a non existence result for low energy sign changing solutions of the brezis nirenberg problem in dimensions 4 5 and 6 abstract we consider the brezis niremberg problem p e u u p 1 u e u in u 0 on where is a smooth bounded domain in r n n 4 5 6 p 1 2 n n 2 is the critical sobolev exponent and e is a positive parameter the main result of the paper generalizes the result of a iacopetti and f pacella 10 precisely we show that there are no low energy sign changing solutions u e with max u e min u e 0 or as e goes to zero

uniqueness and nondegeneracy of positive solutions of delta s u u u p rm in mathbb r n when s is close to 1 we consider the equation delta s u u u p with s in 0 1 in the subcritical range of p we prove that if s is sufficiently close to 1 the equation possesses a unique minimizer which is nondegenerate

spherical conjugacy classes and involutions in the weyl group let g be a simple algebraic group over an algebraically closed field of characteristic zero or positive odd good characteristic let b be a borel subgroup of g we show that the spherical conjugacy classes of g intersect only the double cosets of b in g corresponding to involutions in the weyl group of g this result is used in order to prove that for a spherical conjugacy class mathcal o with dense b orbit v0 bwb there holds ell w rm rk 1 w dim mathcal o extending to the case of groups over fields of odd good characteristic a characterization of spherical conjugacy classes obtained by cantarini costantini and the author it is also shown that the weights occurring in the g module decomposition of the ring of regular functions on mathcal o are self adjoint and they lie in the 1 eigenspace of the element w

frequency operator for anharmonic oscillators by perturbation theory we obtain approximate analytical expressions for the frequency operator for anharmonic oscillators by means of perturbation theory and a unitary transformation

generalized quantum anharmonic oscillator using an operator ordering approach we construct a generalized expression for the normal ordering of a a m for integral values of m and use the result to study the quantum anharmonic oscillator problem in the heisenberg approach in particular we derive generalized expressions for energy eigenvalues and frequency shifts for the hamiltonian h x2 2 2 2 m xm we also derive a closed form first order multiscale perturbation theoretic operator solution of this hamiltonian with a view to generalize some recent results of bender and bettencourt 1996 phys rev lett 77 4114 1996 phys rev d 54 1710

uniqueness of non linear ground states for fractional laplacians in mathbb r we prove uniqueness of ground state solutions q q x 0 of the non linear equation sq q q 1 0inr s q q q 1 0 i n r where 0 s 1 and 0 4s 1 2s for s 12 s 1 2 and 0 for s 12 s 1 2 here s denotes the fractional laplacian in one dimension in particular we answer affirmatively an open question recently raised by kenig martel robbiano and we generalize by completely different techniques the specific uniqueness result obtained by amick and toland for s 12 s 1 2 and 1 in 5 for the benjamin ono equation as a technical key result in this paper we show that the associated linearized operator l s 1 1 q is non degenerate i e its kernel satisfies ker l span q this result about l proves a spectral assumption which plays a central role for the stability of solitary waves and blowup analysis for non linear dispersive pdes with fractional laplacians such as the generalized benjamin ono bo and benjamin bona mahony bbm water wave equations

precise determination of the energy levels of the anharmonic oscillator from the quantization of the angle variable using an ansatz motivated by the classical form of el phi where phi is the angle variable we construct operators which satisfy the commutation relations of the creation annihilation operators for the anharmonic oscillator the matrix elements of these operators can be expressed in terms of entire functions in the position complex plane these functions provide solutions of the ricatti equation associated with the time independent schrodinger equation we relate the normalizability of the eigenstates to the global properties of the flows of this equation these exact results yield approximations which complement the wkb approximation and allow an arbitrarily precise determination of the energy levels we give numerical results for the first 10 levels with 30 digits we address the question of the quantum integrability of the system

on the coordinate ring of spherical conjugacy classes let g be a simple algebraic group over an algebraically closed field k of characteristic zero and mathcal o be a spherical conjugacy class of g we determine the decomposition of the coordinate ring k mathcal o of mathcal o into simple g modules

