Related papers: Spectral problems for operators with crossed magne…

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…

Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function $e_{h,\varepsilon}(x,x,\lambda)$ for a scalar operator \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*}…

We study the spectral shift function (SSF) $\xi(\lambda)$ and the resonances of the operator $H_V := \big( \sigma \cdot (-i\nabla - \textbf{A}) \big)^{2} + V$ in $L^2(\mathbb{R}^3)$ near the origin. Here $\sigma :=…

We explore connections between Krein's spectral shift function $\xi(\lambda,H_0,H)$ associated with the pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift…

We consider the 3D Pauli operator with nonconstant magnetic field B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy…

We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $\lambda^\circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be "along" the…

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field and subject to an electric potential $v_0$ depending only on the variable along the magnetic field $x_3$. The operator $H_0$ has infinitely many eigenvalues of…

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded…

We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of $h$-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint…

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schr\"odinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative…

We consider a 3-dimensional Dirac operator H_0 with non-constant magnetic field of constant direction, perturbed by a sign-definite matrix-valued potential V decaying fast enough at infinity. Then we determine asymptotics, as the energy…

Under certain assumptions (including $d\ge 2)$ we prove that the spectrum of a scalar operator in $\mathscr{L}^2(\mathbb{R}^d)$ \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*} covers interval…

Let $H_0$ be a periodic operator on $\R^+$(or periodic Jacobi operator on $\N$). It is known that the absolutely continuous spectrum of $H_0$ is consisted of spectral bands $\cup[\alpha_l,\beta_l]$. Under the assumption that $\limsup_{x\to…

We consider the self-adjoint operator $H=H_0+V$, where $H_0$ is the free semi-classical Dirac operator on $R^3$. We suppose that the smooth matrix-valued potential $V=O(<x>^{-\delta}), \delta>0,$ has an analytic continuation in a complex…

We consider the 3D Schr\"odinger operator $H = H_0 + V$ where $H_0 = (-i\nabla - A)^2$, $A$ is a magnetic potential generating a constant magnetic field of strength $b>0$, and $V$ is a short-range electric potential which decays…

Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator $H_\epsilon = -\partial_x^2 + x^2 + i\epsilon^{-1}f(x)$ on $L^2(R)$, where $f$ is a real-valued…

An asymptotic equality of the form $\operatorname{Tr}_{L^2} e^{-t(L+V)}=Ct^{-\alpha}+o(t^{-\alpha})$ as $t\rightarrow 0$ is given for the trace of the heat semigroup generated by operators on compact manifolds of the form…

We derive two main results: First, assume that $A$, $B$, $A_n$, $B_n$ are self-adjoint operators in the Hilbert space $\mathcal{H}$, and suppose that $A_n$ converges to $A$ and $B_n$ to $B$ in strong resolvent sense as $n \to \infty$. Fix…

Is considered the asymptotical behavior of spectral function $\rho(\lambda, \epsilon),\epsilon > 0$, of one family of self adjoint differential operators of second order, defined in space $L_2[0,+\infty)$ with potentials, depending on…

We extend the concept of Lifshits--Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Our main result is the following. Let…