asymptotic completeness for n body stark hamiltonians we prove asymptotic completeness for short and long rangen body stark hamiltonians with local singularities of at most coulomb type our results include the usual models for atoms and molecules

new exact solutions for polynomial oscillators in large dimensions a new type of exact solvability is reported the schrodinger equation is considered in a very large spatial dimension d 1 and its central polynomial potential is allowed to depend on many 2q coupling constants in a search for its bound states possessing an exact and elementary wavefunction proportional to a harmonic oscillator like polynomial of a freely varying i e not just small degree n the solvability conditions are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of d q and n assuming that d is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all n and up to q 5 at least

gamow vectors and borel summability in a class of quantum systems we analyze the detailed time dependence of the wave function x t for one dimensional hamiltonians h partial x 2 v x where v for example modeling barriers or wells and x 0 are compactly supported

gamow states as continuous linear functionals over analytical test functions the space of analytical test functions rapidly decreasing on the real axis i e schwartz test functions of the type s on the real axis is used to construct the rigged hilbert space rhs h gamow states gs can be defined in rhs starting from dirac s formula it is shown that the expectation value of a self adjoint operator acting on a gs is real we have computed exactly the probability of finding a system in a gs and found that it is finite the validity of recently proposed approximations to calculate the expectation value of self adjoint operators in a gs is discussed

space of test functions for higher order field theories the fundamental space is defined as the set of entire analytic functions test functions z which are rapidly decreasing on the real axis the variable z corresponds to the complex energy plane the conjugate or dual space is the set of continuous linear functionals distributions on among those distributions are the propagators determined by the poles implied by the equations of motion and the contour of integration implied by the boundary conditions all propagators can be represented as linear combinations of elementary one pole functionals the algebra of convolution products is also determined the fourier transformed space contains test functions x these functions are extra rapidly decreasing so that the exponentially increasing solutions of higher order equations are distributions on

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

growth of sobolev norms in linear schrodinger equations with quasi periodic potential in this paper we consider the following problem let iu t u v x t u 0 be a linear schrodinger equation periodic boundary conditions where v is a real bounded real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector denote s t the corresponding flow map thus s t preserves the l 2 norm and our aim is to study its behaviour on h s t d s 0 our main result is the growth in time is at most logarithmic thus if h s then more precisely is proven in 1d and 2d when v is small we also exhibit examples showing that a growth of higher sobolev norms may occur in this context and is thus essentially best possible

the generalized anharmonic oscillator in three dimensions exact eigenvalues and eigenfunctions we study the generalized anharmonic oscillator in three dimensions described by the potentials of the form 2m 1k 1bkr 2k an asymptotic analysis of the schrodinger equation yields the leading asymptotic behavior of the energy eigenfunctions in terms of the dominant m 1 coupling constants bk m 1 k 2m 1 using an ansatz which incorporates this asymptotic behavior we reduce the eigenvalue equation to an m 2 term difference equation the corresponding hill determinant may be made to factorize with a finite determinant as a factor if a set of constraints on the couplings is satisfied an infinite sequence of such sets exists the exact energy eigenvalues appear as the real roots of the finite factor of the hill determinant the corresponding wavefunctions are gaussian weighted polynomials we consider the potentials 31bkr 2k and 51bkr 2k explicitly potentials of the form 2m1bjr j and 2m1bjr j r containing both even and odd terms are also considered finally we show that this method of constructi

strichartz estimates for the schrodinger equation with time periodic ln 2 potentials abstract we prove strichartz estimates for the schrodinger operator h v t x with time periodic complex potentials v belonging to the scaling critical space l x n 2 l t in dimensions n 3 this is done directly from estimates on the resolvent rather than using dispersive bounds as the latter generally require a stronger regularity condition than what is stated above in typical fashion we project onto the continuous spectrum of the operator and must assume an absence of resonances eigenvalues are permissible at any location in the spectrum including at threshold energies provided that the associated eigenfunction decays sufficiently rapidly

scaling generator coordinate method and the eigensolutions of the schrodinger equation an extremely simple approximate variational method for the study of the eigensolutions of the one dimensional schrodinger equation is introduced based upon the idea of employing scaled eigenstates of the harmonic oscillator as generating functions for the generator coordinate method the approach is successfully applied to a variety of situations

