view B takes the role of the curve and F that of the discshaped cross-section of a conventional quantum waveguide. Our approach allows, among other things, for waveguides whose cross-sections F are deformed along B and also the study of the Laplacian on the boundaries of such waveguides. By applying recent results on the adiabatic limit of Schrödinger operators on fibre bundles we show, in particular, that for small energies the dynamics and the spectrum of the Laplacian on M are reflected by the adiabatic approximation associated to the ground state band of the normal Laplacian. We give explicit formulas for the according effective operator on L 2 (B) in various scenarios, thereby improving and extending many of the known results on quantum waveguides and quantum layers in R 3 .
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