^{1}, P. G. Kevrekidis

^{2}and S. Lepri

^{3}

### Abstract

We consider the eigenvalue problem for one-dimensional linear Schrödinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples, we discuss both linear and nonlinear, Hamiltonian and -symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomial equation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.

We consider the time evolution of a quantum mechanical wave function as governed by the Schrödinger equation. The wave function is distributed spatially on a discrete one-dimensional lattice, i.e., a chain of nodes indexed by integers, so that the spatial derivatives are replaced by differences. The potential function is nonzero only at a few center sites on the lattice, representing either physical impurities or other obstacles such as an external field or a nonlinear material. Since the general solution of the zero potential problem is well-known, we begin by constructing it on the outer (left and right) portions of the lattice. Working our way toward the impurity sites using the restraints of the discrete Schrödinger equation, we find that appropriately defined portions of the outer solution must satisfy a polynomialequation. Our method is also applied to the (again) discrete but (now) nonlinear Schrödinger equation. Here, known stationary solutions are acted upon by a time-dependent perturbation, and we find that appropriately defined portions of the perturbation must satisfy polynomialequations. The main point is to show that the polynomial conditions we derive accurately determine the dynamical stability of the solutions. The success of our method in tracking the associated linear and nonlinear spectra is presented throughout the linear and nonlinear cases. In order to demonstrate the generality of our approach, we show examples using both real valued Hamiltonians and complex parity-time symmetric potentials.

I. INTRODUCTION

II. LINEAR CASE

A. Theoretical analysis

1. Oligomer of length two

2. Oligomer of length three or greater

B. Numerical results

III. NONLINEAR CASE

A. Theoretical analysis

1. Oligomer of length two

2. Oligomer of length three or greater

B. Numerical results

IV. CONCLUSIONS AND FUTURE CHALLENGES

### Key Topics

- Eigenvalues
- 37.0
- Numerical solutions
- 9.0
- Polynomials
- 9.0
- Boundary value problems
- 5.0
- Dispersion relations
- 5.0

## Figures

The left set of panels corresponds to the linear -symmetric dimer with N = 2 and while the right set of panels to the case of the trimer with N = 3 and . Each set contains a (top) plot of linear stability eigenfrequencies as a function of increasing γ as computed an a lattice of length 200. The eigenvalue agreement between numeric (circles) and exact (x's) computations is shown for (bottom left) and (bottom right).

The left set of panels corresponds to the linear -symmetric dimer with N = 2 and while the right set of panels to the case of the trimer with N = 3 and . Each set contains a (top) plot of linear stability eigenfrequencies as a function of increasing γ as computed an a lattice of length 200. The eigenvalue agreement between numeric (circles) and exact (x's) computations is shown for (bottom left) and (bottom right).

Profiles of modulus of unstable eigenvectors are shown for the linear dimer (left) and the linear trimer (right). These eigenvectors correspond to the eigenvalues with negative imaginary part for , which are seen in the bottom right panels of Fig. 1 . Again, numerical computations (circles) are shown to agree with the exact computation (x's) and the lattice length is 200.

Profiles of modulus of unstable eigenvectors are shown for the linear dimer (left) and the linear trimer (right). These eigenvectors correspond to the eigenvalues with negative imaginary part for , which are seen in the bottom right panels of Fig. 1 . Again, numerical computations (circles) are shown to agree with the exact computation (x's) and the lattice length is 200.

The left set of plots corresponds to the linear Hamiltonian dimer with N = 2 and while the right set of plots to the case of the linear trimer with N = 3 and . Each set contains a (top) plot of linear stability eigenvalues and a (bottom) plot of the eigenvector for the defect (point spectrum) mode. All plots show agreement between the numerical (circles) and semi-analytical (x's) results. Here, the lattice length is 40.

The left set of plots corresponds to the linear Hamiltonian dimer with N = 2 and while the right set of plots to the case of the linear trimer with N = 3 and . Each set contains a (top) plot of linear stability eigenvalues and a (bottom) plot of the eigenvector for the defect (point spectrum) mode. All plots show agreement between the numerical (circles) and semi-analytical (x's) results. Here, the lattice length is 40.