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

time dependence of operators in anharmonic quantum oscillators explicit perturbative analysis an explicit order by order perturbative solution valid over extended time scales for the time dependence of operators of anharmonic oscillators is developed within the framework of the method of normal forms the freedom of choice of the zeroth order term and concurrently in the higher order corrections is exploited to develop a minimal normal form mnf the expansion for the eigenvalues of the perturbed hamiltonian in a standard form is independent of the choice however the simple form obtained for the time dependence of the perturbative solution is more suitable than any other choice for application to high lying excited states as it offers a renormalized form for the propagator

random matrix ensembles with column row constraints ii we analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column row sum of the matrix elements of its hermitian operators the presence of additional constraints besides real symmetric nature leads to new correlations among their eigenfunctions hinders a complete delocalization of dynamics and affects the eigenvalues too the statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of brownian ensemble appearing between poisson and gaussian orthogonal ensemble

quantum oscillator of quartic anharmonicity a taylor series method is derived to construct an analytical solution for the quantum anharmonic oscillator the expression presented is found to yield in a rather natural way the corresponding special result obtained by using perturbation theoretic techniques attempts are made to give a possible physical interpretation for the various terms originating from nonlinear effects numerical results are presented to develop a feeling for their relative importance

the pohozaev identity for the fractional laplacian in this paper we prove the pohozaev identity for the semilinear dirichlet problem delta s u f u in omega u equiv0 in mathbb r n backslash omega here s in 0 1 s is the fractional laplacian in mathbb r n and is a bounded c1 1 domain to establish the identity we use among other things that if u is a bounded solution then u delta s omega is c up to the boundary where x dist x in the fractional pohozaev identity the function u delta s partial omega plays the role that u plays in the classical one surprisingly from a nonlocal problem we obtain an identity with a boundary term an integral over which is completely local as an application of our identity we deduce the nonexistence of nontrivial solutions in star shaped domains for supercritical nonlinearities

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

some amazing properties of spherical nilpotent orbits let g be a simple algebraic group defined over an algebraically closed field k of characteristic zero write g for its lie algebra let x g be a nilpotent element and g x g the corresponding nilpotent orbit the maximal number m such that adx 6 0 is called the height of x or of g x denoted ht x recall that an irreducible g variety x is called g spherical if a borel subgroup of g has an open orbit in x it was shown in pa1 that g x is g spherical if and only if adx 0 this means that the spherical nilpotent orbits are precisely the orbits of height 2 and 3 unfortunately whereas my proof for the orbits of height 2 and height 4 was completely general the argument for the orbits of height 3 explicitly used their classification in this paper we give a proof of sphericity that does not rely on the classification of nilpotent orbits see theorem 3 3 we begin with some properties of invariants of symplectic representations for instance we prove that 1 if h sp v is an irreducible representation without nonconstant invariants then h sp v and 2 if h has non constant invariants then it has an invariant of degree 4 applying these results to nilpotent orbits we prove that the centraliser zg x has a rather specific structure whenever ht x is odd from this description we deduce a conceptual proof of sphericity in case ht x 3 as another application we compute the index of zg x it will be shown that ind zg x rk g if ht x 3 this confirms elashvili s conjecture for such x see pa5 sect 3 about this conjecture in section 4 we prove that if the highest root of g is fundamental then g always has a specific orbit of height 3 which is denoted by o this orbit satisfies several arithmetical relations namely if g 3 i 3 g i is the z grading associated with an sl2 triple containing e o then dim g 3 2 and dim g 1 2 dim g 2 furthermore the weighted dynkin diagram of o can explicitly be described let be the unique simple root that is not orthogonal to and let i be all simple roots adjacent to on the dynkin diagram of g then one has to put 1 at all i s and 0 at all other simple roots theorem 4 5 it is curious that these properties of o enables us to give an intrinsic construction of g2 grading in each simple g whose highest root is fundamental