on the rhs description of resonances and virtual states we discuss the role of the m ller wave operators in a rigged hilbert space rhs model for resonances we also construct a new rhs of analytic functions valid for the description of resonances and virtual states these become generalized state vectors in a general physically realizable representation

fredholm s equations for subwavelength focusing subwavelength focusing sf is a very useful tool that can be carried out with the use of left hand materials for optics that involve the range of the microwaves many recent works have described a successful alternative procedure using time reversal methods the advantage is that we do not need devices which require the complicated manufacture of left hand materials nevertheless the theoretical mathematical bases are far from complete because before now we lacked an adequate easy to apply frame in this work we give for a broad class of discrete systems a solid support for the theory of electromagnetic sf that can be applied to communications and nanotechnology the very central procedure is the development of vector matrix formalism vmf based on exploiting both the inhomogeneous and homogeneous fredholm s integral equations in cases where the last two kinds of integral equations are applied to some selected discrete systems to this end we first establish a generalized newmann series for the fourier transform of the green s function in the inhomogeneous fredholm s equation of the problem then we go from an integral operator equation to a vector matrix algebraic one in this way we explore the inhomogeneous case and later on also the very interesting one about the homogeneous equation thus on the one hand we can relate in a simple manner the arriving electromagnetic signals with those at their sources and we can use them to perform a sf on the other hand we analyze the homogeneous version of the equations finding resonant solutions that have analogous properties to their counterparts in quantum mechanical scattering that can be used in a proposed very powerful way in communications also we recover quantum mechanical operator relations that are identical for classical electromagnetics finally we prove two theorems that formalize the relation between the theory of fredholm s integral equations and the vmf we present here

the anharmonic oscillator problem a new algebraic solution the second order one dimensional or radial differential schrodinger equation with the potential v r mu r2 nu r4 may be re interpreted as a difference equation of the fourth order indeed the hamiltonian is a pentadiagonal matrix in the standard harmonic oscillator basis mod n n 0 1 thus the authors construct its four independent general solutions by purely algebraic means via expansions in powers of n 1 14 next preserving the analogy between difference and differential equations the physical wavefunctions n mod psi and their energies are determined by the matching of the jost and regular solutions finally using the simplest matching conditions at the origin the resulting numerical algorithm is demonstrated to be both stable and quickly convergent

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

a class of exact solutions for anharmonically coupled oscillators the author studies the solutions of the two dimensional schrodinger equation for anharmonically coupled oscillators the author obtains a finite set of exact solutions of the form psi x y polynomial x y exp polynomial x y provided that certain constraints on the potential parameters are satisfied

spectral singularities of non hermitian hamiltonians and susy transformations simple examples of non hermitian hamiltonians with purely real spectra defined in l2 r having spectral singularities inside the continuous spectrum are given it is shown that such hamiltonians may appear by shifting the independent variable of a real potential into the complex plane also they may be created as susy partners of hermitian hamiltonians in the latter case spectral singularities of a non hermitian hamiltonian are ordinary points of the continuous spectrum for its hermitian susy partner conditions for transformation functions are formulated when a complex potential with complex eigenenergies and spectral singularities has a susy partner with a real spectrum without spectral singularities finally we shortly discuss why hamiltonians with spectral singularities are bad

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

on optimal truncation of divergent series solutions of nonlinear differential systems we prove that for divergent series solutions of nonlinear or linear differential systems near a generic irregular singularity the common prescription of summation to the least term is if properly interpreted meaningful and correct and we extend this method to trans series solutions in every direction in the complex plane at the singularity stokes9 directions not excepted there exists a non empty set of solutions whose difference from the optimally i e near the least term truncated asymptotic series is of the same exponentially small order of magnitude as the least term of the series there is a family of generalized borel summation formulae b which commute with the usual algebraic and analytic operations addition multiplication differentiation etc we show that there is exactly one of them b 0 such that for any formal series solution f b 0 f differs from the optimal truncation of f by at most the order of the least term of f we show in addition that the berry smoothing phenomenon is universal within this class of differential systems whenever the terms beyond all orders change in crossing a stokes line these terms vary smoothly on the berry scale x x 1 2 and the transition is always given by the error function under the same conditions we show that dingle9 rule of signs for stokes9 transitions holds

