Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semiclassical Resolvent Estimates for the Magnetic Schr{Ö}dinger Operator

Georgi Vodev

TL;DR

This work establishes explicit semiclassical resolvent estimates for the magnetic Schrödinger operator $P(h)=(ih\nabla+b)^2+V$ on $\mathbb{R}^d$, $d\ge3$, with potentials allowed to be long- or short-range and to have Hölder or Lipschitz regularity. The core method is a global Carleman estimate built from carefully designed phase and weight functions, enabling control of weighted solutions to $(P(h)-E\pm i\varepsilon)u=f$. Depending on the regularity and decay of the potentials, the authors obtain a general bound $g_s^{\pm}(h,\varepsilon)\le C h^{-2}\log(h^{-1})$, with refinements to $g_s^{\pm}(h,\varepsilon)\le C h^{-1}$ when short-range parts vanish, and even exponential-type bounds in the long-range Lipschitz regime. Extensions to Hölder and $L^{\infty}$ potentials yield bounds that depend on the Hölder exponents and decay rates, e.g., $g_s^{\pm}(h,\varepsilon)\le C h^{-n}\log(h^{-1})$ with $n=\max\{4/(\alpha+3),2/(\alpha'+1)\}$ in the Hölder case, or $g_s^{\pm}(h,\varepsilon)\le C h^{-2/(\alpha'+1)}\log(h^{-1})$ and $h^{-(2\delta)/(2\delta-1)-\varepsilon}$ in the $L^{\infty}$ setting. The results are complemented by a resolvent-bound framework via a key Lemma (7.1) and a detailed proof (Lemma 7.1) using resolvent identities and commutator estimates. Altogether, the paper provides a comprehensive, quantitative picture of how magnetic fields and potential regularity shape high-frequency resolvent behavior.

Abstract

We obtain semiclassical resolvent estimates for the Schr{ö}dinger operator (ih$\nabla$ + b)^2 + V in R^d , d $\ge$ 3, where h is a semiclassical parameter, V and b are real-valued electric and magnetic potentials independent of h. Under quite general assumptions, we prove that the norm of the weighted resolvent is bounded by exp(Ch^{-2} log(h^{ -1} )) . We get better resolvent bounds for electric potentials which are H{ö}lder with respect to the radial variable and magnetic potentials which are H{ö}lder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form exp(Ch^{-1}) .

Semiclassical Resolvent Estimates for the Magnetic Schr{Ö}dinger Operator

TL;DR

This work establishes explicit semiclassical resolvent estimates for the magnetic Schrödinger operator on , , with potentials allowed to be long- or short-range and to have Hölder or Lipschitz regularity. The core method is a global Carleman estimate built from carefully designed phase and weight functions, enabling control of weighted solutions to . Depending on the regularity and decay of the potentials, the authors obtain a general bound , with refinements to when short-range parts vanish, and even exponential-type bounds in the long-range Lipschitz regime. Extensions to Hölder and potentials yield bounds that depend on the Hölder exponents and decay rates, e.g., with in the Hölder case, or and in the setting. The results are complemented by a resolvent-bound framework via a key Lemma (7.1) and a detailed proof (Lemma 7.1) using resolvent identities and commutator estimates. Altogether, the paper provides a comprehensive, quantitative picture of how magnetic fields and potential regularity shape high-frequency resolvent behavior.

Abstract

We obtain semiclassical resolvent estimates for the Schr{ö}dinger operator (ih + b)^2 + V in R^d , d 3, where h is a semiclassical parameter, V and b are real-valued electric and magnetic potentials independent of h. Under quite general assumptions, we prove that the norm of the weighted resolvent is bounded by exp(Ch^{-2} log(h^{ -1} )) . We get better resolvent bounds for electric potentials which are H{ö}lder with respect to the radial variable and magnetic potentials which are H{ö}lder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form exp(Ch^{-1}) .
Paper Structure (9 sections, 14 theorems, 215 equations)

This paper contains 9 sections, 14 theorems, 215 equations.

Table of Contents

  1. Introduction and statement of results
  2. Construction of the phase and weight functions
  3. Carleman estimates in the general case
  4. Carleman estimates when $V_S\equiv 0$ and $b^S\equiv 0$
  5. Carleman estimates for Hölder potentials
  6. Carleman estimates for $L^\infty$ potentials $V_S$ and Hölder potentials $b^S$
  7. Resolvent estimates
  8. Proof of Lemma \ref{['7.1']}
  9. Proof of Lemma \ref{['3.2']}

Key Result

Theorem 1.1

In addition to the above conditions we assume either the condition or we suppose that the function $x\cdot b^L(x)$ is Lipschitz in $x$ and satisfies Then, there exist constants $C>0$, $0<h_0\ll 1$, independent of $h$ and $\varepsilon$, such that the bound holds for all $0<h\le h_0$. If $V_S\equiv 0$ and $b^S\equiv 0$, under the condition (eq:1.8), we have the better bound

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 6 more