TITLE

Lower bound on the spectrum of the two-dimensional Schr�dinger operator with a d-perturbation on a curve

AUTHOR(S)
Lobanov, I. S.; Lotoreichik, V. Yu.; Popov, I. Yu.
PUB. DATE
March 2010
SOURCE
Theoretical & Mathematical Physics;Mar2010, Vol. 162 Issue 3, p332
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We consider the two-dimensional Schr�dinger operator with a d-potential supported by curve. For the cases of infinite and closed finite smooth curves, we obtain lower bounds on the spectrum of the considered operator that are expressed explicitly in terms of the interaction strength and a parameter characterizing the curve geometry. We estimate the bottom of the spectrum for a piecewise smooth curve using parameters characterizing the geometry of the separate pieces. As applications of the obtained results, we consider curves with a finite number of cusps and general �leaky� quantum graph as the support of the d-potential.
ACCESSION #
49157260

 

Related Articles

  • Infinitesimal form boundedness and Trudinger�s subordination for the Schr�dinger operator. Maz�ya, V. G.; Verbitsky, I. E. // Inventiones Mathematicae;Oct2005, Vol. 162 Issue 1, p81 

    We give explicit analytic criteria for two problems associated with the Schr�dinger operator H=-?+ Q on L2(R n) where Q? D�(R n) is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on Q so that the quadratic form $\langle{Q}\cdot,\...

  • Variational Estimates for Discrete Schr�dinger Operators with Potentials of Indefinite Sign. Damanik, D.; Hundertmark, D.; Killip, R.; Simon, B. // Communications in Mathematical Physics;Jul2003, Vol. 238 Issue 3, p545 

    Let H be a one-dimensional discrete Schr�dinger operator. We prove that if sess ? [-2, 2], then H - H0 is compact and sess (H) = [-2, 2].We also prove that if H0 + �V� has at least one bound state, then the same is true for H0 + V . Further, if H0+ �V� has infinitely...

  • Schrödinger Operator Levels for a Crystal Film with a Nonlocal Potential. Smetanina, M. S.; Chuburin, Yu. P. // Theoretical & Mathematical Physics;Aug2004, Vol. 140 Issue 2, p1146 

    For a crystal film, we consider the Schrödinger operator defined on Bloch functions (with respect to two variables) in a cell. The potential is the sum of two small terms: a function decreasing with respect to the third variable and an operator of rank one. We prove the existence of two...

  • OPTIMAL CONTROL OF SYSTEMS INVOLVING SCHRÖDINGER OPERATORS. Serag, H. M. // Control & Intelligent Systems;2004, Vol. 32 Issue 3, p154 

    The existence and uniqueness of solutions for cooperative systems involving Schrodinger operators are proved under conditions on the coefficients stated by the principal eigenvalue of Schrodinger eigenvalue problem with weight function; then the necessary and sufficient conditions of optimality...

  • On the absolutely continuous and negative discrete spectra of Schrödinger operators on the line with locally integrable globally square summable potentials. Rybkin, Alexei // Journal of Mathematical Physics;Apr2004, Vol. 45 Issue 4, p1418 

    For one-dimensional Schrödinger operators with potentials q subject to ∑[sub n=-∞][sup ∞](∫[sub n][sup n+1]|q(x)|dx)[sup 2]<∞, we prove that the absolutely continuous spectrum is [0,∞), extending the 1999 result due to Dieft–Killip. As a by-product we...

  • Conditional Wegner Estimate for the Standard Random Breather Potential. Täufer, Matthias; Veselić, Ivan // Journal of Statistical Physics;Nov2015, Vol. 161 Issue 4, p902 

    We prove a conditional Wegner estimate for Schrödinger operators with random potentials of breather type. More precisely, we reduce the proof of the Wegner estimate to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in...

  • Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension. Damanik, David; Tcheremchantsev, Serguei // Communications in Mathematical Physics;Jun2003, Vol. 236 Issue 3, p513 

    : We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schr�dinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the...

  • Some Improvements in the Method of the Weakly Conjugate Operator. Richard, Serge // Letters in Mathematical Physics;Apr2006, Vol. 76 Issue 1, p27 

    We present some improvements in the method of the weakly conjugate operator, one variant of the Mourre theory. When applied to certain two-body Schrödinger operators, this leads to a limiting absorption principle that is uniform on the positive real axis.

  • Dynamical Analysis of Schr�dinger Operators with Growing Sparse Potentials. Tcheremchantsev, Serguei // Communications in Mathematical Physics;Jan2005, Vol. 253 Issue 1, p221 

    We consider discrete half-line Schr�dinger operatorsHwith potentials of the formV(n)=S(n)+Q(n). HereQis any compactly supported real function,ifn=L N andS(n)=0 otherwise, where?? (0,1) andL N is a very fast growing sequence. We study in a rather detailed manner the time-averaged dynamics...

Share

Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics