Геометрические свойства собственных функций@@@Geometric properties of eigenfunctions

[1]  Jochen Brüning,et al.  Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators , 1978 .

[2]  T. Paul,et al.  Weighted trace formula near a hyperbolic trajectory and complex orbits , 1998 .

[3]  Kaufman,et al.  Wave chaos in the stadium: Statistical properties of short-wave solutions of the Helmholtz equation. , 1988, Physical review. A, General physics.

[4]  Dennis A. Hejhal,et al.  On the Topography of Maass Waveforms for PSL(2, Z) , 1992, Exp. Math..

[5]  T. Hoffmann-Ostenhof,et al.  The nodal line of the second eigenfunction of the Laplacian in $\mathbb{R}^2$ can be closed , 1997 .

[6]  A. Good.  Cusp forms and eigenfunctions of the Laplacian , 1984 .

[7]  E. Bogomolny.  Smoothed wave functions of chaotic quantum systems , 1988 .

[8]  P. Bérard,et al.  Volume des ensembles nodaux des fonctions propres du laplacien , 1985 .

[9]  Y. Sinai,et al.  Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae , 1995 .

[10]  Riemannian manifolds with uniformly bounded eigenfunctions , 2000, math-ph/0002038.

[11]  P. Bérard.  On the wave equation on a compact Riemannian manifold without conjugate points , 1977 .

[12]  T. Prosen,et al.  Quantization of generic chaotic 3D billiard with smooth boundary II: structure of high-lying eigenstates , 1996, chao-dyn/9611016.

[13]  Analytic continuation of representations and estimates of automorphic forms , 1999, math/9907202.

[14]  Taylor,et al.  Born-Oppenheimer adiabatic mechanism for regularity of states in the quantum stadium billiard. , 1985, Physical review. A, General physics.

[15]  F. Steiner,et al.  Temporal Quantum Chaos , 1999 .

[16]  O. Bohigas,et al.  Quantum chaotic dynamics and random polynomials , 1996 .

[17]  B. Helffer,et al.  Ergodicité et limite semi-classique , 1987 .

[18]  Dieter Ullmann.  Chladni und die Entwicklung der Akustik von 1750-1860 , 1996 .

[19]  S. Wolpert.  The Modulus of Continuity¶for Γ0(m)\? Semi-Classical Limits , 2001 .

[20]  Y. C. Verdière.  Quasi-modes sur les varietes Riemanniennes , 1977 .

[21]  Daniel Grieser,et al.  Asymptotics of the first nodal line of a convex domain , 1996 .

[22]  P. Sarnak.  Estimates for Rankin–Selberg L-Functions and Quantum Unique Ergodicity , 2001 .

[23]  J. Vanderkam.  Correction to: “L∞ norms and quantum ergodicity on the sphere” , 1998 .

[24]  T. Hoffmann-Ostenhof,et al.  Interior hölder estimates for solutions of schrödinger equations and the regularity of nodal sets , 1994 .

[25]  J. Hannay.  The chaotic analytic function , 1998 .

[26]  Hans Lewy.  On the mininum number of domains in which the nodal lines of spherical harmonics divide the sphere , 1977 .

[27]  H. Donnelly.  On the wave equation asymptotics of a compact negatively curved surface , 1978 .

[28]  A. Voros,et al.  Chaotic Eigenfunctions in Phase Space , 1997, chao-dyn/9711016.

[29]  S. Zelditch.  Quantum ergodicity on the sphere , 1992 .

[30]  K. Burdzy,et al.  On the “Hot Spots” Conjecture of J. Rauch , 1999 .

[31]  C. Fefferman,et al.  Nodal Sets of Eigenfunctions: Riemannian Manifolds With Boundary , 1990 .

[32]  J. Vanderkam.  L∞ norms and quantum ergodicity on the sphere , 1997 .

[33]  T. Prosen.  Parametric statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials , 1996, chao-dyn/9612006.

[34]  Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles , 1998, math/9803052.

[35]  Arke Pleijel,et al.  Remarks on courant's nodal line theorem , 1956 .

[36]  On the pointwise behavior of semi-classical measures , 1996 .

