arXiv subjects
Publications and source records attributed to Michal Jex.
We prove necessary and sufficient conditions for the Schr\"odinger operators to have zero-energy bound states at the threshold of the essential spectrum such that they have bounded $k$-th moment. This result is the extension of the results published in D. Hundertmark, M. Jex, and M. Lange [Forum Mathematics, Sigma 11(2023)].
We prove necessary and sufficient conditions for lattice Schr\"{o}dinger operators to have a zero energy bound state in arbitrary dimension. The two criteria are sharp, complementary, and depend crucially on both the dimension and asymptotic behaviour of the potential. The method relies on a discrete variant of Agmon's comparison principle which is also proven. Our results represent a discrete variant of the recent criteria obtained in the continuous setting by D. Hundertmark, M. Jex, and M. Lange [$\textit{Forum Mathematics, Sigma}$ $\textbf{11}$ (2023)].
We prove that the lowest free energy of a classical interacting system at temperature $T$ with a prescribed density profile $\rho(x)$ can be approximated by the local free energy $\int f_T(\rho(x))dx$, provided that $\rho$ varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here $f_T$ is the free energy per unit volume of an infinite homogeneous gas of the corresponding uniform density. The proof uses quantitative Ruelle bounds (estimates on the local number of particles in a large system), which are derived in an appendix.
We provide upper and lower bounds on the lowest free energy of a classical system at given one-particle density $ρ(x)$. We study both the canonical and grand-canonical cases, assuming the particles interact with a pair potential which decays fast enough at infinity.
We present a method to calculate the asymptotic behavior of eigenfunctions of Schr\"odinger operators that also works at the threshold of the essential spectrum. It can be viewed as a higher order correction to the well-known WKB method which does need a safety distance to the essential spectrum. We illustrate its usefulness on examples of quantum particles in a potential well with a long-range repulsive term outside the well.
One of the crucial properties of a quantum system is the existence of bound states. While the existence of eigenvalues below zero, i.e., below the essential spectrum, is well understood, the situation of zero energy bound states at the edge of the essential spectrum is far less understood. We present necessary and sufficient conditions for Schrödinger operators to have a zero energy bound state. Our sharp criteria show that the existence and non-existence of zero energy ground states depends strongly on the dimension and the asymptotic behavior of the potential. There is a spectral phase transition with dimension four being critical.
We give a rigorous argument that long--range repulsion stabilizes quantum systems; ground states of such quantum systems exist even when the ground state energy is precisely at the ionization threshold. For atomic systems at the critical nuclear charge, our bounds show that the ground state falls off like $\exp(-c\sqrt{|x|})$ for large $|x|$. This is much slower than what the WKB method predicts for bound states with energies strictly below the ionization threshold. For helium type systems at critical nuclear charge, we show that our upper bounds are sharp. This rigorously confirms predictions by quantum chemists.
We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Σ\subset\mathbb R^d$, $d \ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in terms of the geodesic distance to the boundary of $Σ$. We provide its applications to the heat equation on $\mathbb R^d$ with an insulating cut at $Σ$ and to the Schrödinger operator with a $δ'$-interaction supported on $Σ$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.
It is well known that $N$-electron atoms undergoes unbinding for a critical charge of the nucleus $Z_c$, i.e. the atom has eigenstates for the case $Z> Z_c$ and it has no bound states for $Z<Z_c$. In the present paper we derive upper bound for the bound state for the case $Z=Z_c$ under the assumption $Z_c<N-K$ where $K$ is the number of electrons to be removed for atom to be stable for $Z=Z_c$ without any change in the ground state energy. We show that the eigenvector decays faster as $\exp\left(-C\sum\sqrt{|x|_{k}}\right)$ where we sum K largest values of $|x_j|$, $j\in\{1,\ldots,N\}$. Our method do not require Born-Oppenheimer approximation.
