Negative Laplacian Research Articles

Self-similar drums and generalized Weierstrass functions

We consider the number N (λ;Ω) of eigenvalues less than λ of the negative Laplacian with Dirichlet boundary conditions in a domain Ω⊂ R n with fractal boundary ∂Ω. It is known that for λ→∞, N (λ;Ω)= C n |Ω| n λ n/2 + O (λ D/2 ), where D is the Minkowski dimension of Ω. For a certain class of self-similar domains (“drums”) we obtain for N (λ;Ω) a second term of the form - F (ln λ)λ D/2 with a bounded periodic function F . F contains a generalized Weierstrass function with a self-similar fractal graph. A number of examples with n=1,2,3,… has been studied, where more information about F is available. Finally, a possible physical application will be sketched.

An inverse eigenvalue problem for an arbitrary multiply connected bounded region: an extension to higher dimensions

The basic problem in this paper is that of detemnining the geometry of an arbitrary multiply connected bounded region inR3together with the mixed boundary conditions, from the complete knowledge of the eigenvalues{λj}j=1∞for the negative Laplacian, using the asymptotic expansion of the spectral functionθ(t)=∑j=1∞exp(−tλj)ast→0.

Projective block spin transformations in lattice gauge theories

A block spin definition for Higgs fields and non-abelian lattice gauge fields is examined which is in the spirit of the projective multigrid procedure of Hulsebos et al., and Brower et al. Their procedure is generalizable to a systematic multigrid method for non-abelian gauge theories in any number of space-time dimensions. The block spin transformation involves a gauge-covariant kernel C which makes the fine-to-coarse transition, and a kernel A which makes the coarse-to-fine interpolation. These kernels could be used in a projective multigrid computation of propagators, for instance. Vectorizable algorithms for the computation of C and A are presented. For SU (2) lattice gauge theory in four dimensions, the required CPU time for computing C or A on the whole lattice is comparable to that for one standard Monte Carlo sweep through the lattice. Numerical results obtained after performing one blocking transformation from a 9 4 to a 3 4 lattice is presented. The block spin computation yields auxiliary quantities of interest, such as the lowest eigenvalues λ 0, λ′ 0 of the negative gauge-covariant laplacian with Neumann and Dirichlet boundary conditions on block boundaries. λ 0 and λ′ 0 are measures of disorder and their renormalizatiob group flow is therefore instructive.

Order of energy levels in relativistic one-electron models

We prove that if a particle satisfying the Dirac equation is submitted to a potential composed of a pure Coulomb part and of a vector-like perturbation with negative laplacian the energy levels belonging to a Coulomb multiplet are completely ordered: the energies increase with increasing J for fixed L and increasing L for fixed J, J and L being respectively the total and orbital angular momenta.

Heat equation for a general convex domain inR3with a finite number of piecewise impedance boundary conditions

The spectral function , where are are the eigenvalues of the negative Laplacian –space, is studied for agiven arbitrary simply connected bounded domain together with a finite number of piecewise smooth impedance boundary conditions on the parts Si (i≡l,…,m) of its bounding surface S such that .

Confinement in the color dielectric model

A block spin definition for Higgs fields and non-abelian lattice gauge fields is examined which is in the spirit of the projective multigrid procedure of Hulsebos et al., and Brower et al. Their procedure is generalizable to a systematic multigrid method for non-abelian gauge theories in any number of space-time dimensions. The block spin transformation involves a gauge-covariant kernel C which makes the fine-to-coarse transition, and a kernel A which makes the coarse-to-fine interpolation. These kernels could be used in a projective multigrid computation of propagators, for instance. Vectorizable algorithms for the computation of C and A are presented. For SU (2) lattice gauge theory in four dimensions, the required CPU time for computing C or A on the whole lattice is comparable to that for one standard Monte Carlo sweep through the lattice. Numerical results obtained after performing one blocking transformation from a 94 to a 34 lattice is presented. The block spin computation yields auxiliary quantities of interest, such as the lowest eigenvalues λ0, λ′0 of the negative gauge-covariant laplacian with Neumann and Dirichlet boundary conditions on block boundaries. λ0 and λ′0 are measures of disorder and their renormalizatiob group flow is therefore instructive.

Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions

Nonrelativistic Schrödinger operators are perturbations of the negative Laplacian and the connection with stochastic processes (and Brownian motion in particular) is well known and usually goes under the name of Feynman and Kac. We present a similar connection between a class of relativistic Schrödinger operators and a class of processes with stationary independent increments. In particular, we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur.

Spectral theory for a generalized cauchy-riemann operator

In this paper it is shown how the Dirichlet problem for the negative Laplacian in a bounded open domain of the cuclidean space leads to a characterization of the specturm and eigenvectors of a generalized Cauchy - Riemann operator.

Numerical search for a breakdown of the Waterman algorithm for approximating the T matrix of a rigid obstacle

The convergence of the Waterman algorithm for approximating the T matrix [J. Acoust. Soc. Am. 45, 1417–1429 (1969)] has not been established for scatterers of general shape. One can prove that the convergence criteria of Kristensson, Ramm, and Ström [J. Math. Phys. 24, 2619–2631 (1983)] are never fulfilled if the obstacle is nonspherical and the spherical wave functions have their usual normalization; the proof of this fact shall appear elsewhere. One of the coordinate sequences of this algorithm is the family of regular spherical waves, which fails to be complete in L2 (Γ), Γ denoting the obstacle boundary, when k2 is an interior Dirichlet eigenvalue for the negative Laplacian. These circumstances lead one to conjecture that the method will fail at such frequencies (as it does for the sphere). The results of computations near various of these frequencies for spheroidal scatterers are reported. For the sphere, the widths of the ka intervals of instability are examined, while for the nonspherical case, the object is to determine whether a breakdown occurs at all.

Schrödinger semigroups for vector fields

Suppose H is the Hamiltonian that generates time evolution in an N-body, spin-dependent, nonrelativistic quantum system. If r is the total number of independent spin components and the particles move in three dimensions, then the Hamiltonian H is an r×r matrix operator given by the sum of the negative Laplacian −Δx on the (d=3N)-dimensional Euclidean space Rd plus a Hermitian local matrix potential W(x). Uniform higher-order asymptotic expansions are derived for the time-evolution kernel, the heat kernel, and the resolvent kernel. These expansions are, respectively, for short times, high temperatures, and high energies. Explicit formulas for the matrix-valued coefficient functions entering the asymptotic expansions are determined. All the asymptotic expansions are accompanied by bounds for their respective error terms. These results are obtained for the class of potentials defined as the Fourier image of bounded complex-valued matrix measures. This class is suitable for the N-body problem since interactions of this type do not necessarily decrease as ‖x‖→∞. Furthermore, this Fourier image class also contains periodic, almost periodic, and continuous random potentials. The method employed is based upon a constructive series representation of the kernels that define the analytic semigroup {e−zH‖Re z>0}. The asymptotic expansions obtained are valid for all finite coordinate space dimensions d and all finite vector space dimensions r, and are uniform in Rd×Rd. The order of expansion is solely a function of the smoothness properties of the local potential W(x).

Schrödinger spectral kernels: High order asymptotic expansions

The large energy behavior of the spectral kernel for the N-body Schrödinger Hamiltonian is obtained. In a setting of a d-dimensional Euclidean space without boundaries, the Schrödinger Hamiltonian H is the sum of the negative Laplacian plus a real-valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N-body problem, since they do not necessarily decrease as ‖x‖→∞. Let {e(x,y;λ): λ∈R} be the family of spectral kernels generated by H. In the λ→∞ limit, explicit higher order asymptotic expansions are obtained for e(x,y;λ) and its associated Riesz means. The asymptotic expansion is uniform in x and y and is accompanied by estimates of the error term.