extended continued fractions and energies of the anharmonic oscillators we describe the analytic solution to the schrodinger eigenvalue problem for the class of the central potentials v r za r where a 2 1 4 amax 0 z is an arbitrary finite set of the integer or rational exponents 2 1 0 the formalism is based on an analysis of the asymptotic behavior of the explicit regular solution r and issues in the formulation of the secular equation 1 l1 e 0 which determines the binding energies the main result is the rigorous construction of l1 e as a generalized extended and convergent continued fraction the method cannot be applied to the a i 1 0 cases this disproves the closely related hill determinant approach as conjectured recently by singh et al for the simplest potentials with z 2 2 4 6 and z 2 1 1 2

analytic treatment of the αx2 βx4 oscillator we carry out an investigation of the schrodinger equation with the x2 x4 interaction for 0 re x and exp i in the sectors 0 arg x 3 6 0 and 6 arg x 3 6 0 a subdominant solution is constructed essentially in terms of the laplace transform of a function f s 0 s which is expressible as a converging power series in s the solution y x thus obtained is compared and contrasted with the one valid in the usual case where re x 0 is included and it is argued that y x may not be accessible to perturbative approaches

some finite p groups with bounded index of every cyclic subgroup in its normal closure let g be a finite p group g is called a bi pm group if a g a pm for every a g in this paper we mainly study the properties of bi 2 groups and bi p2 groups for p 3 and we study some bi pm groups m 3 with some additional requirement

quantum fluid dynamics description of a multiterm potential nonlinear schrodinger langevin equation via quantum fluid dynamics the author presents a general method to obtain a solution of the nonlinear schrodinger langevin equation with a multiterm potential for the description of interactions in non conservative systems the nonlinear complete set of equations forming the basis of the model are derived the corresponding solutions are exhibited in terms of auxiliary ordinary differential equations which provide relations between the potential coefficients and other coefficients prescribing the fluid particle behaviour

quantum states of a sextic potential hidden symmetry and quantum monodromy a known terminating polynomial ansatz for a finite sub set of states of the sextic central potential v a b c r ar2 br4 cr6 c 0 subject to integer related parameter constraints is confirmed to apply in two as well as three dimensions a hidden symmetry relating the eigenstates of v a b c r to those of v a b c r is used to establish the boundary between the ansatz sub set and the infinite set of higher states connections between the radial wavefunctions rn a b c r and rn a b c r also provide semiclassical insight into the origin of the parameter constraint finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

the rigged hilbert space approach to the lippmann schwinger equation ii the analytic continuation of the lippmann schwinger bras and kets the analytic continuation of the lippmann schwinger bras and kets is obtained and characterized it is shown that the natural mathematical setting for the analytic continuation of the solutions of the lippmann schwinger equation is the rigged hilbert space rather than just the hilbert space it is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution resulting in a semigroup time evolution physically the semigroup time evolution is simply a retarded or advanced propagator

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

exact solutions for anharmonic oscillators completes the author s investigation see phys lett a vol 81 p 17 8 1981 and presents exact solutions and eigenvalues for anharmonic oscillators the solutions being given in the form of definite integrals the conditions for the validity of these results are investigated

further polynomial type eigenfunctions the method developed previously by the author for obtaining eigenfunctions in the form polynomial exponential polynomial for a linear second order differential equation in normal form is extended to embrace the general case when the interaction potential or square of the refractive index profile has the form polynomial polynomial various situations arise in the development of the theory some of which can be concluded algebraically but the majority require numerical calculations regarding the rank of quite a general matrix containing many unknown parameters