[37]  R. Aurich,et al.  On the rate of quantum ergodicity on hyperbolic surfaces and for billiards , 1997, chao-dyn/9707016.

[38]  V. M. Babich,et al.  Eigenfunctions Concentrated Near a Closed Geodesic , 1968 .

[39]  H. Maass.  Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichlet scher Reihen durch Funktionalgleichungen , 1949 .

[40]  F. Toscano,et al.  Geometrical approach to the distribution of the zeros for the Husimi function , 1999, chao-dyn/9903028.

[41]  C. Fefferman,et al.  Nodal sets of eigenfunctions on Reimannian manifolds , 1988 .

[42]  J. Toth.  Various Quantum Mechanical Aspects of Quadratic Forms , 1995 .

[43]  A. M. Almeida.  The Weyl representation in classical and quantum mechanics , 1998 .

[44]  S. V. Ngoc,et al.  Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type , 2000 .

[45]  Christopher D. Sogge,et al.  Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds , 1988 .

[46]  J. Toth.  On the quantum expected values of integrable metric forms , 1999 .

[47]  Fishman,et al.  Semiclassical criterion for scars in wave functions of chaotic systems. , 1994, Physical review letters.

[48]  M. Berry.  Evolution of semiclassical quantum states in phase space , 1979 .

[49]  A. Voros,et al.  Chaos-revealing multiplicative representation of quantum eigenstates , 1990 .

[50]  M. Srednicki,et al.  Random matrix elements and eigenfunctions in chaotic systems , 1997, chao-dyn/9711020.

[51]  P. Sarnak,et al.  $L^\infty$ norms of eigenfunctions of arithmetic surfaces , 1995 .

[52]  Peter Sarnak,et al.  Integrals of products of eigenfunctions , 1994 .

[53]  L. Bers.  Local behavior of solutions of general linear elliptic equations , 1955 .

[54]  R. Aurich,et al.  Statistical properties of highly excited quantum eigenstates of a strongly chaotic system , 1993 .

[55]  Uzy Smilansky,et al.  Nodal domains statistics: a criterion for quantum chaos. , 2001, Physical review letters.

[56]  David Jerison,et al.  The diameter of the first nodal line of a convex domains , 1995 .

[57]  David Jerison,et al.  The “hot spots” conjecture for domains with two axes of symmetry , 2000 .

[58]  R. Salem,et al.  Some properties of trigonometric series whose terms have random signs , 1954 .

[59]  J. Toth.  Eigenfunction decay estimates in the quantum integrable case , 1998 .

[60]  Steve Zelditch,et al.  Uniform distribution of eigenfunctions on compact hyperbolic surfaces , 1987 .

[61]  V. Guillemin.  Lectures on spectral theory of elliptic operators , 1977 .

[62]  N. Nadirashvili,et al.  Quasi-symmetry of $L^p$ norms of eigenfunctions , 2002 .

[63]  Quantum Ergodicity of Eisenstein Series for Arithmetic 3-Manifolds , 2000 .

[64]  M. Berry,et al.  Quantum scars of classical closed orbits in phase space , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[65]  Eric Leichtnam,et al.  Ergodic properties of eigenfunctions for the Dirichlet problem , 1993 .

[66]  J. Toth.  Eigenfunction localization in the quantized rigid body , 1996 .

[67]  Universality and scaling of correlations between zeros on complex manifolds , 1999, math-ph/9904020.

[68]  Lev Kaplan,et al.  Linear and Nonlinear Theory of Eigenfunction Scars , 1998, chao-dyn/9809011.

[69]  A. Good.  The square mean of Dirichlet series associated with cusp forms , 1982 .

[70]  S. Zelditch.  On the rate of Quantum Ergodicity, II: Lower bounds , 1994 .

[71]  Poincaré–Lelong Approach to Universality and Scaling of Correlations Between Zeros , 1999, math-ph/9903012.

[72]  B. S'evennec.  Multiplicité du spectre des surfaces : une approche topologique , 1994 .

[73]  A. Shnirelman.  Addendum On the Asymptotic Properties of Eigenfunctions in the Regions of Chaotic Motion , 1993 .

[74]  Xiaojun Di,et al.  Correlations between zeros of a random polynomial , 1997 .

[75]  Y. C. Verdière.  Spectre conjoint d'opérateurs pseudo-différentiels qui commutent , 1980 .

[76]  A M Ozorio de Almeida,et al.  The Wigner function for two dimensional tori: Uniform approximation and projections , 1983 .

[77]  P. Shukla.  On the distribution of zeros of chaotic wavefunctions , 1997, cond-mat/9705298.

[78]  J. Bernstein,et al.  Sobolev norms of automorphic functionals and Fourier coefficients of cusp forms , 1998 .

[79]  Random Analytic Chaotic Eigenstates , 1999, chao-dyn/9901019.

[80]  Peter Sarnak,et al.  Spectra and eigenfunctions of laplacians , 1997 .

[81]  Szego kernels and a theorem of Tian , 2000, math-ph/0002009.

[82]  Nikolai Nadirashvili,et al.  Metric properties of eigenfunctions of the Laplace operator on manifolds , 1991 .

[83]  F. Leyvraz,et al.  Distribution of eigenvalues for the modular group , 1995 .

[84]  S. Zelditch.  Spectral Theory and Geometry: Lectures on wave invariants , 1999 .

[85]  M. Berry.  Semi-classical mechanics in phase space: A study of Wigner’s function , 1977, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[86]  Shiu-yuen Cheng.  Eigenfunctions and nodal sets , 1976 .

[87]  R. Hardt,et al.  Nodal sets for solutions of elliptic equations , 1989 .

[88]  Antonios D. Melas.  On the nodal line of the second eigenfunction of the Laplacian in $\mathbf{R}^2$ , 1992 .

[89]  P. Sarnak,et al.  Quantum unique ergodicity for SL $ _2({\cal O})\!\setminus\! {\bf H}^3 $ and estimates for L-functions , 2001 .

[90]  A. Bäcker,et al.  On the number of bouncing ball modes in billiards , 1997 .

[91]  S. Zelditch,et al.  Ergodicity of eigenfunctions for ergodic billiards , 1996 .

[92]  L. Hörmander,et al.  The spectral function of an elliptic operator , 1968 .

[93]  A. Good.  Beitraege zur theorie der Dirichletreihen, die spitzenformen zugeordnet sind , 1981 .

[94]  D. Armitage.  Spherical Extrema of Harmonic Polynomials , 1979 .

[95]  Integrable geodesic flows on n-step nilmanifolds , 2000 .

[96]  Y. C. Verdière,et al.  Ergodicité et fonctions propres du laplacien , 1985 .

[97]  P. Leboeuf,et al.  Universal fluctuations of zeros of chaotic wavefunctions , 1996 .

[98]  B. Chirikov,et al.  Quantum chaos: Quantum nodal points as fingerprints of classical chaos , 1995 .

[99]  D. Jakobson.  QUANTUM LIMITS ON FLAT TORI , 1997 .

[100]  Quantum ergodicity of Eigenfunctions on PSL2(Z)/H2 , 1995 .

[101]  M. Berry.  Fringes decorating anticaustics in ergodic wavefunctions , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[102]  Y. C. Verdière,et al.  Sur la multiplicité de la première valeur propre non nulle du Laplacien , 1986 .

[103]  N. Nadirashvili,et al.  Bounds on the Multiplicity of Eigenvalues for Fixed Membranes , 1998 .

[104]  P. Kröger.  On the Ranges of Eigenfunctions on Compact Manifolds , 1998 .

[105]  M. Berry,et al.  The quantum phase 2-form near degeneracies: two numerical studies , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[106]  A. Bäcker,et al.  RATE OF QUANTUM ERGODICITY IN EUCLIDEAN BILLIARDS , 1997, chao-dyn/9709030.

[107]  Peter Sarnak,et al.  Perspectives on the Analytic Theory of L-Functions , 2000 .

[108]  M. Berry,et al.  Intensity moments of semiclassical wavefunctions , 1983 .

[109]  M. Srednicki,et al.  Gaussian Fluctuations in Chaotic Eigenstates , 1996, chao-dyn/9603012.

[110]  R. Aurich,et al.  Exact theory for the quantum eigenstates of a strongly chaotic system , 1991 .

