Related papers: An infinite dimensional saddle point theorem and a…

We consider quasilinear Schr\"{o}dinger equations in $\mathbb{R}^{N}$ of the form% \[ -\Delta u+V(x)u-u\Delta(u^{2})=g(u)\text{,}% \] where $g(u)$ is $4$-superlinear. Unlike all known results in the literature, the Schr\"{o}dinger operator…

We consider physical interpretations of non-trivial boundary conditions of self-adjoint extensions for one-dimensional Schr\"odinger operator of free spinless particle. Despite its model and rather abstract character this question is worth…

We extend the well-known trace formula for Hill's equation to general one-dimensional Schr\"odinger operators. The new function $\xi$, which we introduce, is used to study absolutely continuous spectrum and inverse problems.

We construct and study an explicit simultaneous $\mathscr{Y}$-eigenbasis of Ion and Wu's standard representation of the $^+$stable-limit double affine Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis arises as a…

We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of the form \begin{align*}…

The classical $L^2$ estimate for the $\overline{\partial}$ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this estimate to infinite dimensional complex analysis, but this is a…

We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We arrive at this result using transforms…

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear"…

If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a…

The main goal of this dissertation is to find conditions which will guarantee the existence of solutions in the Hilbert space $H$ of semilinear equation \[ L u+N(u)=h \] where $L$ is a linear and self-adjoint operator, $N$ a non-linear…

Reference [1] established an index theory for a class of linear selfadjoint operator equations covering both second order linear Hamiltonian systems and first order linear Hamiltonian systems as special cases. In this paper based upon this…

A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schr\"odinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of…

We study global uniqueness in an inverse problem for the fractional semilinear Schr\"{o}dinger equation $(-\Delta)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data…

We look for solutions to generic nonlinear Schr\"odinger equations build upon solitons and kinks. Solitons are localized solitary waves and kinks are their non localized counter-parts. We prove the existence of infinite soliton trains, i.e.…

Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the ``shift'' operator give time dependence…

We prove optimal pointwise bounds on quasimodes of semiclassical Schrodinger operators with arbitrary smooth real potentials in dimension two. This end-point estimate was left open in the general study of semiclassical Lp bounds conducted…

The indefinite Sturm-Liouville operator $A = (\sgn x)(-d^2/dx^2+q(x))$ is studied. It is proved that similarity of $A$ to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of…

In recent years tamed schemes have become an important technique for simulating SDEs and SPDEs whose continuous coefficients display superlinear growth. The taming method, which involves curbing the growth of the coefficients as a function…

Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\bf C}$ and state space $H$. The function $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$ on $L^2((0, \infty…

Schauder's fixed point theorem is used to derive the existence of solutions to a semilinear heat equation. The equation features a nonlinear term that depends on the time-integral of the unknown on the whole, a priori given, interval of…