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Published: 8 June 2024

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1

Benedikt, Jiří. "On the discreteness of the spectra of the Dirichlet and Neumannp-biharmonic problems." Abstract and Applied Analysis 2004, no. 9 (2004): 777–92. http://dx.doi.org/10.1155/s1085337504311115.

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We are interested in a nonlinear boundary value problem for(|u″|p−2u″)′′=λ|u|p−2uin[0,1],p>1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to thenth eigenvalue, has preciselyn−1zero points in(0,1). Eigenvalues of the Neumann problem are nonnegative and isolated,0is an eigenvalue which is not simple, and the positive eigenvalues are simple and they form an increasing unbounded sequence. An eigenfunction, corresponding to thenth positive eigenvalue, has preciselyn+1zero points in(0,1).

2

Lamberti, Pier Domenico, and Luigi Provenzano. "Neumann to Steklov eigenvalues: asymptotic and monotonicity results." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 2 (January 16, 2017): 429–47. http://dx.doi.org/10.1017/s0308210516000214.

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We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.

3

Karpukhin, Mikhail A. "The Steklov Problem on Differential Forms." Canadian Journal of Mathematics 71, no. 2 (January 7, 2019): 417–35. http://dx.doi.org/10.4153/cjm-2018-028-6.

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AbstractIn this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

4

Diyab, Farah, and B. Surender Reddy. "Comparison of Laplace Beltrami Operator Eigenvalues on Riemannian Manifolds." European Journal of Mathematics and Statistics 3, no. 5 (October 23, 2022): 55–60. http://dx.doi.org/10.24018/ejmath.2022.3.5.143.

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Let $\Delta_{g}$ be the Laplace Beltrami operator on a manifold $M$ with Dirichlet (resp.,Neumann) boundary conditions. We compare the spectrum of on a Riemannian manifold for Neumann boundary condition and Dirichlet boundary condition . Then we construct aneffective method of obtaining small eigenvalues for Neumann's problem.

5

BARBU, LUMINIŢA, and GHEORGHE MOROŞANU. "On a Steklov eigenvalue problem associated with the (p,q)-Laplacian." Carpathian Journal of Mathematics 37, no. 2 (June 9, 2021): 161–71. http://dx.doi.org/10.37193/cjm.2021.02.02.

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"Consider in a bounded domain \Omega \subset \mathbb{R}^N, N\ge 2, with smooth boundary \partial \Omega, the following eigenvalue problem (1) \begin{eqnarray*} &~&\mathcal{A} u:=-\Delta_p u-\Delta_q u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega, \nonumber \\ &~&\big(\mid \nabla u\mid ^{p-2}+\mid \nabla u\mid ^{q-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) \mid u\mid ^ {r-2}u ~ \mbox{ on} ~ \partial \Omega, \nonumber \end{eqnarray*} where 1<r<q<p<\infty or 1<q<p<r<\infty; r\in \Big(1, \frac{p(N-1)}{N-p}\Big) if p<N and r\in (1, \infty) if p\ge N; a\in L^{\infty}(\Omega),~ b\in L^{\infty}(\partial\Omega) are given nonnegative functions satisfying \[ \int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0. \] Under these assumptions we prove that the set of all eigenvalues of the above problem is the interval [0, \infty). Our result complements those previously obtained by Abreu, J. and Madeira, G., [Generalized eigenvalues of the (p, 2)-Laplacian under a parametric boundary condition, Proc. Edinburgh Math. Soc., 63 (2020), No. 1, 287–303], Barbu, L. and Moroşanu, G., [Full description of the eigenvalue set of the (p,q)-Laplacian with a Steklov-like boundary condition, J. Differential Equations, in press], Barbu, L. and Moroşanu, G., [Eigenvalues of the negative (p,q)– Laplacian under a Steklov-like boundary condition, Complex Var. Elliptic Equations, 64 (2019), No. 4, 685–700], Fărcăşeanu, M., Mihăilescu, M. and Stancu-Dumitru, D., [On the set of eigen-values of some PDEs with homogeneous Neumann boundary condition, Nonlinear Anal. Theory Methods Appl., 116 (2015), 19–25], Mihăilescu, M., [An eigenvalue problem possesing a continuous family of eigenvalues plus an isolated eigenvale, Commun. Pure Appl. Anal., 10 (2011), 701–708], Mihăilescu, M. and Moroşanu, G., [Eigenvalues of -\triangle_p-\triangle_q under Neumann boundary condition, Canadian Math. Bull., 59 (2016), No. 3, 606–616]."

6

Li, Wei, and Ping Yan. "Various Half-Eigenvalues of Scalarp-Laplacian with Indefinite Integrable Weights." Abstract and Applied Analysis 2009 (2009): 1–27. http://dx.doi.org/10.1155/2009/109757.

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Consider the half-eigenvalue problem(ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0a.e.t∈[0,1], where1<p<∞,ϕp(x)=|x|p−2x,x±(⋅)=max⁡{±x(⋅),0}forx∈𝒞0:=C([0,1],ℝ), anda(t)andb(t)are indefinite integrable weights in the Lebesgue spaceℒγ:=Lγ([0,1],ℝ),1≤γ≤∞. We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in(a,b)∈(ℒγ,wγ)2, wherewγdenotes the weak topology inℒγspace. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in(a,b)∈(ℒγ,‖⋅‖γ)2, where‖⋅‖γis theLγnorm ofℒγ.

7

Legendre, Eveline. "Extrema of Low Eigenvalues of the Dirichlet–Neumann Laplacian on a Disk." Canadian Journal of Mathematics 62, no. 4 (August 1, 2010): 808–26. http://dx.doi.org/10.4153/cjm-2010-042-8.

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AbstractWe study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet–Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

8

Mihăilescu, Mihai, and Gheorghe Moroşanu. "Eigenvalues of −Δp − Δq Under Neumann Boundary Condition." Canadian Mathematical Bulletin 59, no. 3 (September 1, 2016): 606–16. http://dx.doi.org/10.4153/cmb-2016-025-2.

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AbstractThe eigenvalue problem −Δpu − Δqu = λ|u|q−2u with p ∊ (1,∞), q ∊ (2,∞), p ≠ q subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from ℝN with N ≥ 2. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval (λ1, ∞) plus an isolated point λ = 0. This comprehensive result is strongly related to our framework, which is complementary to the well-known case p = q ≠ 2 for which a full description of the set of eigenvalues is still unavailable.

9

Ma, Ruyun, Chenghua Gao, and Yanqiong Lu. "Spectrum of Discrete Second-Order Neumann Boundary Value Problems with Sign-Changing Weight." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/280508.

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We study the spectrum structure of discrete second-order Neumann boundary value problems (NBVPs) with sign-changing weight. We apply the properties of characteristic determinant of the NBVPs to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to thejth positive/negative eigenvalue changes its sign exactlyj-1times.

10

D'AGUÌ, GIUSEPPINA, and GIOVANNI MOLICA BISCI. "THREE NON-ZERO SOLUTIONS FOR ELLIPTIC NEUMANN PROBLEMS." Analysis and Applications 09, no. 04 (October 2011): 383–94. http://dx.doi.org/10.1142/s021953051100190x.

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In this note we obtain a multiplicity result for an eigenvalue Neumann problem. Precisely, a recent critical point result for differentiable functionals is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three non-zero weak solutions.

11

Kijowski, M., and L. Klinkenbusch. "Eigenmode analysis of the electromagnetic field scattered by an elliptic cone." Advances in Radio Science 9 (July 29, 2011): 31–37. http://dx.doi.org/10.5194/ars-9-31-2011.