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

resonance theory for schrodinger operators resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods a general theory is developed with new and weaker conditions allowing for perturbations of threshold eigenvalues and relaxed fermi golden rule the exponential decay rate of resonances is addressed its uniqueness in the time dependent picture is shown in certain cases the relation to the existence of meromorphic continuation of the properly weighted green s function to time dependent resonance is further elucidated by giving an equivalent time dependent asymptotic expansion of the solutions of the schrodinger equation

direct demonstration of the completeness of the eigenstates of the schrodinger equation with local and nonlocal potentials bearing a coulomb tail demonstrating the completeness of wave function solutions of the radial schrodinger equation is a very difficult task existing proofs relying on operator theory are often very abstract and far from intuitive comprehension however it is possible to obtain rigorous proofs amenable to physical insight if one restricts the considered class of schrodinger potentials one can mention in particular unbounded potentials yielding a purely discrete spectrum and short range potentials however those possessing a coulomb tail very important for physical applications have remained problematic due to their long range character the method proposed in this paper allows to treat them correctly provided that the non coulomb part of potentials vanishes after a finite radius nonlocality of potentials can also be handled the main idea in the proposed demonstration is that regular solutions behave like sine cosine functions for large momenta so that their expansions verify fourier transform properties the high

exceptional points of non hermitian operators exceptional points associated with non hermitian operators i e operators being non hermitian for real parameter values are investigated the specific characteristics of the eigenfunctions at the exceptional point are worked out within the domain of real parameters the exceptional points are the points where eigenvalues switch from real to complex values these and other results are exemplified by a classical problem leading to exceptional points of a non hermitian matrix

reply to comment on on the inconsistency of the bohm gadella theory with quantum mechanics in this reply we show that when we apply standard distribution theory to the lippmann schwinger equation the resulting spaces of test functions would comply with the hardy axiom only if classic results of paley and wiener of gelfand and shilov and of the theory of ultradistributions were wrong also we point out several differences between the standard method of constructing rigged hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory

the rigged hilbert space of the algebra of the one dimensional rectangular barrier potential the rigged hilbert space of the algebra of the one dimensional rectangular barrier potential is constructed the one dimensional rectangular potential provides another opportunity to show that the rigged hilbert space fully accounts for dirac s bra ket formalism the analogy between dirac s formalism and fourier methods is pointed out

resonance poles and gamow vectors in the rigged hilbert space formulation of quantum mechanics after a summary of the rigged hilbert space formulation of quantum mechanics and a brief statement of its advantages over von neumann s formulation a mathematically correct definition of gamow s exponentially decaying vectors as generalized energy eigenvectors is suggested it is shown that exponentially decaying vectors are obtained from the s matrix poles in the lower half of the second sheet and exponentially growing vectors from the s matrix poles in the upper half of the second sheet decaying state vectors are defined as functionals over half of the space of physical states and growing state vectors are defined as functionals over the other half on functionals over these subspaces the dynamical group of time development splits into two semigroups one for t 0 and the other for t 0 the generalized basis system connected with the spectrum of the hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated

semiclassical dynamics with exponentially small error estimates we construct approximate solutions to the time dependent schrodinger equation for small values of if v satisfies appropriate analyticity and growth hypotheses and these solutions agree with exact solutions up to errors whose norms are bounded by for some c and 0 under more restrictive hypotheses we prove that for sufficiently small t implies the norms of the errors are bounded by for some c 0 and 0

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

the fractionary schrodinger equation green functions and ultradistributions abstract in this work we generalize previous results about the fractionary schrodinger equation within the formalism of the theory of tempered ultradistributions several examples of the use of this theory are given in particular we evaluate the green function for a free particle in the general case for an arbitrary order of the derivative index

nonlocal potentials isolated states and levinson s theorem a compact expression for the calculation of phase shifts is derived for a potential which is the sum of local and nonlocal parts nonlocal potentials can support positive energy bound states that is states embedded in the continuous energy spectrum these states sometimes referred to as isolated states are not associated with any poles of the s matrix some controversy exists in the literature on how such bound states are included in levinson s theorem it is found that the phase shift should be taken continuous at the energy of the bound state rather than taken to have a discontinuity of for simplicity the analysis is restricted to the radial s wave schrodinger equation and separable nonlocal potentials

on the classical theory of ordinary linear differential equations of the second order and the schrodinger equation for power law potentials the power law potentials in the schrodinger equation solved recently are shown to come from the classical treatment of the singularities of a linear second order differential equation this allows one to enlarge the class of solvable power law potentials

