On semiclassical and universal inequalities for eigenvalues of quantum graphs

We study the spectra of quantum graphs with the method of trace identities (sum rules), which are used to derive inequalities of Lieb-Thirring, Payne-Pólya-Weinberger, and Yang types, among others. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false.

[1]  K. Ruedenberg,et al.  Free‐Electron Network Model for Conjugated Systems. I. Theory , 1953 .

[2]  Rupert L. Frank,et al.  Eigenvalue estimates for Schrödinger operators on metric trees , 2007, 0710.5500.

[3]  Evans M. Harrell,et al.  Universal Bounds and Semiclassical Estimates for Eigenvalues of Abstract Schrödinger Operators , 2010, SIAM J. Math. Anal..

[4]  D. Hundertmark.  Some Bound State Problems in Quantum Mechanics , 2007 .

[5]  Sharp Lieb-Thirring inequalities in high dimensions , 1999, math-ph/9903007.

[6]  H. Weyl.  Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) , 1912 .

[7]  L. Hermi,et al.  On Riesz Means of Eigenvalues , 2007, 0712.4088.

[8]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[9]  Linus Pauling,et al.  The Diamagnetic Anisotropy of Aromatic Molecules , 1936 .

[10]  W. Thirring,et al.  Quantum mathematical physics : atoms, molecules and large systems , 2002 .

[11]  E. Harrell,et al.  Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues , 2007, 0705.3673.

[12]  A. Laptev,et al.  Recent results on Lieb-Thirring inequalities , 2000 .

[13]  P. Kuchment,et al.  Quantum Graphs and Their Applications , 2006 .

[14]  G. Pólya,et al.  ON THE RATIO OF CONSECUTIVE EIGENVALUES , 1956 .

[15]  Peter Kuchment,et al.  Analysis on graphs and its applications , 2008 .

[16]  E. Lieb,et al.  Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities , 2002 .

[17]  E. Harrell,et al.  On trace identities and universal eigenvalue estimates for some partial differential operators , 1997 .

[18]  E. Lieb,et al.  On semi-classical bounds for eigenvalues of Schrödinger operators , 1978 .

[19]  E. Lieb.  The Number of Bound States of One-Body Schroedinger Operators and the Weyl Problem (代数解析学の最近の発展) , 1979 .

[20]  T. Weidl.  On the Lieb-Thirring constants L ?,1 for ??1/2 , 1996 .

[21]  Mark S. Ashbaugh,et al.  The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H C Yang , 2002 .

[22]  Pavel Kurasov.  Schrödinger operators on graphs and geometry I: Essentially bounded potentials , 2008 .

[23]  E. Harrell,et al.  Trace identities for commutators, with applications to the distribution of eigenvalues , 2009, 0903.0563.

[24]  M. Birman,et al.  The spectrum of singular boundary problems , 1961 .

[25]  M. Cwikel.  Weak Type Estimates for Singular Values and the Number of Bound States of Schrodinger Operators , 1977 .