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Abstract. The vector spherical-multipole analysis is applied to determine the scattering of a plane electromagnetic wave by a perfectly electrically conducting (PEC) semi-infinite elliptic cone. From the eigenfunction expansion of the total field in the space outside the elliptic cone, the scattered far field is obtained as a multipole expansion of the free-space type by a single integration over the induced surface currents. As for the evaluation of the free-space-type expansion it is necessary to apply suitable series transformation techniques, a sufficient number of eigenfunctions has to be considered. The eigenvalues of the underlying two-parametric eigenvalue problem with two coupled Lamé equations belong to the Dirichlet- or the Neumann condition and can be arranged as so-called eigenvalue curves. It has been observed that the eigenvalues are in two different domains: In the first one Dirichlet- and Neumann eigenvalues are either nearly coinciding, while in the second one they are strictly separated. The eigenfunctions of the first (coinciding) type look very similar to free-space modes and do not contribute to the scattered field. This observation allows to significantly improve the determination of diffraction coefficients.

12

Ben Haddouch, Khalil, Zakaria El Allali, Najib Tsouli, Siham El Habib, and Fouad Kissi. "Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions." Boletim da Sociedade Paranaense de Matemática 34, no. 1 (January 1, 2016): 253–72. http://dx.doi.org/10.5269/bspm.v34i1.25626.

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In this work we will study the eigenvalues for a fourth order elliptic equation with $p(x)$-growth conditions $\Delta^2_{p(x)} u=\lambda |u|^{p(x)-2} u$, under Neumann boundary conditions, where $p(x)$ is a continuous function defined on the bounded domain with $p(x)>1$. Through the Ljusternik-Schnireleman theory on $C^1$-manifold, we prove the existence of infinitely many eigenvalue sequences and $\sup \Lambda =+\infty$, where $\Lambda$ is the set of all eigenvalues.

13

Dudko, Anastasia, and Vyacheslav Pivovarchik. "Three spectra problem for Stieltjes string equation and Neumann conditions." Proceedings of the International Geometry Center 12, no. 1 (February 28, 2019): 41–55. http://dx.doi.org/10.15673/tmgc.v12i1.1367.

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Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.

14

van den Berg, M., D. Bucur, and K. Gittins. "Maximising Neumann eigenvalues on rectangles." Bulletin of the London Mathematical Society 48, no. 5 (August 8, 2016): 877–94. http://dx.doi.org/10.1112/blms/bdw049.

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15

Ammari, Habib, Kthim Imeri, and Nilima Nigam. "Optimization of Steklov-Neumann eigenvalues." Journal of Computational Physics 406 (April 2020): 109211. http://dx.doi.org/10.1016/j.jcp.2019.109211.

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16

Huang, Min-Jei, and Tzong-Mo Tsai. "The eigenvalue gap for one-dimensional Schrödinger operators with symmetric potentials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 2 (March 25, 2009): 359–66. http://dx.doi.org/10.1017/s0308210507000388.

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We consider the eigenvalue gap for Schrödinger operators on an interval with Dirichlet or Neumann boundary conditions. For a class of symmetric potentials, we prove that the gap between the two lowest eigenvalues is maximized when the potential is constant. We also give some related results for doubly symmetric potentials.

17

Edward, Julian. "Jacobi Matrices and the Spectrum of the Neumann Operator on a Family of Riemann Surfaces." Canadian Journal of Mathematics 45, no. 4 (August 1, 1993): 709–26. http://dx.doi.org/10.4153/cjm-1993-040-0.

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AbstractThe Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. The spectrum of the Neumann operator is studied on the curves bounding a family of Riemann surfaces. The Neumann operator is shown to be isospectral to a direct sum of symmetric Jacobi matrices, each acting on l2(ℤ). The Jacobi matrices are shown to be isospectral to generators of bilateral, linear birth-death processes. Using the connection between Jacobi matrices and continued fractions, it is shown that the eigenvalues of the Neumann operator must solve a certain equation involving hypergeometric functions. Study of the equation yields uniform bounds on the eigenvalues and also the asymptotics of the eigenvalues as the curves degenerate into a wedge of circles.

18

Gol’dshtein, Vladimir, Ritva Hurri-Syrjänen, and Alexander Ukhlov. "Space quasiconformal mappings and Neumann eigenvalues in fractal type domains." Georgian Mathematical Journal 25, no. 2 (June 1, 2018): 221–33. http://dx.doi.org/10.1515/gmj-2018-0025.