The left set of panels corresponds to the -symmetric dimer with N = 2, , and while the right set of panels to the case of the trimer with N = 3, , and . Each set contains a (top) plot of linear stability eigenfrequencies of the stationary solution corresponding to T = 0.7 and as computed on a lattice of length 20. Plots of the eigenvalue ν indicating agreement between numeric (circles) and exact (x's) results are shown for the dimer with (bottom left) and (bottom right), and for the trimer with (bottom left) and (bottom middle) and (bottom right). For the trimer, when γ is small the dominant pair of (unstable) eigenfrequencies of negative imaginary part increases in magnitude as γ increases until . At this point, the complex pair recedes and a single dominant purely imaginary emerges at . This single eigenfrequency increases in magnitude until when it begins to decrease. At , a new dominant complex pair increases in magnitude and continues to increase as γ increases.

The left set of panels corresponds to the -symmetric dimer with N = 2, , and while the right set of panels to the case of the trimer with N = 3, , and . Each set contains a (top) plot of linear stability eigenfrequencies of the stationary solution corresponding to T = 0.7 and as computed on a lattice of length 20. Plots of the eigenvalue ν indicating agreement between numeric (circles) and exact (x's) results are shown for the dimer with (bottom left) and (bottom right), and for the trimer with (bottom left) and (bottom middle) and (bottom right). For the trimer, when γ is small the dominant pair of (unstable) eigenfrequencies of negative imaginary part increases in magnitude as γ increases until . At this point, the complex pair recedes and a single dominant purely imaginary emerges at . This single eigenfrequency increases in magnitude until when it begins to decrease. At , a new dominant complex pair increases in magnitude and continues to increase as γ increases.

Profiles of the moduli of unstable eigenvectors are shown for the nonlinear dimer. The left four plots show eigenvectors for and the right four plots for . These eigenvectors correspond to eigenvalues with negative imaginary part which are seen in Fig. 4 . Again, numerical computations are shown to agree with the exact (x's) results and the length of the lattice is 20. Notice the localization in both components of the eigenvector in the case of the -symmetry breaking case of .

Profiles of the moduli of unstable eigenvectors are shown for the nonlinear dimer. The left four plots show eigenvectors for and the right four plots for . These eigenvectors correspond to eigenvalues with negative imaginary part which are seen in Fig. 4 . Again, numerical computations are shown to agree with the exact (x's) results and the length of the lattice is 20. Notice the localization in both components of the eigenvector in the case of the -symmetry breaking case of .

Profiles of the moduli of unstable eigenvectors are shown for the nonlinear trimer with lattice length 20. The left four plots show eigenvectors for and the right six plots for . These eigenvectors correspond to eigenvalues with negative imaginary part which are seen in Fig. 4 . The agreement is similar to those of the earlier figures.

Profiles of the moduli of unstable eigenvectors are shown for the nonlinear trimer with lattice length 20. The left four plots show eigenvectors for and the right six plots for . These eigenvectors correspond to eigenvalues with negative imaginary part which are seen in Fig. 4 . The agreement is similar to those of the earlier figures.

The left set of panels correspond to the -symmetric dimer with N = 2, , and while the right set of panels to the case of the trimer with N = 3, , and . Each set contains contour plots of extremal stability eigenfrequencies for extended solutions as in Eq. (28) on a lattice of length 20 with (top left), (top right), (bottom left), and (bottom right).

The left set of panels correspond to the -symmetric dimer with N = 2, , and while the right set of panels to the case of the trimer with N = 3, , and . Each set contains contour plots of extremal stability eigenfrequencies for extended solutions as in Eq. (28) on a lattice of length 20 with (top left), (top right), (bottom left), and (bottom right).

The left set of panels correspond to the Hamiltonian dimer with N = 2, , and while the right set of panels to the case of the trimer with N = 3, , and . Each set contains contour plots of extremal stability eigenfrequencies for extended solutions as in Eq. (28) on a lattice of length 20 with (top left), (top right), (bottom left), and (bottom right).

The left set of panels correspond to the Hamiltonian dimer with N = 2, , and while the right set of panels to the case of the trimer with N = 3, , and . Each set contains contour plots of extremal stability eigenfrequencies for extended solutions as in Eq. (28) on a lattice of length 20 with (top left), (top right), (bottom left), and (bottom right).

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