A Riesz basis criterion for Schrödinger operators with boundary conditions dependent on the eigenvalue parameter

Guliyev, Namig J.

doi:10.1007/s13324-019-00348-0

Cited by 9 publications

(5 citation statements)

References 12 publications

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“…One example is the Orr-Sommerfeld equation for a liquid film flowing over an inclined plane, with a surface tension gradient which involves boundary conditions that depend linearly on λ; see [22,25]. Other problems with λ-dependent boundary conditions can be found in [28,29,37].…”

Section: Numerical Illustrationmentioning

confidence: 99%

Least-Squares Spectral Methods for ODE Eigenvalue Problems

Hashemi¹,

Nakatsukasa²

2022

SIAM J. Sci. Comput.

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We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices and objects combining quasimatrices and matrices. The strength of the approach is its flexibility that lies in the quasimatrix formulation allowing the basis functions to be chosen arbitrarily, a good choice (e.g., those obtained by solving nearby problems) leading to rapid convergence, and often giving high accuracy. We also show how our algorithm can easily be modified to solve problems with eigenvalue-dependent boundary conditions, and discuss reformulations as an integral equation, which often improves the accuracy.

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Section: Numerical Illustrationmentioning

confidence: 99%

Least-Squares Spectral Methods for ODE Eigenvalue Problems

Hashemi¹,

Nakatsukasa²

2022

SIAM J. Sci. Comput.

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“…Generally speaking, the eigenparameter only appears in the equation, but in many actual phenomena, it is necessary for the eigenparameter to appear in the boundary conditions, such as heat conduction at the liquid-solid interface [6], and so on. Due to its physical significance, many scholars have studied the problem of boundary conditions containing a spectral parameter [7][8][9][10][11][12][13][14]. In recent decades, more researchers have studied eigenparameter-dependent SLPs with discontinuity, including the asymptotic behavior of eigenvalues, the inverse spectral theory, the finite spectrum, the oscillation of eigenfunctions, etc., see [9,10,[15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Matrix Representations for a Class of Eigenparameter Dependent Sturm–Liouville Problems with Discontinuity

Cai

2023

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Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem of a certain type and an eigenparameter-dependent condition, a class of Sturm–Liouville problems with this specified condition was constructed. It has been proven that each Sturm–Liouville problem is equivalent to the given matrix eigenvalue problem.

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“…Various applications in physics and other fields such as the vibration of loaded strings, diffusion processes in probability theory and so on yield such problems [1]. A large number of literature have devoted to the study of such problems for Sturm-Liouville (S-L) problems and fourth-order beam equations, and numerous significance results are obtained (see, for example [2][3][4][5][6][7][8][9][10][11][12]).…”

Section: Introductionmentioning

confidence: 99%