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Abstract We study the variation of Neumann eigenvalues of the p-Laplace operator under quasiconformal perturbations of space domains. This study allows us to obtain the lower estimates of Neumann eigenvalues in fractal type domains. The proposed approach is based on the geometric theory of composition operators in connection with the quasiconformal mapping theory.

19

Aliyev, Araz, and Elshad H. Eyvazov. "On the asymptotics of eigenvalues of the Neumann problem for the Schrödinger operator." Baku Mathematical Journal 3, no. 1 (March 1, 2024): 3–11. http://dx.doi.org/10.32010/j.bmj.2024.01.

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In this paper, we study the change in the eigenvalues of the Neumann problem for the Schr¨odinger equation with respect to the radius of the ball. We prove the self-adjointness of the Schr¨odinger operator with a spherically symmetric homogeneous potential and obtain asymptotic formulas for the eigenvalues of the Neumann problem as the radius of the ball tends to zero.

20

Xing, Hui, Hongbin Chen, and Ruofei Yao. "Reversed S-Shaped Bifurcation Curve for a Neumann Problem." Discrete Dynamics in Nature and Society 2018 (August 1, 2018): 1–8. http://dx.doi.org/10.1155/2018/5376075.

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We study the bifurcation and the exact multiplicity of solutions for a class of Neumann boundary value problem with indefinite weight. We prove that all the solutions obtained form a smooth reversed S-shaped curve by topological degree theory, Crandall-Rabinowitz bifurcation theorem, and the uniform antimaximum principle in terms of eigenvalues. Moreover, we obtain that the equation has exactly either one, two, or three solutions depending on the real parameter. The stability is obtained by the eigenvalue comparison principle.

21

CHEN, I. L., Y. T. LEE, P. S. KUO, and J. T. CHEN. "ON THE TRUE AND SPURIOUS EIGENVALUES BY USING THE REAL OR THE IMAGINARY-PART OF THE METHOD OF FUNDAMENTAL SOLUTIONS." International Journal of Computational Methods 10, no. 02 (March 2013): 1341003. http://dx.doi.org/10.1142/s021987621341003x.

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In this paper, the method of fundamental solutions (MFS) of real-part or imaginary-part kernels is employed to solve two-dimensional eigenproblems. The occurring mechanism of spurious eigenvalues for circular and elliptical membranes is examined. It is found that the spurious eigensolution using the MFS depends on the location of the fictitious boundary where the sources are distributed. By employing the singular value decomposition technique, the common left unitary vectors of the true eigenvalue for the single- and double-layer potential approaches are found while the common right unitary vectors of the spurious eigenvalue are obtained. Dirichlet and Neumann eigenproblems are both considered. True eigenvalues are dependent on the boundary condition while spurious eigenvalues are different in the different approach, single-layer or double-layer potential MFS. Two examples of circular and elliptical membranes are numerically demonstrated to see the validity of the present method and the results are compared well with the theoretical prediction.

22

Friedlander, Leonid. "Remarks on Dirichlet and Neumann Eigenvalues." American Journal of Mathematics 117, no. 1 (February 1995): 257. http://dx.doi.org/10.2307/2375044.

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23

Gol'dshtein, Vladimir, Valerii Pchelintsev, and Alexander Ukhlov. "Sobolev extension operators and Neumann eigenvalues." Journal of Spectral Theory 10, no. 1 (February 24, 2020): 337–53. http://dx.doi.org/10.4171/jst/295.

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24

Rudnick, Zeév, Igor Wigman, and Nadav Yesha. "Differences Between Robin and Neumann Eigenvalues." Communications in Mathematical Physics 388, no. 3 (November 2, 2021): 1603–35. http://dx.doi.org/10.1007/s00220-021-04248-y.

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AbstractLet $$\Omega {\subset } {\mathbb {R}}^2$$ Ω ⊂ R 2 be a bounded planar domain, with piecewise smooth boundary $$\partial \Omega $$ ∂ Ω . For $$\sigma >0$$ σ > 0 , we consider the Robin boundary value problem $$\begin{aligned} -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \text{ on } \partial \Omega \end{aligned}$$ - Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where $$ \frac{\partial f}{\partial n} $$ ∂ f ∂ n is the derivative in the direction of the outward pointing normal to $$\partial \Omega $$ ∂ Ω . Let $$0<\lambda ^\sigma _0\le \lambda ^\sigma _1\le \ldots $$ 0 < λ 0 σ ≤ λ 1 σ ≤ … be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps $$\begin{aligned} d_n(\sigma ):=\lambda _n^\sigma -\lambda _n^0 . \end{aligned}$$ d n ( σ ) : = λ n σ - λ n 0 . For a wide class of planar domains we show that there is a limiting mean value, equal to $$2{\text {length}}(\partial \Omega )/{\text {area}}(\Omega )\cdot \sigma $$ 2 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of $$d_n(\sigma )\le C(\Omega ) n^{1/3}\sigma $$ d n ( σ ) ≤ C ( Ω ) n 1 / 3 σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

25

Fan, Xianling. "Eigenvalues of the -Laplacian Neumann problems." Nonlinear Analysis: Theory, Methods & Applications 67, no. 10 (November 2007): 2982–92. http://dx.doi.org/10.1016/j.na.2006.09.052.

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26

Levine, Howard A., and Hans F. Weinberger. "Inequalities between dirichlet and Neumann eigenvalues." Archive for Rational Mechanics and Analysis 94, no. 3 (1986): 193–208. http://dx.doi.org/10.1007/bf00279862.

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27

Gottlieb, H. P. W. "Eigenvalues of the Laplacian with Neumann boundary conditions." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 26, no. 3 (January 1985): 293–309. http://dx.doi.org/10.1017/s0334270000004525.

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AbstractVarious grometrical properties of a domain may be elicited from the asymptotic expansion of a spectral function of the Laplacian operator for that region with apporpriate boundary conditions. Explicit calculations, using analytical formulae for the eigenvalues, are performed for the cases fo Neumann and mixed boundary conditions, extending earlier work involving Dirichet boundary conditions. Two- and three-dimensional cases are considered. Simply-connected regions dealt with are the rectangle, annular sector, and cuboid. Evaluations are carried out for doubly-connected regions, including the narrow annulus, annular cylinder, and thin concentric spherical cavity. The main summation tool is the Poission summation formula. The calculations utilize asymptotic expansions of the zeros of the eigenvalue equations involving Bessel and related functions, in the cases of curved boundaries with radius ratio near unity. Conjectures concerning the form of the contributions due to corners, edges and vertices in the case of Neumann and mixed boundary conditions are presented.

28

Hassannezhad, Asma, and Ari Laptev. "Eigenvalue bounds of mixed Steklov problems." Communications in Contemporary Mathematics 22, no. 02 (March 12, 2019): 1950008. http://dx.doi.org/10.1142/s0219199719500081.

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We study bounds on the Riesz means of the mixed Steklov–Neumann and Steklov–Dirichlet eigenvalue problem on a bounded domain [Formula: see text] in [Formula: see text]. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain.

29

Hu, Yi-Teng, and Murat Şat. "Trace Formulae for Second-Order Differential Pencils with a Frozen Argument." Mathematics 11, no. 18 (September 20, 2023): 3996. http://dx.doi.org/10.3390/math11183996.

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This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, the trace formulae have the same form which reveals the impact of the frozen argument.

30

Lu, Junjie, Tobias Hofmann, Ulrich Kuhl, and Hans-Jürgen Stöckmann. "Implications of Spectral Interlacing for Quantum Graphs." Entropy 25, no. 1 (January 4, 2023): 109. http://dx.doi.org/10.3390/e25010109.

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Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed.

31

KOROTYAEV, EVGENY. "REMARK ON ESTIMATE OF A POTENTIAL IN TERMS OF EIGENVALUES OF THE STURM–LIOUVILLE OPERATOR." Modern Physics Letters B 22, no. 23 (September 10, 2008): 2177–80. http://dx.doi.org/10.1142/s0217984908016959.

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We consider the Sturm–Liouville operator on the unit interval. We obtain two-sided a priori estimates of potential in terms of Dirichlet and Neumann eigenvalues and eigenvalues for 2 types of mixed boundary conditions.

