Nth Eigenfunction Research Articles

The dual eigenvalue problems of the conformable fractional Sturm\u2013Liouville problems

In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has n-1 zero in ( 0,pi ) for nin mathbb{N}. Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

On Pleijel\u2019s Nodal Domain Theorem for Quantum Graphs

We establish metric graph counterparts of Pleijel’s theorem on the asymptotics of the number of nodal domains nu _n of the nth eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with L^1-potentials and a variety of vertex conditions as well as the p-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. Among other things, these results characterise the accumulation points of the sequence (frac{nu _n}{n})_{nin mathbb {N}}, which are shown always to form a finite subset of (0, 1]. This extends the previously known result that nu _nsim ngenerically, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs any graph with pairwise commensurable edge lengths and at least one cycle, one can find eigenfunctions thereon for which {nu _n}not sim {n}; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.

Topological properties of eigenoscillations in mathematical physics

Courant proved that the zeros of the nth eigenfunction of the Laplace operator on a compact manifold M divide this manifold into at most n parts. He conjectured that a similar statement is also valid for any linear combination of the first n eigenfunctions. However, later it was found out that some corollaries to this generalized statement contradict the results of quantum field theory. Later, explicit counterexamples were constructed by O. Viro. Nevertheless, the one-dimensional version of Courant’s theorem is apparently valid; to prove it, I.M. Gel’fand proposed a method based on the ideas of quantum mechanics and the analysis of the actions of permutation groups. This leads to interesting questions of describing the statistical properties of group representations that arise from their action on eigenfunctions of the Laplace operator. The analysis of these questions entails, among other things, problems of singularity theory.

Estimates of sums of zero multiplicities for eigenfunctions of the Laplace-Beltrami operator

We obtain an upper estimate N−χ(M) for the sum Q N of singular zero multiplicities of the Nth eigenfunction of the Laplace-Beltrami operator on the two-dimensional, compact, connected Riemann manifold M, where χ M is the Euler characteristic ofM. Stronger estimates, but equivalent asymptotically (N a ∞), are given for the cases of the sphere S 2 and the projective plane ℝ2. Asymptotically sharper estimates are shown for the case of a domain on the plane.

Eigenvalue and nodal properties on quantum graph trees*

We consider the Schrödinger operator on a finite tree with linear separate (e.g. Dirichlet or Neumann) conditions at the boundary. The potential q is assumed to be real-valued, positive and finite. At the interior vertices the solution has to be continuous and smooth in a certain way (Kirchhoff's or δ-type condition). Using oscillation theory we derive first for radial and then for general trees the interlacing property: the eigenvalues of the tree problem lie between the eigenvalues of the problems of subtrees joining at an arbitrary vertex. From this we can derive some eigenvalue asymptotics. Moreover we can generalize Sturm's Theorem from the interval to a tree: the nth eigenfunction has n nodal domains. *In memoriam Z. E. Schapotschnikow.

Courant's Nodal Line Theorem and Its Discrete Counterparts

Summary Courant's nodal line theorem (CNLT) states that if the Dirichlet eigenvalues of the Helmholtz equationu + λρ u = 0 for D ∈ R m are ordered increasingly, the nodal set of the nth eigenfunction un, which consists of hypersurfaces of dimension m − 1, divides D into no more than n sign domains in which un has one sign. We formulate and prove a discrete CNLT for a piecewise linear finite element discretization on a triangular/tetrahedral mesh.

On the Nodal Sets of the Eigenfunctions of the String Equation

In this paper the nodal sets of the eigenfunctions of the string equation are investigated. For n sufficiently large it is found that the shortest nodal domain of the nth eigenfunction must be one of the neighboring nodal domains of the maximum points, and the longest nodal domain of the nth eigenfunction must be one of the neighboring nodal domains of the minimum points of the density function of the string equation. A limit formula for the ratio of the longest length and the shortest length of the nodal domains of the nth eigenfunction is also proved, and some average formulae for the nodal domains are derived.

Sturm–Liouville eigenproblems with an interior pole

The eigenvalues and eigenfunctions of self-adjoint Sturm–Liouville problems with a simple pole on the interior of the interval [A, B] are investigated. Three general theorems are proved and it is shown that as n→∞, the eigenfunctions more and more closely resemble those of an ordinary Sturm–Liouville problem and λn ∼−m2π2/(B−A)2, just as if there were no singularity. The low-order modes, however, differ drastically from those of a nonsingular eigenproblem in that (i) both eigenvalues and eigenfunctions are complex (despite the fact the problem is self-adjoint), (ii) the real and imaginary parts of the nth eigenfunction may both have ever-increasing numbers of interior zeros as B→∞, instead of just (n−1) zeros, and (iii) as B→∞, the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. These results are general, but in order to present quantitative solutions for the low-order modes, too, special attention is given to the particular case u″+(1/x−λ)u = 0, (1) with u(A) = u(B) = 0 where λ is the eigenvalue and A and B are of opposite signs. For small n, one can obtain the approximation λn∼exp[(1+31/2i)dn/(2B1/3)]/B, (2) where dn is the nth root of the Airy function Ai(−z). The imaginary part of (2) shows explicitly how profoundly the interior pole has modified the structure of the eigenproblem. The WKB method, which was used to derive (2), is shown to be accurate for all n. The WKB analysis is of some interest in and of itself. Although the number of WKB ’’transition’’ points is the same as for the half-century old quantum harmonic oscillator (two), the substitution of the interior pole for one of the turning points has a profound (and fascinating) impact on both the WKB formalism and the numerical results. Thus, although this problem was motivated by the physics of hydrodynamic waves, it is also an extension to both classical Sturm–Liouville theory and to the WKB treatment of eigenvalue problems.

Eigenfunctions for a class of nonlinear differential equations

Y (1.2) and olrw + bY(b) = 07 yy’(a) + Sy’(b) = 0. (1.3) A solutiony =y(X, x) of (1.1) + (1.2) [or (1.1) + (1.3)] will be called an nth eigenfunction of the boundary-value problem if it has 11 1 simple zeros in (a, b). In Sections 2 and 3 we treat the boundary conditions y(u) = y(b) = 0. (1.4) Moroney [7] examined this case and proved the existence of an infinite sequence of characteristic functions under certain conditions. We will show that these conditions can considerably be weakened (Section 3). In Sections 4 and 5 we prove analogous theorems for the boundary conditions (1.2) and (1.3). Finally, in Section 6 we show how the method of Section 4 may be extended to nonhomogeneous equations.

Instability of Ekman Flow at large Taylor Number

Instability of Ekman Flow at large Taylor Number