[111]  Voros.  Wentzel-Kramers-Brillouin method in the Bargmann representation. , 1989, Physical review. A, General physics.

[112]  Michael V Berry,et al.  Regular and irregular semiclassical wavefunctions , 1977 .

[113]  Hidekazu Ito.  Convergence of Birkhoff normal forms for integrable systems , 1989 .

[114]  R. Hardt,et al.  Critical sets of solutions to elliptic equations , 1999 .

[115]  Fishman,et al.  Approach to ergodicity in quantum wave functions. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[116]  S. Zelditch.  Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series , 1991 .

[117]  V. Guillemin.  Some Classical Theorems in Spectral Theory Revisited , 1979 .

[118]  J. Bourgain.  Eigenfunction bounds for the Laplacian on the n -torus , 1993 .

[119]  Wendelin Werner,et al.  A counterexample to the “hot spots” conjecture , 1998 .

[120]  E. Heller,et al.  Measuring scars of periodic orbits. , 1998, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[121]  S. Zelditch,et al.  Classical Limits of Eigenfunctions for Some Completely Integrable Systems , 1999 .

[122]  A. V. Volovoy.  Improved two–term asymptotics for the eigenvalue distribution function of an elliptic operator on a compact manifold , 1990 .

[123]  Chaotic eigenfunctions in momentum space , 1999, chao-dyn/9905015.

[124]  V. Ivrii,et al.  Spectral Asymptotics with Highly Accurate Remainder Estimates , 1990 .

[125]  Gregor Tanner,et al.  How chaotic is the stadium billiard? A semiclassical analysis , 1996, chao-dyn/9610013.

[126]  A. Zygmund.  On Fourier coefficients and transforms of functions of two variables , 1974 .

[127]  I Petridis,et al.  Fourier coefficients of cusp forms , 1999 .

[128]  Eigenvalues of the Laplacian for Bianchi Groups , 1999 .

[129]  M. Berry,et al.  Quantum states without time-reversal symmetry: wavefront dislocations in a non-integrable Aharonov-Bohm billiard , 1986 .

[130]  F. Steiner,et al.  Quantum eigenstates of a strongly chaotic system and the scar phenomenon , 1995 .

[131]  S. Zelditch.  On the rate of quantum ergodicity I: Upper bounds , 1994 .

[132]  Quantum integrability for the Kovalevsky top , 1998 .

[133]  Y. Petridis.  On squares of eigenfunctions for the hyperbolic plane and a new bound on certain L-series , 1995 .

[134]  S. Fishman,et al.  Fredholm method for scars , 1996 .

[135]  S. Yau.  A Note on the Distribution of Critical Points of Eigenfunctions , 1997 .

[136]  T. Hoffmann-Ostenhof,et al.  Critical sets of smooth solutions to elliptic equations in dimension 3 , 1996 .

[137]  E. Hecke.  Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I , 1937 .

[138]  Quantum ergodicity ofC* dynamical systems , 2000, math-ph/0002008.

[139]  Eric J. Heller,et al.  Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits , 1984 .

[140]  R. Cooke,et al.  The Cantor-Lebesgue Theorem , 1979 .

[141]  Peter Sarnak,et al.  The behaviour of eigenstates of arithmetic hyperbolic manifolds , 1994 .

[142]  P. Sarnak,et al.  Number variance for arithmetic hyperbolic surfaces , 1994 .

[143]  A. Voros,et al.  Normal modes of billiards portrayed in the stellar (or nodal) representation , 1995 .

[144]  N. Nadirashvili,et al.  Eigenfunctions with few critical points , 1999 .

[145]  S. Fishman,et al.  Quantum eigenfunctions in terms of periodic orbits of chaotic systems , 1993 .

[146]  G. Besson.  On the multiplicy of the eigenvalues of the Laplacian , 1988 .

[147]  Richard F. Bass,et al.  Fiber Brownian motion and the `hot spots''problem Duke Math , 2000 .

[148]  S. Fishman,et al.  Exact eigenfunctions of a chaotic system , 1997, chao-dyn/9707021.

[149]  Lev Kaplan,et al.  Weak quantum ergodicity , 1998, chao-dyn/9810002.