32

Girouard, Alexandre, Mikhail Karpukhin, and Jean Lagacé. "Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems." Geometric and Functional Analysis 31, no. 3 (June 2021): 513–61. http://dx.doi.org/10.1007/s00039-021-00573-5.

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AbstractWe associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is $$8\pi k$$ 8 π k , which is the best upper bound for the $$k^{\text {th}}$$ k th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $$k=1$$ k = 1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

33

Abatangelo, L., V. Felli, and C. Léna. "Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 39. http://dx.doi.org/10.1051/cocv/2019022.

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We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet–Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion’s leading term. This allows inferring some remarkable consequences for Aharonov–Bohm eigenvalues when the singular part of the operator has two coalescing poles.

34

Li, Wei, and Stephen P. Shipman. "Embedded eigenvalues for the Neumann-Poincare operator." Journal of Integral Equations and Applications 31, no. 4 (August 2019): 505–34. http://dx.doi.org/10.1216/jie-2019-31-4-505.

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Mugnai, Dimitri, and Edoardo Proietti Lippi. "Neumann fractionalp-Laplacian: Eigenvalues and existence results." Nonlinear Analysis 188 (November 2019): 455–74. http://dx.doi.org/10.1016/j.na.2019.06.015.

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36

Friedlander, Leonid. "Some inequalities between Dirichlet and neumann eigenvalues." Archive for Rational Mechanics and Analysis 116, no. 2 (1991): 153–60. http://dx.doi.org/10.1007/bf00375590.

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37

Siudeja, Bartłomiej. "On mixed Dirichlet-Neumann eigenvalues of triangles." Proceedings of the American Mathematical Society 144, no. 6 (October 14, 2015): 2479–93. http://dx.doi.org/10.1090/proc/12888.

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38

Henrot, Antoine, and Marco Michetti. "A comparison between Neumann and Steklov eigenvalues." Journal of Spectral Theory 12, no. 4 (May 17, 2023): 1405–42. http://dx.doi.org/10.4171/jst/429.

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39

Soeharyadi, Yudi, Janny Lindiarni, Pilipus Neri Agustima, and Mohammad Januar Ismail Burhan. "Invariant eigenvalues of Laplacian on complex star metric graphs." Hilbert Journal of Mathematical Analysis 1, no. 2 (August 11, 2023): 81–92. http://dx.doi.org/10.62918/hjma.v1i2.13.

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Abstract:

In this article eigenvalues of Laplacian acting on complex star metric graphs is considered. The operator is coupled with Neumann-Kirchhoff vertex condition, implying self adjointness of the operator. We exhibit the invariance of the eigenvalues over the number of the bonds of the star metric graphs. Moreover, the eigenvalues are also invariant over parallel bonds of the star metric multigraphs.

40

Wen, Zhiyuan. "On principal eigenvalues of measure differential equations and a patchy Neumann eigenvalue problem." Journal of Differential Equations 286 (June 2021): 710–30. http://dx.doi.org/10.1016/j.jde.2021.03.040.

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41

Hasanov, Mahir. "On the Travelling Waves for the Generalized Nonlinear Schrödinger Equation." Abstract and Applied Analysis 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/181369.

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This paper is devoted to the analysis of the travelling waves for a class of generalized nonlinear Schrödinger equations in a cylindric domain. Searching for travelling waves reduces the problem to the multiparameter eigenvalue problems for a class of perturbedp-Laplacians. We study dispersion relations between the eigenparameters, quantitative analysis of eigenfunctions and discuss some variational principles for eigenvalues of perturbedp-Laplacians. In this paper we analyze the Dirichlet, Neumann, No-flux, Robin and Steklov boundary value problems. Particularly, a “duality principle” between the Robin and the Steklov problems is presented.

42

Yang, Yisong. "A Comparison of Eigenvalues of Two Sturm-Liouville Problems." Canadian Mathematical Bulletin 33, no. 4 (December 1, 1990): 386–90. http://dx.doi.org/10.4153/cmb-1990-063-0.