In the present paper we study two challenging problems for helium-type systems. Existence of eigenvalues at thresholds and the asymptotic behavior of the corresponding eigenfunctions. Since the usual methods for addressing these problems need a safety distance to the essential spectrum, they cannot be applied in critical cases, when an eigenvalue enters the continuum. We develop a method to address both problems and derive sharp upper and lower bounds for the asymptotic behavior of the ground state of critical helium-type systems at the threshold of the essential spectrum. This is the first proof of the precise asymptotic behavior of the ground state for this benchmark problem in quantum chemistry. Moreover, our bounds describe precisely how the asymptotic decay of the ground state changes, when the system becomes critical. In addition, we show the existence of a ground state of this quantum critical system with a finite nuclear mass. Previously this had been known only in the Born-Oppenheimer approximation of infinite nuclear mass.
We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the negative eigenvalues of Schr\"odinger operators, which significantly improve the so far existing bounds.
Let $Λ\subset\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let $u_\pm|_Λ\in L^2(Λ)$ be the traces of a function $u$ in the Sobolev space $H^1({\mathbb R}^2\setminus Λ)$ onto two faces of $Λ$. We prove that for a wide class of shapes of $Λ$ the Schrödinger operator $\mathsf{H}_ω^Λ$ with $δ^\prime$-interaction supported on $Λ$ of strength $ω\in L^\infty(Λ;\mathbb{R})$ associated with the quadratic form \[ H^1(\mathbb{R}^2\setminusΛ)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u \big|^2 \mathsf{d} x - \int_Λω\big| u_+|_Λ- u_-|_Λ\big|^2 \mathsf{d} s \] has no negative spectrum provided that $ω$ is pointwise majorized by a strictly positive function explicitly expressed in terms of $Λ$. If, additionally, the domain $\mathbb{R}^2\setminusΛ$ is quasi-conical, we show that $σ(\mathsf{H}_ω^Λ) = [0,+\infty)$. For a bounded curve $Λ$ in our class and non-varying interaction strength $ω\in\mathbb{R}$ we derive existence of a constant $ω_* > 0$ such that $σ(\mathsf{H}_ω^Λ) = [0,+\infty)$ for all $ω\in (-\infty, ω_*]$; informally speaking, bound states are absent in the weak coupling regime.
We consider a generalized Schrödinger operator in $L^2(\mathbb R^2)$ describing an attractive $δ'$ interaction in a strong coupling limit. $δ'$ interaction is characterized by a coupling parameter $β$ and it is supported by a $C^4$-smooth infinite asymptotically straight curve $Γ$ without self-intersections. It is shown that in the strong coupling limit, $β\to 0_+$, the eigenvalues for a non-straight curve behave as $-\frac{4}{β^2} +μ_j+\mathcal O(β|\lnβ|)$, where $μ_j$ is the $j$-th eigenvalue of the Schrödinger operator on $L^2(\mathbb R)$ with the potential $-\frac14 γ^2$ where $γ$ is the signed curvature of $Γ$.
We consider a generalized Schrödinger operator in $L^2(\R^2)$ with an attractive strongly singular interaction of $δ'$ type characterized by the coupling parameter $β>0$ and supported by a $C^4$-smooth closed curve $Γ$ of length $L$ without self-intersections. It is shown that in the strong coupling limit, $β\to 0_+$, the number of eigenvalues behaves as $\frac{2L}{πβ} + \OO(|\lnβ|)$, and furthermore, that the asymptotic behaviour of the $j$-th eigenvalue in the same limit is $-\frac{4}{β^2} +μ_j+\OO(β|\lnβ|)$, where $μ_j$ is the $j$-th eigenvalue of the Schrödinger operator on $L^2(0,L)$ with periodic boundary conditions and the potential $-\frac14 γ^2$ where $γ$ is the signed curvature of $Γ$.
We study relations between the ground-state energy of a quantum graph Hamiltonian with attractive $δ$ coupling at the vertices and the graph geometry. We derive a necessary and sufficient condition under which the energy increases with the increase of graph edge lengths. We show that this is always the case if the graph has no branchings while both change signs are possible for graphs with a more complicated topology.