the general structure of eigenvalues in nonlinear oscillators hilbert spaces of bounded one dimensional nonlinear oscillators are studied it is shown that the eigenvalue structure of all such oscillators have the same general form they depend only on the ground state energy of the system and a single function h of the hamiltonian operator h it is also found that the hilbert space of the nonlinear oscillator is unitarily inequivalent to the hilbert space of the simple harmonic oscillator providing an explicit example of haag s theorem a number operator for the nonlinear oscillator is constructed and the general form of the partition function and average energy of a nonlinear oscillator in contact with a heat bath is determined connection with the wkb result in the semiclassical limit is made the analysis is then applied to the case of the x4 anharmonic oscillator as an explicit example

on the structure of the nodal set and asymptotics of least energy sign changing radial solutions of the fractional brezis nirenberg problem abstract in this paper we study the asymptotic and qualitative properties of least energy radial sign changing solutions of the fractional brezis nirenberg problem ruled by the s laplacian in a ball of r n when s 0 1 and n 6 s as usual is the positive parameter in the linear part in u we prove that for sufficiently small such solutions cannot vanish at the origin we show that they change sign at most twice and their zeros coincide with the sign changes moreover when s is close to 1 such solutions change sign exactly once finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a tower of bubbles as 0 i e the positive and negative parts concentrate at the same point with different concentration speeds

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

blow up and symmetry of sign changing solutions to some critical elliptic equations abstract in this paper we continue the analysis of the blow up of low energy sign changing solutions of semi linear elliptic equations with critical sobolev exponent started in m ben ayed k el mehdi f pacella blow up and nonexistence of sign changing solutions to the brezis nirenberg problem in dimension three ann inst h poincare anal non lineaire in press in addition we prove axial symmetry results for the same kind of solutions in a ball

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

replica equivalence in the edwards anderson model after introducing and discussing the link overlap between spin configurations we show that the edwards anderson model has a replica equivalent quenched equilibrium state a property introduced by parisi in the description of the mean field spin glass phase which generalizes ultrametricity our method is based on the control of fluctuations through the property of stochastic stability and works for all the finite dimensional spin glass models

perturbation theory free from secular terms for the equations of motion of anharmonic oscillators we develop a time independent perturbation theory in operator form for the equations of motion of classical and quantum mechanical anharmonic oscillators the method focuses on the frequency of the motion in the former case and in a frequency operator in the latter producing a power series in a small coupling constant the resulting expressions are free from secular instabilities that commonly appear in perturbation treatments of nonlinear problems

calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential we choose the squeezed vacuum state as a one parameter trial wavefunction to minimize the energy of an anharmonic oscillator with a polynomial perturbation potential using the optimal vacuum state obtained we generate the optimal generalized number state basis and set up the hamiltonian in the normal ordered form then truncating the basis we produce finite dimensional matrices in a simple manner and find their eigenvalues by standard methods we will apply our method to the morse potential whose exact eigenvalues are known as a check on the accuracy of the results that we have derived two models of the double well and the quartic potential are considered as examples

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

asymptotic behavior of solutions for nonlinear elliptic problems with the fractional laplacian abstract in this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional laplacians and the critical exponents this work can be seen as a nonlocal analog of the results of han 1991 24 and rey 1990 35

a direct method of moving planes for the fractional laplacian abstract in this paper we develop a direct method of moving planes for the fractional laplacian instead of using the conventional extension method introduced by caffarelli and silvestre we work directly on the non local operator using the integral defining the fractional laplacian by an elementary approach we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space such as strong maximum principles for anti symmetric functions narrow region principles and decay at infinity then using simple examples semi linear equations involving the fractional laplacian we illustrate how this new method of moving planes can be employed to obtain symmetry and non existence of positive solutions we firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities

frequency operator for anharmonic oscillators we develop a simple approach based on time independent perturbation theory that leads to a frequency operator for quantum mechanical models this method is suitable for the calculation of time dependent physical properties free from secular terms in particular we obtain the first order perturbation correction to the frequency operator for a class of anharmonic oscillators