Narrow Results

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L²(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

In this paper, we consider the boundary value problem

This paper provides the optimal exponential decay rate of the lower bound for the first positive eigenvalue of the Laplacian operator on a compact Riemannian manifold with a negative lower bound on the Ricci curvature and with large diameter. For manifolds with boundary, suitable convexity conditions are assumed.

The Klein–Gordon and Dirac equations are considered in a semiinfinite laboratory (x > 0) in the presence of background metrics ds² = u²(x)η_μν dx^μ dx^ν and ds² = -dt² + u²(x)η_ij dxⁱ dx^j with u(x) = e^±gx. These metrics have nonconstant scalar-curvatures. Various aspects of the solutions are studied. For the first metric with u(x) = e^gx, it is shown that the spectra are discrete, with the ground state energy for spin-0 particles. For u(x) = e^-gx, the spectrums are found to be continuous. For the second metric with u(x) = e^-gx, each particle, depends on its transverse-momentum, can have continuous or discrete spectrum. For Klein–Gordon particles, this threshold transverse-momentum is , while for Dirac particles it is g/2. There is no solution for u(x) = e^gx case. Some geometrical properties of these metrics are also discussed.

In this work, we illustrate the use of eigenfunction method (EFM) in quantum mechanics by taking a specific example namely the splitting of energy levels under perturbation. Initially, the symmetry group of the Hamiltonian is taken to be $C_{4v}$ which under the action of a perturbing potential changes to a lower symmetry namely $C_{2v}$. We have constructed an irreducible basis using the EFM for both the symmetries and show the simplifications that ensure the calculation of the expectation value of the perturbing Hamiltonian.

Consider a family of bounded domains Ω_t in the plane (or more generally any Euclidean space) that depend analytically on the parameter t, and consider the ordinary Neumann Laplacian Δ_t on each of them. Then we can organize all the eigenfunctions into continuous families with eigenvalues also varying continuously with t, although the relative sizes of the eigenvalues will change with t at crossings where . We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has Ω₀ equal to a square and Ω₁ equal to a circle; in both cases the eigenfunctions are known explicitly, so our homotopies connect these two explicit families. In the second example we approximate the Sierpinski carpet starting with a square, and we continuously delete subsquares of varying sizes. (Data available in full at ).

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains $\mathrm{\Omega }\subset {ℝ}^d$ with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by $H_0^s(\overline{\mathrm{\Omega }})$, and Sobolev–Slobodeckij̆ spaces, denoted by $H^s(\mathrm{\Omega })$, where $s>\frac{d}{2}$. Our goal is accomplished by obtaining the Green’s solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green’s kernels satisfying these problems are symmetric and positive definite reproducing kernels of $H_0^s(\overline{\mathrm{\Omega }})$ and $H^s(\mathrm{\Omega })$, respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains.

The radial Schrödinger equation was solved with the combination of three important potentials with $q$ as deformed parameter via the parametric Nikiforov–Uvarov method and the energy equation as well as the corresponding normalized radial wave function were obtained in close and compact form. The energy equation obtained was used to study eight molecules. The effect of the deformed parameter on energy eigenvalues was also studied numerically. The subset of the combined potential was also studied numerically and the results were found to be in agreement with the previous results. To extend the application of our work, the wave function obtained was used to calculate some theoretic quantities such as the Tsallis entropy, Rényi entropy and information energy. By putting the Tsallis index to 2, we deduced the information energy from Tsallis entropy. Finally, the effect of the deformed parameter and screening parameter, respectively, on the theoretic quantities were also studied.

In this note, we establish a sharp remainder formula for the Poincaré inequality for the Baouendi–Grushin vector fields. We give a simple proof for it without using the variational principle. As an application, we obtain a blow-up result for solutions to the Dirichlet initial-boundary value problem for the Baouendi–Grushin heat operator.

The eigenfunctions of the modified Helmholtz operator have no orthogonality in a bounded domain with a perfectly matched layer (PML), which makes it difficult to calculate the coordinates under two different local bases when the marching algorithm is applied. In this paper, we derive the conjugate eigenfunctions of the operator and discuss the cross orthogonality between the eigenfunctions and their conjugate eigenfunctions. On the other hand, we derive a simple formula for calculating the coordinates under the local base. The numerical results indicate that this method is effective.

A Hamiltonian model is developed for analyzing viscoelastic problems with annular-sector domain by using of the polar coordinate system and elastic-viscoelastic correspondence principle. Making use of the state space method, a complete eigensolution space, including zero eigenfunctions and non zero eigenfunctions, is successfully established. Due to the completeness of the solution space, the final solution of any viscoelastic problem can be expressed in terms of linear combination of the eigenfunctions. The corresponding unknown coefficients are determined by using prescribed conditions on the circular boundary. Further, using the variable substitution method, the lateral boundary conditions are transformed into a particular solution which can be obtained by applying the eigenfunction expansion.

A weighted eigenvalue problem of generalized Tricomi equation is investigated, generalizing the case considered by ponomarev⁷⁾. All the eigenfunctions are exhibited.