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43

Hui, Kin Ming. "Comparison Theorems for the Eigenvalues of the Laplacian in the Unit Ball in RN." Canadian Mathematical Bulletin 35, no. 2 (June 1, 1992): 214–17. http://dx.doi.org/10.4153/cmb-1992-030-0.

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44

Persson, Mikael. "Eigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann Hamiltonian." Advances in Mathematical Physics 2009 (2009): 1–15. http://dx.doi.org/10.1155/2009/873704.

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Abstract:

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in , . The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We give asymptotic formulas for the rate of accumulation of eigenvalues in these clusters. When the compact is a Reinhardt domain we are able to show a more precise asymptotic formula.

45

Williams, Fred W., Andrew Watson, W. Paul Howson, and Antonia J. Jones. "Exact solutions for Sturm–Liouville problems on trees via novel substitute systems and the Wittrick–Williams algorithm." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2088 (September 25, 2007): 3195–224. http://dx.doi.org/10.1098/rspa.2007.1905.

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Abstract:

Since 1970, the Wittrick–Williams algorithm has been applied with increasing sophistication in structural mechanics to guarantee that eigenvalues cannot be missed and are calculated accurately. The underlying theorem enables its application to any discipline requiring eigenvalues of self-adjoint systems of differential equations. Its value in mathematics was recently illustrated by studying Sturm–Liouville equations on large homogeneous trees with Dirichlet boundary conditions and n (≤43) levels. Recursive subsysteming was applied n −1 times to assemble the tree progressively from sub-trees. Hence, numerical results confirmed the recent theoretical bounds of Sobolev & Solomyak for n →∞. In addition, a structural mechanics analogy yielded a proof that many eigenvalues had high multiplicities determined by n and the branching number b . Inspired by the structural mechanics analogy, we now prove that all eigenvalues of the tree are obtainable from n substitute chains r (=1, 2, …, n ) which involve only r consecutively linked differential equations and which have only singlefold eigenvalues. Equations are also derived for the multiplicities these eigenvalues have for the tree. Hence, double precision calculations on a PC readily gave eigenvalues for n =10 6 and b =10, i.e. ≃10 999 999 linked Sturm–Liouville equations. Moreover, a simple equation is derived which gives all the eigenvalues of uniform trees with Dirichlet conditions at both ends, and band-gap spectra are numerically demonstrated and theoretically justified for trees with the Dirichlet conditions at either end replaced by Neumann ones. Additionally, even if each multiple eigenvalue would be counted as if it were singlefold, the proportion of eigenvalues that are multiple is proved to approach unity as n →∞.

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Marrocos, Marcus A. M., and Antônio L. Pereira. "Eigenvalues of the Neumann Laplacian in symmetric regions." Journal of Mathematical Physics 56, no. 11 (November 2015): 111502. http://dx.doi.org/10.1063/1.4935300.

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47

Gol’dshtein, V., and A. Ukhlov. "Composition Operators on Sobolev Spaces and Neumann Eigenvalues." Complex Analysis and Operator Theory 13, no. 6 (August 2, 2018): 2781–98. http://dx.doi.org/10.1007/s11785-018-0826-1.

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48

Band, Ram, Michael Bersudsky, and David Fajman. "Courant-sharp eigenvalues of Neumann 2-rep-tiles." Letters in Mathematical Physics 107, no. 5 (November 29, 2016): 821–59. http://dx.doi.org/10.1007/s11005-016-0926-7.

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49

Enache, C., and G. A. Philippin. "Some inequalities involving eigenvalues of the Neumann Laplacian." Mathematical Methods in the Applied Sciences 36, no. 16 (March 21, 2013): 2145–53. http://dx.doi.org/10.1002/mma.2743.

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50

Edward, Julian. "Spectral Theory for the Neumann Laplacian on Planar Domains With Horn-Like Ends." Canadian Journal of Mathematics 49, no. 2 (April 1, 1997): 232–62. http://dx.doi.org/10.4153/cjm-1997-012-8.

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Abstract:

AbstractThe spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at 0 or ∞. The proof uses Mourre theory.