Time evolution kernels: uniform asymptotic expansions

For a wide class of self-adjoint Schrödinger Hamiltonians, a detailed description of the time evolution kernel is obtained. In a setting of a d-dimensional Euclidean space without boundaries, the Schrödinger Hamiltonian H is the sum of the negative Laplacian plus a real-valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N-body problem, since they do not necessarily decrease as ‖x‖→∞. An asymptotic expansion in the complex parameter z, around z=0, is derived for the family of kernels Uz(x,y) corresponding to the analytic semigroup {e−zH:Re z>0}, which is uniform in the coordinate variables x and y. The asymptotic expansion has a simple semiclassical interpretation. Furthermore, an explicit bound for the remainder term in the asymptotic expansion is found. The expansion and the remainder term bound continue to the time axis boundary z=it/ℏ (t≠0) of the analytic semigroup domain.

Scattering by periodic surfaces

If Ω⊆ Rν (ν?2) is an exterior domain containing a half-space and contained in a half space, we prove that the wave operators W± = s-limt→±∞ exp(itH) Jexp(−itH0) are partial isometries and that the invariance principle W±(φ) = W± holds for suitable real functions φ on R (’’admissible’’ functions). Here H0 is the negative (distributional) Laplacian in L2(Rν) (ν?2); H is HD or HN , the negative Dirichlet or Neumann Laplacians in ℋ = L2(Ω), respectively; J is an appropriate identification operator; and W± (φ) are defined as were W± , but with H0 and H replaced by φ(H0) and φ(H), respectively. Suppose, in addition, that Ω has a suitable periodicity property and that when H = HN it has a certain mild local compactness property. Then we prove: (1) that W± are asymptotically complete, in the sense that Ran W± = ℋscatt(H) = ℋ○ℋsurf(H) ; (2) that ℋscatt(φ(H)) = ℋscatt(H) and ℋsurf (φ(H)) = ℋsurf(H) for each ’’admissible’’ function φ, and hence that W± (φ) are asymptotically complete in a similar sense for each such φ. Here, for any self-adjoint operator A in ℋ, ℋscatt(A), and ℋsurf(A) are naturally defined, mutually orthogonal subspaces of scattering and surface states of ℋ, respectively. No smoothness assumptions on ∂Ω are made in this paper. Its results entail the asymptotic completeness, in a physically very satisfactory sense, of wave operators describing acoustic and certain types of electromagnetic scattering by a very wide class of periodic surfaces.

Quantum-mechanical scattering by impenetrable periodic surfaces

In this paper, we investigate the existence and completeness of the wave operators W± = s-limt→±∞ exp (itH) P exp (−itH0) corresponding to the quantum-mechanical scattering of nonrelativistic particles by certain classes of impenetrable noncompact surfaces bounding domains Ω ⊆Rν (ν ⩾2) which contain a half-space and are contained in another half-space. Here, H0 is the usual negative (distributional) Laplacian−Δ in H0 = L2(Rν ), H is the negative Dirichlet Laplacian in H = L2(Ω), and P is an appropriate identification operator. Under these conditions, we prove by elementary methods that W± exist as partially isometric operators whose initial sets have a transparent physical meaning. Suppose now that the domain Ω ⊆Rν also has the periodicity property (x̃,xν)∈Ω→(x̃+l,xν)∈Ω when l ranges over a Bravais lattice in Rν−1 , where we write x∈Rν as (x̃,xν), with x̃∈Rν−1 and xν ∈R. Then (a) RanW± = Hscatt (H) and (b) W± are asymptotically complete, in the sense that H = Hscatt(H)⊕Hsurf(H). Here, Hscatt (H) and Hsurf (H) are suitably defined subspaces of scattering and surface states of H, respectively. Results (a) and (b) are proved by reducing the original scattering problem to a family of ’’scattering’’ problems in a periodicity cell of Ω, using direct-integral methods, and by then using methods analogous to those of Lyford. The present work constitutes a rigorous foundation for the theory of scattering of low-energy atomic beams by crystal surfaces, considered as impenetrable periodic barriers. Our methods should also be applicable to rigorous investigations of classical scattering by periodic surfaces with Dirichlet or Neumann boundary conditions.