Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schrodinger operator $-\partial_x^2+V(x)$ with an interface. The interface is modeled by a jump either in the value or the derivative of $V(x)$ and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can occur only in spectral gaps. We pose the eigenvalue problem as a $C^1$ gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions $R_\pm=\frac{\psi'_\pm(0)}{\psi_\pm(0)}$, where $\psi_\pm$ are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Prufer transformation. The limit values of $R_\pm$ at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at ga...

[1]  Rachel J. Steiner,et al.  The spectral theory of periodic differential equations , 1973 .

[2]  P. Hartman.  Ordinary Differential Equations , 1965 .

[3]  Bernhard Kawohl,et al.  Rearrangements and Convexity of Level Sets in PDE , 1985 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[6]  R. Morandotti,et al.  Optical modes at the interface between two dissimilar discrete meta-materials. , 2007, Optics express.

[7]  H. Ohno,et al.  Tamm states in superlattices , 1992 .

[8]  P. Deift,et al.  On the existence of eigenvalues of the Schrödinger operatorH−λW in a gap of σ(H) , 1986 .

[9]  C. Fort,et al.  Bose-Einstein Condensates in an Optical Lattice , 2004 .

[10]  The spectrum of a periodic operator with a small localized perturbation , 2007 .

[11]  E. Gross.  Structure of a quantized vortex in boson systems , 1961 .

[12]  C. Bandle,et al.  Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions , 2008 .

[13]  J. Garnett,et al.  Gaps and bands of one dimensional periodic Schrödinger operators , 1984 .

[14]  On the spectrum of a periodic operator with a small localized perturbation , 2006, math/0609195.

[15]  E. Korotyaev.  Schrödinger operator with a junction of two 1-dimensional periodic potentials , 2005, math/0507295.

[16]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[17]  G. Stolz,et al.  Localization for random perturbations of periodic Schrödinger operators , 1998 .

[18]  Localization for Random Perturbations of Periodic Schrödinger Operators with Regular Floquet Eigenvalues , 2002, math-ph/0510063.

[19]  Nate Orlow,et al.  Hill ’ s Equation , 2010 .

[20]  Percy Deift,et al.  Eigenvalue branches of the Schrödinger operatorH−λW in a gap of σ(H) , 1989 .

[21]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[22]  Evgeni Korotyaev,et al.  Lattice Dislocations in a 1-Dimensional Model , 2000 .