on the inconsistency of the bohm gadella theory with quantum mechanics the bohm gadella theory sometimes referred to as the time asymmetric quantum theory of scattering and decay is based on the hardy axiom the hardy axiom asserts that the solutions of the lippmann schwinger equation are functionals over spaces of hardy functions the preparation registration arrow of time provides the physical justification for the hardy axiom in this paper it is shown that the hardy axiom is incorrect because the solutions of the lippmann schwinger equation do not act on spaces of hardy functions it is also shown that the derivation of the preparation registration arrow of time is flawed thus hardy functions neither appear when we solve the lippmann schwinger equation nor should they appear it is also shown that the bohm gadella theory does not rest on the same physical principles as quantum mechanics and that it does not solve any problem that quantum mechanics cannot solve the bohm gadella theory must therefore be abandoned

study of anharmonic singular potentials a simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a hamiltonian h with singular potentials closed form analytic expressions in n dimensions are obtained for the matrix elements of h with respect to the eigenfunctions of a soluble singular problem with two free parameters a and b the matrix eigenvalues are then optimized with respect to a and b for a given n applications convergence rates and comparisons with earlier work are discussed in detail

spiked and symmetrized decadic potentials supporting elementary n plets of bound states we show that the potential wells v x x10 ax8 bx6 cx4 dx2 f x2 with a central spike possess arbitrary finite multiplets of elementary exact bound states their strong asymptotic growth implies an ambiguity in their symmetrically generalized quantization via complex boundary conditions but the three eligible methods coincide at our exceptional solutions

finite 2 groups with index of every cyclic subgroup in its normal closure no greater than 4 abstract in this paper we study the finite 2 group g such that a g a 2 2 for every a g we prove that g 8 z g g 4 is abelian and cl g 5 we also get that exp g 2 3

a description of virtual scattering states in the rigged hilbert space formulation of quantum mechanics we define a pair of gel fand triplets i hi i i 1 2 where the linear spaces i are formed by suitably chosen entire analytic functions on the riemann surface associated to the transformation z2 virtual scattering states are well defined as pairs of functionals respectively in 1 and 2

on exact solutions of the schrodinger equation in the one and three dimensional schrodinger equation a broad class of the regular potentials rational functions of the rational powers of r may admit exact bound state solutions in the generalised harmonic oscillator elementary form psi r rsigma polynomial exp polynomial the necessary and sufficient conditions of this phenomenon are derived in the form of coupled algebraic equations the methods of their solution and a few examples are discussed in particular the well known coulombic and oscillator solvability and the similar recent results of singh et al 1978 and whitehead et al 1982 are reproduced as the two simplest special cases

on the applicability of the hill determinant and the analytic continued fraction method to anharmonic oscillators the authors finish a line of argument started in an earlier paper and show that the application of the hill determinant and the associated analytic continued fraction method for the calculation of eigenvalues of anharmonic oscillators is of dubious validity and may thus lead to erroneous results in this context some theorems concerning the eigenvalues of the lambda x4 and the doubly anharmonic oscillators are proved

spectral theory of time periodic many body systems we study the spectrum of the monodromy operator for an n body quantum system in a time periodic external field with time mean equal to zero this includes ac stark and circularly polarized fields and pair potentials with a local singularity up to and including the coulomb singularity in the framework of floquet theory we prove a local commutator estimate and use it to prove a limiting absorption principle for the floquet hamiltonian as well as exponential decay estimates on non threshold eigenfunctions these two results are then used to obtain a second order perturbation theory for embedded eigenvalues the principal tool is a new extended mourre theory

on the αx2 βx4 interaction by applying the laplace transform method the authors obtain a class of exact solutions for the schrodinger equation with the potential v x alpha x2 beta x4 x 0 alpha or 0 beta 0 the corresponding energy spectrum being continuous a particular feature of these eigen solutions is that they may not be accessible to perturbation theory the properties of the solutions obtained are investigated and their relation to the case where x or 0 is included is discussed

a rigged hilbert space of hardy class functions applications to resonances the explicit construction of a dense subspace of square integrable functions on the positive half of the real line is given this space has the properties that 1 it is endowed with a metrizable nuclear topology 2 it is stable under multiplication by x and 3 the functions in have suitable analytical continuation to a half plane the space of functions which are conjugate to elements of is also considered then the triplets l2 0 and l2 0 are used to give a description of resonances

