Low Eigenvalues of Laplace and Schrödinger Operators (original) (raw)

A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Communications in Mathematical Physics, 2006

Let λ i (Ω, V ) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω ⊂ R n and with the positive potential V . Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V ⋆ , we prove that λ 2 (Ω, V ) ≤ λ 2 (S 1 , V ⋆ ). Here S 1 denotes the ball, centered at the origin, that satisfies the condition λ 1 (Ω, V ) = λ 1 (S 1 , V ⋆ ).

A second eigenvalue bound for the Dirichlet Schroedinger operator

2005

Let λ_i(Ω,V) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω⊂^n and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V_, we prove that λ_2(Ω,V) <λ_2(S_1,V_). Here S_1 denotes the ball, centered at the origin, that satisfies the condition λ_1(Ω,V) = λ_1(S_1,V_). Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ_2(B_R, V) / λ_1(B_R, V) decreases when the radius R of the ball B_R increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.

Refined asymptotics for eigenvalues on domains of infinite measure

Journal of Mathematical Analysis and Applications, 2010

In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure. Also, we derive some estimates for the the spectral counting function of the Laplace operator on unbounded two-dimensional domains.

A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of 𝕊ⁿ

Transactions of the American Mathematical Society, 2000

For a domain Ω \Omega contained in a hemisphere of the n n –dimensional sphere S n \mathbb {S}^n we prove the optimal result λ 2 / λ 1 ( Ω ) ≤ λ 2 / λ 1 ( Ω ⋆ ) \lambda _2/\lambda _1(\Omega ) \le \lambda _2/\lambda _1(\Omega ^{\star }) for the ratio of its first two Dirichlet eigenvalues where Ω ⋆ \Omega ^{\star } , the symmetric rearrangement of Ω \Omega in S n \mathbb {S}^n , is a geodesic ball in S n \mathbb {S}^n having the same n n –volume as Ω \Omega . We also show that λ 2 / λ 1 \lambda _2/\lambda _1 for geodesic balls of geodesic radius θ 1 \theta _1 less than or equal to π / 2 \pi /2 is an increasing function of θ 1 \theta _1 which runs between the value ( j n / 2 , 1 / j n / 2 − 1 , 1 ) 2 (j_{n/2,1}/j_{n/2-1,1})^2 for θ 1 = 0 \theta _1=0 (this is the Euclidean value) and 2 ( n + 1 ) / n 2(n+1)/n for θ 1 = π / 2 \theta _1=\pi /2 . Here j ν , k j_{\nu ,k} denotes the k k th positive zero of the Bessel function J ν ( t ) J_{\nu }(t) . This result generalizes the Payne–Pólya–W...

Eigenvalue estimates for magnetic Schrödinger operators in domains

Proceedings of the American Mathematical Society, 2008

Inequalities are derived for sums and quotients of eigenvalues of magnetic Schrödinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit. ogy,

Eigenvalue estimates for a three-dimensional magnetic Schr\"odinger operator

Asymptotic Analysis

We consider a magnetic Schrödinger operator H h = (−ih∇ − A) 2 with the Dirichlet boundary conditions in an open set Ω ⊂ R 3 , where h > 0 is a small parameter. We suppose that the minimal value b 0 of the module | B| of the vector magnetic field B is strictly positive, and there exists a unique minimum point of | B|, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator H h in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.

Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians

Reviews in Mathematical Physics, 2014

Let H 0,D (resp., H 0,N ) be the Schrödinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let H ℓ := H 0,ℓ − V , ℓ = D, N , where the scalar potential V is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of H D and H N below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of H ℓ near inf σ ess (H ℓ ) = inf σ(H 0,ℓ ), ℓ = D, N . Applying these Hamiltonians, we show that σ disc (H D ) is infinite even if V has a compact support, while σ disc (H N ) could be finite or infinite depending on the decay rate of V .

A bound for the eigenvalue counting function for Krein–von Neumann and Friedrichs extensions

Advances in Mathematics, 2017

For an arbritray open, nonempty, bounded set Ω ⊂ R n , n ∈ N, and sufficiently smooth coefficients a, b, q, we consider the minimally defined, strictly positive, higher-order differential operator A min,Ω,2m (a, b, q) in L 2 (Ω) defined onW 2m,2 (Ω), associated with the higher-order differential expression τ 2m (a, b, q) := n j,k=1 (−i∂ j − b j)a j,k (−i∂ k − b k) + q m , m ∈ N, and its Krein-von Neumann extension A K,Ω,2m (a, b, q) in L 2 (Ω). Denoting by N (λ, A K,Ω,2m (a, b, q)), λ > 0, the eigenvalue counting function corresponding to the strictly positive eigenvalues of A K,Ω,2m (a, b, q), we derive the bound N (λ, A K,Ω,2m (a, b, q)) Cvn(2π) −n 1 + 2m 2m + n n/(2m) λ n/(2m) , λ > 0, where C = C(a, b, q, Ω) > 0 (with C(In, 0, 0) = |Ω|) is connected to the eigenfunction expansion of the self-adjoint operator A 2m (a, b, q) in L 2 (R n) defined on W 2m,2 (R n), corresponding to τ 2m (a, b, q). Here vn := π n/2 /Γ((n + 2)/2) denotes the (Euclidean) volume of the unit ball in R n. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying (abstract) buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of A 2m (a, b, q) in L 2 (R n). We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension A F,Ω,2m (a, b, q) in L 2 (Ω) of A min,Ω,2m (a, b, q). No assumptions on the boundary ∂Ω of Ω are made.