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

comment on on the inconsistency of the bohm gadella theory with quantum mechanics rigged hilbert spaces of hardy functions lead to a consistent theory of resonance scattering and decay contrary to the claims of a recent article 8 the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances

completeness in quantum mechanics and the weyl titchmarsh kodaira theorem we discuss the completeness of generalized eigenfunctions in quantum mechanics using the classical theory developed by weyl titchmarsh and kodaira as applications we rigorously prove the completeness of generalized eigenfunctions for the step and well potentials

an algebraic approach to problems with polynomial hamiltonians on euclidean spaces explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi dimensional spaces in a continuous sequence of orthonormal bases for unitary su 1 1 irreps explicit expressions are also given for so n reduced matrix elements of basic orbital observables these developments make it possible to determine the matrix elements of polynomial and other hamiltonians analytically to within so n clebsch gordan coefficients and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent

the generalized continued fractions and potentials of the lennard jones type for a broad class of the strongly singular potentials v r which are defined as superpositions of separate power law components the general solution of the corresponding schrodinger differential equation is constructed as an analog of mathieu functions the analogy is supported by the use of the generalized continued fractions the questions of convergence are analyzed in detail

sharp upper bounds on the number of the scattering poles abstract we study the scattering poles of a compactly supported black box perturbations of the laplacian in r n n odd we prove a sharp upper bound of the counting function n r modulo o r n in terms of the counting function of the reference operator in the smallest ball around the black box in the most interesting cases we prove a bound of the type n r a n r n o r n with an explicit a n we prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula

the rigged hilbert space approach to the gamow states we use the resonances of the spherical shell potential to present a thorough description of the gamow quasinormal states within the rigged hilbert space it will be concluded that the natural setting for the gamow states is a rigged hilbert space whose test functions fall off at infinity faster than gaussians

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

elementary bound states for the power law potentials the author shows how the exact polynomial exp polynomial bound state solutions to the radial schrodinger equation may be constructed for any potential of the form v r sigma delta in d gdelta rdelta where d is a finite set of arbitrary rational numbers and some of the couplings gdelta are not independent

time decay for solutions of schrodinger equations with rough and time dependent potentials in this paper we establish dispersive estimates for solutions to the linear schrodinger equation in three dimensions frac 1 i partial t psi triangle psi v psi 0 qquad psi s f 0 1 where v t x is a time dependent potential that satisfies the conditions sup t v t cdot l frac 3 2 mathbb r 3 sup x in mathbb r 3 int mathbb r 3 int infty infty frac v hat tau x x y d tau dy c 0 here c0 is some small constant and v hat tau x denotes the fourier transform with respect to the first variable we show that under these conditions 0 1 admits solutions l t l2x 3 l2t l6x 3 for any f l2 3 satisfying the dispersive inequality psi t infty le c t s frac32 f 1 text for all times t s 0 2 for the case of time independent potentials v x 0 2 remains true if int mathbb r 6 frac v x v y x y 2 dxdy 4 pi 2 text and v mathcal k sup x in mathbb r 3 int mathbb r 3 frac v y x y dy 4 pi we also establish the dispersive estimate with an e loss for large energies provided v mathcal k v 2 infty

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

on computing the trace of the kernel of the homogeneous fredholm s equation a method for computing the trace of the kernel of the homogeneous fredholm s equation for resonant states arising from nonlocal potentials is proposed we show that this integral formulation is convergent

the lippmann schwinger equations in the rigged hilbert space we give a rigorous meaning to the lippmann schwinger equations relating generalized eigenvectors of the free and perturbed hamiltonians in a scattering process some related formulae are also presented a discussion is made on the context of rigged hilbert space formulation of quantum mechanics

the rigged hilbert space approach to the lippmann schwinger equation part i we exemplify the way the rigged hilbert space deals with the lippmann schwinger equation by way of the spherical shell potential we explicitly construct the lippmann schwinger bras and kets along with their energy representation their time evolution and the rigged hilbert spaces to which they belong it will be concluded that the natural setting for the solutions of the lippmann schwinger equation and therefore for scattering theory is the rigged hilbert space rather than just the hilbert space